One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution

Abstract

We consider a two-player finite horizon linear–quadratic Stackelberg differential game. For this game, we study the case where the control cost of a leader in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. We look for open-loop optimal players’ controls of this game. Using the game’s solvability conditions, the obtaining such controls is reduced to the solution to a proper boundary-value problem. Due to the smallness of the leader’s control cost, this boundary-value problem is singularly perturbed. Asymptotic behavior of the solution to this problem is analyzed. Based on this analysis, the asymptotic behavior of the open-loop optimal players’ controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players’ controls are designed. An illustrative example is presented.

1. Introduction

In this paper, a two-player finite horizon linear–quadratic Stackelberg differential game is considered. A feature of the considered game is that the control cost of a leader in the cost functionals of both players is small, meaning that this game is a cheap control game. More general, a cheap control problem is an extremal control problem in which a control cost of at least one decision maker is much smaller than a cost of the state variable in at least one cost functional of the problem. Cheap control problems are important in qualitative and quantitative analysis of many topics in optimal control and differential games theories. Thus, such problems are important in: (1) the qualitative study and analytical/numerical computation of singular controls and arcs (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14]); (2) the analysis of limiting forms and maximally achievable accuracy of optimal regulators and filters (see, e.g., [15,16,17,18,19,20,21]); (3) inverse optimal control problems (see, e.g., [22]); and (4) optimal control problems with high gain control in dynamics (see, e.g., [23,24]).

Cheap control differential games and closely related singular differential games appear in various applications. For example, such games appear in robust interception problems with uncertainties (see, e.g., [25]), in robust tracking problems with uncertainties (see, e.g., [26,27]), in pursuit-evasion problems (see, e.g., [13,28,29,30,31,32,33]), in robust investment problems (see, e.g., [34]), and in biology processes (see, e.g., [35]).

The smallness of the control cost yields a singularly perturbed nature of a boundary-value problem associated with a cheap control problem by solvability (control optimality) conditions.

As was mentioned, in this paper, we consider a cheap control differential game. Cheap control differential games were extensively studied in the literature. Thus, in [12,13,14,25,26,29,30,31,32,36] various zero-sum cheap control games were analyzed. Different cheap control Nash equilibrium games were studied in [33,37,38]. However, to the best of our knowledge, a cheap control Stackelberg equilibrium differential game has not yet been studied in the literature. It should be noted that, like a zero-sum differential game and a Nash equilibrium differential game, a Stackelberg equilibrium differential game is not a cooperative game. Moreover, like a Nash equilibrium differential game, a Stackelberg equilibrium differential game is a nonzero-sum differential game. In spite of this, a Stackelberg equilibrium differential game differs considerably from a zero-sum differential game and from a Nash equilibrium differential game. Namely, in contrast to a zero-sum differential game and a Nash equilibrium differential game, players in a Stackelberg equilibrium differential game do not act simultaneously. A Stackelberg equilibrium differential game is a hierarchical game. In this game, one of the players (the leader) makes its best decision based on the sets of possible best decisions of the other players (the followers). Then, each of the followers chooses its best decision (from the aforementioned set), which corresponds to the best decision of the leader. Thus, the leader has an advantage in comparison to the followers. More details on a Stackelberg equilibrium differential game can be found in the book [39]. There are different real-life applications of a Stackelberg equilibrium differential game. Thus, such a game is used in analysis and optimization of supply and marketing channels (see, e.g., the work [40] and references therein). In [41,42], the application of a Stackelberg equilibrium differential game to some production-and-sale problems is considered. Application of a Stackelberg equilibrium differential game to a security (defenders–attackers) problem is presented in [43] (see also the references therein). In [44], the problem of a transponder’s communication channel with multiple-user terminals is modeled by a Stackelberg equilibrium differential game. Application of a Stackelberg equilibrium differential game to analysis and optimization of power systems is proposed in [45].

A small cost of the leader’s control in its cost functional (a cheap control of the leader) can considerably increase the aforementioned advantage of this player. Opposite, a small cost of the follower’s control in its cost functional (a cheap control of the follower) can considerably decrease the leader’s advantage. This circumstance can be important; for instance, in a pursuit-evasion Stackelberg equilibrium differential game where the optimization of a state-dependent part of the leader/follower cost functional is much more essential than the optimization of the control energy expenditure part of the cost functional. Such results in the cases of zero-sum differential games and Nash equilibrium differential games were obtained in [13,25,29,31,32,33].

The main motivation of this (purely theoretical) paper is to study a novel nontrivial class of cheap control differential games. More precisely, in this paper, we consider a Stackelberg finite horizon linear–quadratic differential game with two players (a leader and a follower), and with a small cost of the leader’s control in both cost functionals. We study an asymptotic behavior of an open-loop solution to this game. For this purpose, we consider the boundary-value problem associated with the Stackelberg game by the solvability conditions. Due to the small cost of the leader’s control, the aforementioned boundary-value problem is singularly perturbed in the conditionally stable case. An asymptotic solution to this problem is formally constructed and justified. Based on this solution, asymptotic expansions of the optimal controls of the leader and the follower, as well as asymptotic expansions of the optimal values of the cost functionals, are constructed. Using these results, suboptimal controls of the players are formally designed and justified.

Note that a cheap control problem is closely related with a control problem for a singularly perturbed system. Namely, the former can be converted to the latter by a proper change in the control variable with the small (cheap) cost (see, e.g., [46,47,48,49]). Linear–quadratic Stackelberg differential games for singularly perturbed systems were studied in the works [50,51,52,53,54,55]. However, in these works (in contrast with the present paper), the infinite horizon version of the game was considered, and a suboptimal closed-loop (feedback) solution to the considered game was derived.

It should also be noted that the asymptotic method of solution for the considered cheap control Stackelberg differential game is a kind of an iterations’ method. Another iterations’ methods and their applications to approximate solutions to optimal control problems and differential games can be found in [32,56,57,58,59,60,61,62,63] and the references therein.

This paper is organized as follows. In Section 2, the Stackelberg cheap control game is rigorously formulated. The main notions and assumptions are introduced. The objectives of the paper are stated. In Section 3, the solvability conditions of the considered game are presented. An asymptotic solution to the singularly perturbed boundary-value problem, appearing in these solvability conditions, is formally constructed, and justified in Section 4. Section 5 is devoted to obtaining the main results of the paper. In this section, the asymptotic expansion of the game’s solution is formally derived and justified. Also, the suboptimal players’ controls are formally designed and justified. In Section 6, an illustrative example is presented. Conclusions are given in Section 7.

The following main notations are applied in the paper:

• $E^n$ denotes the n-dimensional real Euclidean space;
• $\parallel ·\parallel$ denotes the Euclidean norm either of a vector ($\parallel z\parallel$) or of a matrix ($\parallel A\parallel$);
• The superscript “$T$” denotes the transposition either of a vector ($z^T$) or of a matrix ($A^T$);
• $I_n$ denotes the identity matrix of dimension n;
• $\mathrm{col}\left(x_1,x_2,\dots ,x_k\right)$, where $x_1\in E^{n_1}$, $x_2\in E^{n_2}$,..., $x_k\in E^{n_k}$, denotes the column block-vector of the dimension $n_1+n_2+\dots +n_k$ with the upper block $x_1$, the next block $x_2$, and so on, and the lower block $x_k$;
• $\mathrm{Re}\left(\mu \right)$ denotes the real part of a complex number $\mu$.

2. Problem Statement

2.1. Initial Game Formulation

The following differential equation describes the dynamics of the game:

$${\displaystyle \frac{dZ\left(t\right)}{dt}}=\mathcal{A}\left(t\right)Z\left(t\right)+{\mathcal{B}}_{\mathrm{u}}\left(t\right)u\left(t\right)+{\mathcal{B}}_{\mathrm{v}}\left(t\right)v\left(t\right),t\in [0,t_f],Z\left(0\right)=Z_0,$$

(1)

where $Z\left(t\right)\in E^n$ is the state vector, $u\left(t\right)\in E^r$, ($r<n$), $v\left(t\right)\in E^s$, ($s\le n$) are the players’ controls; $t_f>0$ is a given time-instant; $\mathcal{A}\left(t\right)$ and ${\mathcal{B}}_i\left(t\right)$, $(i=\mathrm{u},\mathrm{v})$, are given matrix-valued functions of corresponding dimensions continuous for $t\in [0,t_f]$; $Z_0\in E^n$ is a given vector.

In what follows, we assume:

• (A1) For any $t\in [0,t_f]$, the matrix ${\mathcal{B}}_{\mathrm{u}}\left(t\right)$ has full column rank r.

The cost functionals of the first player (the player with the control $u\left(t\right)$ is called a leader) and the second player (the player with the control $v\left(t\right)$ is called a follower) are, respectively,

$$\begin{array}{c}\hfill {\mathcal{J}}_{\mathrm{u}}(u,v)={\displaystyle \frac{1}{2}}{\int }_0^{t_f}\left[Z^T\left(t\right){\mathcal{D}}_{\mathrm{u}}\left(t\right)Z\left(t\right)+{\epsilon }^2{u}^T\left(t\right)G_{\mathrm{u},\mathrm{u}}\left(t\right)u\left(t\right)+v^T\left(t\right)G_{\mathrm{u},\mathrm{v}}\left(t\right)v\left(t\right)\right]dt,\end{array}$$

(2)

and

$$\begin{array}{c}\hfill {\mathcal{J}}_{\mathrm{v}}(u,v)={\displaystyle \frac{1}{2}}{\int }_0^{t_f}\left[Z^T\left(t\right){\mathcal{D}}_{\mathrm{v}}\left(t\right)Z\left(t\right)+v^T\left(t\right)G_{\mathrm{v},\mathrm{v}}\left(t\right)v\left(t\right)+{\epsilon }^2{u}^T\left(t\right)G_{\mathrm{v},\mathrm{u}}\left(t\right)u\left(t\right)\right]dt,\end{array}$$

(3)

where, for any $t\in [0,t_f]$, ${\mathcal{D}}_i\left(t\right)$, $G_{i,j}\left(t\right)$, $(i=\mathrm{u},\mathrm{v};j=\mathrm{u},\mathrm{v})$ are given symmetric matrices of corresponding dimensions; the matrix-valued functions ${\mathcal{D}}_i\left(t\right)$, $G_{i,j}\left(t\right)$, $(i=\mathrm{u},\mathrm{v};j=\mathrm{u},\mathrm{v})$ are continuous for $t\in [0,t_f]$; $\epsilon >0$ is a small parameter.

We assume that:

• (A2) For any $t\in [0,t_f]$, the matrices ${\mathcal{D}}_i\left(t\right)$, $(i=\mathrm{u},\mathrm{v})$ are positive semi-definite.
• (A3) For any $t\in [0,t_f]$, the matrices $G_{\mathrm{u},\mathrm{u}}\left(t\right)$, $G_{\mathrm{v},\mathrm{v}}\left(t\right)$, $G_{\mathrm{u},\mathrm{v}}\left(t\right)$ and $G_{\mathrm{v},\mathrm{u}}\left(t\right)$ are positive definite.

Remark 1. 

In what follows, for the sake of technical simplicity (but without loss of generality), we set $G_{\mathrm{u},\mathrm{u}}\left(t\right)\equiv I_r$, $G_{\mathrm{v},\mathrm{v}}\left(t\right)\equiv I_s$, $t\in [0,t_f]$.

We are going to solve the game, given by differential Equation (1) and the cost Functionals (2) and (3), with respect to the open-loop Stackelberg (hierarchical) equilibrium (solution) $\left(u^{*}\left(t\right),v^{*}\left(t\right)\right)$. This equilibrium is defined as follows (for more details see, e.g., [39,64,65]).

First of all, let us note that we look for the Stackelberg solution to the game (1)–(3) in the set of all pairs $\left(u\left(t\right),v\left(t\right)\right)$, $t\in [0,t_f]$, $u\left(t\right)\in E^r$, $v\left(t\right)\in E^s$, and the vector-valued functions $u\left(t\right)$ and $v\left(t\right)$ are continuous in the interval $[0,t_f]$. Now, for any aforementioned $u\left(t\right)$, let

$$v^0\left(t;u\left(t\right)\right)\stackrel{▵}{=}{\mathrm{argmin}}_{v\left(t\right)}{\mathcal{J}}_{\mathrm{v}}\left(u\left(t\right),v\left(t\right)\right)$$

(4)

along the trajectories of (1).

Furthermore, let

$$u^{*}\left(t\right)\stackrel{▵}{=}{\mathrm{argmin}}_{u\left(t\right)}{\mathcal{J}}_{\mathrm{u}}\left(u\left(t\right),v^0\left(t;u\left(t\right)\right)\right)$$

(5)

along the trajectories of (1) with $v\left(t\right)=v^0\left(t;u\left(t\right)\right)$.

Moreover, let

$$v^{*}\left(t\right)\stackrel{▵}{=}{v}^0\left(t;u^{*}\left(t\right)\right).$$

(6)

Finally, the pair $\left(u^{*}\left(t\right),v^{*}\left(t\right)\right)$ is the open-loop Stackelberg solution to the game (1)–(3).

We call the trajectory of differential Equation (1), generated by the pair of controls $\left(u\left(t\right)=u^{*}\left(t\right),v\left(t\right)=v^{*}\left(t\right)\right)$, the Stackelberg optimal trajectory of the game (1)–(3). Also, we call the values ${\mathcal{J}}_{\mathrm{u}}\left(u^{*}\left(t\right),v^{*}\left(t\right)\right)$ and ${\mathcal{J}}_{\mathrm{v}}\left(u^{*}\left(t\right),v^{*}\left(t\right)\right)$ the Stackelberg optimal values of the cost functionals ${\mathcal{J}}_{\mathrm{u}}(u,v)$ and ${\mathcal{J}}_{\mathrm{v}}(u,v)$, respectively, in the game (1)–(3).

2.2. Transformation of the Game (1)–(3)

In what follows, we assume:

• (A4)$det\left({\mathcal{B}}_{\mathrm{u}}^T\left(t\right){\mathcal{D}}_{\mathrm{u}}\left(t\right){\mathcal{B}}_{\mathrm{u}}\left(t\right)\right)\ne 0$$\forall t\in [0,t_f]$.
• (A5) The matrix-valued functions $\mathcal{A}\left(t\right)$, ${\mathcal{B}}_{\mathrm{v}}\left(t\right)$, ${\mathcal{D}}_{\mathrm{v}}\left(t\right)$, $G_{\mathrm{u},\mathrm{v}}\left(t\right)$ and $G_{\mathrm{v},\mathrm{u}}\left(t\right)$ are continuously differentiable in the interval $[0,t_f]$.
• (A6) The matrix-valued functions ${\mathcal{B}}_{\mathrm{u}}\left(t\right)$ and ${\mathcal{D}}_{\mathrm{u}}\left(t\right)$ are twice continuously differentiable in the interval $[0,t_f]$.

Let ${\mathcal{B}}_c\left(t\right)$ be a complement matrix to the matrix ${\mathcal{B}}_{\mathrm{u}}\left(t\right)$ for any $t\in [0,t_f]$. Thus, the dimension of the matrix ${\mathcal{B}}_c\left(t\right)$ is $n\times (n-r)$, and the block-form matrix $\left({\mathcal{B}}_c\left(t\right),{\mathcal{B}}_{\mathrm{u}}\left(t\right)\right)$ is invertible for all $t\in [0,t_f]$. Due to Assumption (A6) and the results from [14] (Section 3.3), the matrix-valued function ${\mathcal{B}}_c\left(t\right)$ can be chosen as twice continuously differentiable in the interval $[0,t_f]$.

Consider the following matrix-valued functions of $t\in [0,t_f]$:

$${\mathcal{L}}_{\mathrm{u}}\left(t\right)={\mathcal{B}}_c\left(t\right)-{\mathcal{B}}_{\mathrm{u}}\left(t\right){\left({\mathcal{B}}_{\mathrm{u}}^T\left(t\right){\mathcal{D}}_{\mathrm{u}}\left(t\right){\mathcal{B}}_{\mathrm{u}}\left(t\right)\right)}^{-1}{\mathcal{B}}_{\mathrm{u}}^T\left(t\right){\mathcal{D}}_{\mathrm{u}}\left(t\right){\mathcal{B}}_c\left(t\right),{\mathcal{R}}_{\mathrm{u}}\left(t\right)=\left({\mathcal{L}}_{\mathrm{u}}\left(t\right),{\mathcal{B}}_{\mathrm{u}}\left(t\right)\right).$$

(7)

Remark 2. 

Due to the results of [14] (Section 3.3), the matrix ${\mathcal{R}}_{\mathrm{u}}\left(t\right)$ is invertible for all $t\in [0,t_f]$. Moreover, the matrix-valued function ${\mathcal{R}}_{\mathrm{u}}\left(t\right)$ is twice continuously differentiable with respect to $t\in [0,t_f]$.

Using the matrix ${\mathcal{R}}_{\mathrm{u}}\left(t\right)$, we transform the state in the game (1)–(3) as

$$Z\left(t\right)={\mathcal{R}}_{\mathrm{u}}\left(t\right)z\left(t\right),t\in [0,t_f],$$

(8)

where $z\left(t\right)\in E^n$ is a new state variable.

Using Remark 1, we obtain (quite similarly to the results of [14] (Section 3.3)) the following assertion.

Proposition 1. 

Let Assumptions (A1)–(A6) be valid. Then, Transformation (8) converts System (1) and Functionals (2) and (3) into the following new system and functionals:

$${\displaystyle \frac{dz\left(t\right)}{dt}}=A\left(t\right)z\left(t\right)+B_{\mathrm{u}}\left(t\right)u\left(t\right)+B_{\mathrm{v}}\left(t\right)v\left(t\right),t\in [0,t_f],z\left(0\right)=z_0,$$

(9)

$$J_{\mathrm{u}}(u,v)={\displaystyle \frac{1}{2}}{\int }_0^{t_f}\left[z^T\left(t\right)D_{\mathrm{u}}\left(t\right)z\left(t\right)+{\epsilon }^2{u}^T\left(t\right)u\left(t\right)+v^T\left(t\right)G_{\mathrm{u},\mathrm{v}}\left(t\right)v\left(t\right)\right]dt,$$

(10)

$$J_{\mathrm{v}}(u,v)={\displaystyle \frac{1}{2}}{\int }_0^{t_f}\left[z^T\left(t\right)D_{\mathrm{v}}\left(t\right)z\left(t\right)+v^T\left(t\right)v\left(t\right)+{\epsilon }^2{u}^T\left(t\right)G_{\mathrm{v},\mathrm{u}}\left(t\right)u\left(t\right)\right]dt,$$

(11)

where

$$A\left(t\right)={\mathcal{R}}_{\mathrm{u}}^{-1}\left(t\right)\left[\mathcal{A}\left(t\right){\mathcal{R}}_{\mathrm{u}}\left(t\right)-d{\mathcal{R}}_{\mathrm{u}}\left(t\right)/dt\right],$$

(12)

$$B_{\mathrm{u}}\left(t\right)={\mathcal{R}}_{\mathrm{u}}^{-1}\left(t\right){\mathcal{B}}_{\mathrm{u}}\left(t\right)=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill I_r\hfill \end{array}\right),B_{\mathrm{v}}\left(t\right)={\mathcal{R}}_{\mathrm{u}}^{-1}\left(t\right){\mathcal{B}}_{\mathrm{v}}\left(t\right),$$

(13)

$$\begin{array}{c}\hfill D_{\mathrm{u}}\left(t\right)={\mathcal{R}}_{\mathrm{u}}^T\left(t\right){\mathcal{D}}_{\mathrm{u}}\left(t\right){\mathcal{R}}_{\mathrm{u}}\left(t\right)=\left(\begin{array}{c}\hfill D_{\mathrm{u},1}\left(t\right)0\hfill \\ \hfill 0D_{\mathrm{u},2}\left(t\right)\hfill \end{array}\right),\hfill \\ \\ \hfill D_{\mathrm{u},1}\left(t\right)={\mathcal{L}}_{\mathrm{u}}^T\left(t\right){\mathcal{D}}_{\mathrm{u}}\left(t\right){\mathcal{L}}_{\mathrm{u}}\left(t\right),D_{\mathrm{u},2}\left(t\right)={\mathcal{B}}_{\mathrm{u}}^T\left(t\right){\mathcal{D}}_{\mathrm{u}}\left(t\right){\mathcal{B}}_{\mathrm{u}}\left(t\right),\hfill \end{array}$$

(14)

$$D_{\mathrm{v}}\left(t\right)={\mathcal{R}}_{\mathrm{u}}^T\left(t\right){\mathcal{D}}_{\mathrm{v}}\left(t\right){\mathcal{R}}_{\mathrm{u}}\left(t\right),z_0={\mathcal{R}}_{\mathrm{u}}^{-1}\left(0\right)Z_0.$$

(15)

The matrix $D_{\mathrm{u},1}\left(t\right)$ is symmetric positive semi-definite, while the matrix $D_{\mathrm{u},2}\left(t\right)$ is symmetric positive definite for all $t\in [0,t_f]$. Moreover, the matrix-valued functions $A\left(t\right)$, $B_{\mathrm{v}}\left(t\right)$, $D_{\mathrm{v}}\left(t\right)$, $G_{\mathrm{u},\mathrm{v}}\left(t\right)$, and $G_{\mathrm{v},\mathrm{u}}\left(t\right)$ are continuously differentiable in the interval $[0,t_f]$, while the matrix-valued function $D_{\mathrm{u}}\left(t\right)$ is twice continuously differentiable in the interval $[0,t_f]$.

Remark 3. 

First of all, let us note that the open-loop Stackelberg solution to the differential game (9)–(11) is defined similarly to such a solution to the game (1)–(3). Furthermore, since Transformation (8) is invertible, the games (1)–(3) and (9)–(11) have the same open-loop Stackelberg solution (if it exists), i.e., these games are equivalent to each other. Moreover, due to the form of the matrices $B_{\mathrm{u}}\left(t\right)$ and $D_{\mathrm{u}}\left(t\right)$, the game (9)–(11) is simpler than the game (1)–(3). Therefore, in the following, we will deal with the game (9)–(11).

2.3. Main Objectives of the Paper

The objectives of the paper are:

(i)

To construct and justify an asymptotic expansion with respect to $\epsilon$ of the open-loop Stackelberg solution $\left(u^{*}(t,\epsilon ),v^{*}(t,\epsilon )\right)$ to the game (9)–(11);

(ii)

To construct and justify asymptotic expansions with respect to $\epsilon$ of the Stackelberg optimal values $J_{\mathrm{u}}^{*}\left(\epsilon \right)=J_{\mathrm{u}}\left(u^{*}(t,\epsilon ),v^{*}(t,\epsilon )\right)$, $J_{\mathrm{v}}^{*}\left(\epsilon \right)=J_{\mathrm{v}}\left(u^{*}(t,\epsilon ),v^{*}(t,\epsilon )\right)$ of the cost functionals in the game (9)–(11);

(iii)

To derive an asymptotically suboptimal Stackelberg solution to the game (9)–(11), i.e., the pair of the admissible controls $\left(\tilde{u}(t,\epsilon ),\tilde{v}(t,\epsilon )\right)$, such that

$$\begin{array}{c}\hfill \underset{\epsilon \to +0}{lim}|J_{\mathrm{u}}\left(\tilde{u}(t,\epsilon ),\tilde{v}(t,\epsilon )\right)-J_{\mathrm{u}}^{*}\left(\epsilon \right)|=0,\\ \hfill \underset{\epsilon \to +0}{lim}|J_{\mathrm{v}}\left(\tilde{u}(t,\epsilon ),\tilde{v}(t,\epsilon )\right)-J_{\mathrm{v}}^{*}\left(\epsilon \right)|=0.\end{array}$$

(16)

3. $\epsilon$-Dependent Solvability Conditions of the Stackelberg Game (9)–(11)

For a given $\epsilon >0$, let us consider the following boundary-value problem in the time-interval $[0,t_f]$:

$$\begin{array}{c}\hfill {\displaystyle \frac{dz\left(t\right)}{dt}}=A\left(t\right)z\left(t\right)-S_{\mathrm{u}}(t,\epsilon ){\lambda }_{\mathrm{u}}\left(t\right)-S_{\mathrm{v}}\left(t\right){\lambda }_{\mathrm{v}}\left(t\right),z\left(0\right)=z_0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}}\left(t\right)}{dt}}=-D_{\mathrm{u}}\left(t\right)z\left(t\right)-A^T\left(t\right){\lambda }_{\mathrm{u}}\left(t\right)+D_{\mathrm{v}}\left(t\right){\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right),{\lambda }_{\mathrm{u}}\left(t_f\right)=0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{v}}\left(t\right)}{dt}}=-D_{\mathrm{v}}\left(t\right)z\left(t\right)-A^T\left(t\right){\lambda }_{\mathrm{v}}\left(t\right),{\lambda }_{\mathrm{v}}\left(t_f\right)=0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right)}{dt}}=S_{\mathrm{v}}\left(t\right){\lambda }_{\mathrm{u}}\left(t\right)-S_{\mathrm{u},\mathrm{v}}\left(t\right){\lambda }_{\mathrm{v}}\left(t\right)+A\left(t\right){\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right),{\lambda }_{\mathrm{u},\mathrm{v}}\left(0\right)=0,\end{array}$$

(17)

where

$$\begin{array}{c}\hfill S_{\mathrm{u}}(t,\epsilon )={\displaystyle \frac{1}{{\epsilon }^2}}{B}_{\mathrm{u}}\left(t\right)B_{\mathrm{u}}^T\left(t\right)=\left(\begin{array}{c}\hfill 00\hfill \\ \hfill 0{\displaystyle \frac{1}{{\epsilon }^2}}{I}_r\hfill \end{array}\right),\\ \hfill S_{\mathrm{v}}\left(t\right)=B_{\mathrm{v}}\left(t\right)B_{\mathrm{v}}^T\left(t\right),S_{\mathrm{u},\mathrm{v}}\left(t\right)=B_{\mathrm{v}}\left(t\right)G_{\mathrm{u},\mathrm{v}}\left(t\right)B_{\mathrm{v}}^T\left(t\right).\end{array}$$

(18)

Based on the results of the book [39] (Section 7.6), we directly have the following assertion.

Proposition 2. 

Let Assumptions (A1)–(A6) be valid. Then, for any given $\epsilon >0$, the boundary-value Problem (17) has the unique solution $\mathrm{col}\left(z(t,\epsilon ),{\lambda }_{\mathrm{u}}(t,\epsilon ),{\lambda }_{\mathrm{v}}(t,\epsilon ),{\lambda }_{\mathrm{u},\mathrm{v}}(t,\epsilon )\right)$. Moreover, the Stackelberg game (9)–(11) has the unique open-loop solution $\left(u^{*}(t,\epsilon ),v^{*}(t,\epsilon )\right)$, where

$$u^{*}(t,\epsilon )=-{\displaystyle \frac{1}{{\epsilon }^2}}{B}_{\mathrm{u}}^T\left(t\right){\lambda }_{\mathrm{u}}(t,\epsilon ),v^{*}(t,\epsilon )=-B_{\mathrm{v}}^T\left(t\right){\lambda }_{\mathrm{v}}(t,\epsilon ),t\in [0,t_f].$$

(19)

The component $z(t,\epsilon )$ of the solution to the boundary-value Problem (17) is the Stackelberg optimal trajectory of the game (9)–(11).

4. Asymptotic Solution to the Boundary-Value Problem (17)

4.1. Transformation of Problem (17)

Let us partition the matrices $A\left(t\right)$ and $D_{\mathrm{v}}\left(t\right)$ into blocks as

$$A\left(t\right)=\left(\begin{array}{c}\hfill A_1\left(t\right)A_2\left(t\right)\hfill \\ \hfill A_3\left(t\right)A_4\left(t\right)\hfill \end{array}\right),D_{\mathrm{v}}\left(t\right)=\left(\begin{array}{c}\hfill D_{\mathrm{v},1}\left(t\right)\hfill \\ \hfill D_{\mathrm{v},2}\left(t\right)\hfill \end{array}\right),t\in [0,t_f],$$

(20)

where the matrices $A_1\left(t\right)$, $A_2\left(t\right)$, $A_3\left(t\right)$, and $A_4\left(t\right)$ are of the dimensions $(n-r)\times (n-r)$, $(n-r)\times r$, $r\times (n-r)$ and $r\times r$, respectively; the matrices $D_{\mathrm{v},1}\left(t\right)$ and $D_{\mathrm{v},2}\left(t\right)$ are of the dimensions $(n-r)\times n$ and $r\times n$, respectively.

Furthermore, let us partition the matrix $B_{\mathrm{v}}\left(t\right)$ into blocks as

$$B_{\mathrm{v}}\left(t\right)=\left(\begin{array}{c}{B}_{\mathrm{v},1}\left(t\right)\\ B_{\mathrm{v},2}\left(t\right)\end{array}\right),t\in [0,t_f],$$

(21)

where the matrices $B_{\mathrm{v},1}\left(t\right)$ and $B_{\mathrm{v},2}\left(t\right)$ are of the dimensions $(n-r)\times s$ and $r\times s$, respectively.

Using the block form (21) of the matrix $B_{\mathrm{v}}\left(t\right)$ and the expressions of the matrices $S_{\mathrm{v}}\left(t\right)$ and $S_{\mathrm{u},\mathrm{v}}\left(t\right)$ (see Equation (18)), we directly have

$$S_{\mathrm{v}}\left(t\right)=\left(\begin{array}{c}\hfill S_{\mathrm{v},1}\left(t\right)\hfill \\ \hfill S_{\mathrm{v},2}\left(t\right)\hfill \end{array}\right),S_{\mathrm{u},\mathrm{v}}\left(t\right)=\left(\begin{array}{c}\hfill S_{\mathrm{uv},1}\left(t\right)\hfill \\ \hfill S_{\mathrm{uv},2}\left(t\right)\hfill \end{array}\right),t\in [0,t_f],$$

(22)

where

$$\begin{array}{c}\hfill S_{\mathrm{v},1}\left(t\right)=B_{\mathrm{v},1}\left(t\right)B_{\mathrm{v}}^T\left(t\right),S_{\mathrm{v},2}\left(t\right)=B_{\mathrm{v},2}\left(t\right)B_{\mathrm{v}}^T\left(t\right),\\ \hfill S_{\mathrm{uv},1}\left(t\right)=B_{\mathrm{v},1}\left(t\right)G_{\mathrm{u},\mathrm{v}}\left(t\right)B_{\mathrm{v}}^T\left(t\right),S_{\mathrm{uv},2}\left(t\right)=B_{\mathrm{v},2}\left(t\right)G_{\mathrm{u},\mathrm{v}}\left(t\right)B_{\mathrm{v}}^T\left(t\right).\end{array}$$

(23)

Also, we partition the vector $z_0$ into blocks as

$$z_0=\left(\begin{array}{c}{z}_{01}\\ z_{02}\end{array}\right),z_{01}\in E^{n-r},z_{02}\in E^r.$$

(24)

Due to the expression of $S_{\mathrm{u}}(t,\epsilon )$ (see Equation (18)), the right-hand side of the first differential equation in (17) is singular for $\epsilon =0$. Taking into account this singularity, we look for the components $z\left(t\right)$ and ${\lambda }_{\mathrm{u}}\left(t\right)$ of the solution to the boundary-value Problem (17) in the following block form:

$$\begin{array}{c}\hfill z\left(t\right)=\left(\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right),{\lambda }_{\mathrm{u}}\left(t\right)=\left(\begin{array}{c}{\lambda }_{\mathrm{u}1}\left(t\right)\\ \epsilon {\lambda }_{\mathrm{u}2}\left(t\right)\end{array}\right),\end{array}$$

(25)

where the vectors $z_1\left(t\right)$ and ${\lambda }_{\mathrm{u}1}\left(t\right)$ are of the dimension $(n-r)$, and the vectors $z_2\left(t\right)$ and ${\lambda }_{\mathrm{u}2}\left(t\right)$ are of the dimension r.

Substitution of the block forms of the matrices $D_{\mathrm{u}}\left(t\right)$, $S_{\mathrm{u}}(t,\epsilon )$, $A\left(t\right)$, $D_{\mathrm{v}}\left(t\right)$, $S_{\mathrm{v}}\left(t\right)$, $S_{\mathrm{u},\mathrm{v}}\left(t\right)$ (see Equations (14), (18), (20) and (22)), and the block forms of the vectors $z_0$, $z\left(t\right)$, ${\lambda }_{\mathrm{u}}\left(t\right)$ (see Equations (24) and (25)), into the boundary-value Problem (17), yields after a routine algebra the following equivalent boundary-value problem in the time interval $[0,t_f]$:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_1\left(t\right)}{dt}}=A_1\left(t\right)z_1\left(t\right)+A_2\left(t\right)z_2\left(t\right)-S_{\mathrm{v},1}\left(t\right){\lambda }_{\mathrm{v}}\left(t\right),\\ \hfill \epsilon {\displaystyle \frac{d{z}_2\left(t\right)}{dt}}=\epsilon A_3\left(t\right)z_1\left(t\right)+\epsilon A_4\left(t\right)z_2\left(t\right)-{\lambda }_{\mathrm{u}2}\left(t\right)-\epsilon S_{\mathrm{v},2}\left(t\right){\lambda }_{\mathrm{v}}\left(t\right),\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}1}\left(t\right)}{dt}}=-D_{\mathrm{u},1}\left(t\right)z_1\left(t\right)-A_1^T\left(t\right){\lambda }_{\mathrm{u}1}\left(t\right)-\epsilon A_3^T\left(t\right){\lambda }_{\mathrm{u}2}\left(t\right)+D_{\mathrm{v},1}\left(t\right){\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right),\\ \hfill \epsilon {\displaystyle \frac{d{\lambda }_{\mathrm{u}2}\left(t\right)}{dt}}=-D_{\mathrm{u},2}\left(t\right)z_2\left(t\right)-A_2^T\left(t\right){\lambda }_{\mathrm{u}1}\left(t\right)-\epsilon A_4^T\left(t\right){\lambda }_{\mathrm{u}2}\left(t\right)+D_{\mathrm{v},2}\left(t\right){\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right),\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{v}}\left(t\right)}{dt}}=-D_{\mathrm{v}}\left(t\right)\left(\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right)-A^T\left(t\right){\lambda }_{\mathrm{v}}\left(t\right),\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right)}{dt}}=S_{\mathrm{v}}\left(t\right)\left(\begin{array}{c}{\lambda }_{\mathrm{u}1}\left(t\right)\\ \epsilon {\lambda }_{\mathrm{u}2}\left(t\right)\end{array}\right)-S_{\mathrm{u},\mathrm{v}}\left(t\right){\lambda }_{\mathrm{v}}\left(t\right)+A\left(t\right){\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right),\end{array}$$

(26)

$$\begin{array}{c}\hfill z_1\left(0\right)=z_{01},z_2\left(0\right)=z_{02},{\lambda }_{\mathrm{u}1}\left(t_f\right)=0,{\lambda }_{\mathrm{u}2}\left(t_f\right)=0,{\lambda }_{\mathrm{v}}\left(t_f\right)=0,{\lambda }_{\mathrm{u},\mathrm{v}}\left(0\right)=0.\end{array}$$

(27)

Remark 4. 

It is important to note that the above made partitioning the matrices $A\left(t\right)$, $D_v\left(t\right)$, $B_v\left(t\right)$ and the vector $z_0$ (see Equations (20), (21) and (24)), as well as the representation of the state variables $z\left(t\right)$ and ${\lambda }_u\left(t\right)$ in the block form (see Equation (25)), allow us to partition the first and second equations of the differential system in (17) into two considerably different modes. Namely, the differential equation with respect to $z\left(t\right)$ is partitioned into two differential equations with respect to $z_1\left(t\right)$ and $z_2\left(t\right)$ in System (26). The derivative $d{z}_2\left(t\right)/dt$ is multiplied by the small parameter $\epsilon >0$, while the derivative $d{z}_1\left(t\right)/dt$ is not. This means that the state variable $z_2\left(t\right)$ varies much faster than the state variable $z_1\left(t\right)$. Similarly, the differential equation with respect to ${\lambda }_{\mathrm{u}}\left(t\right)$ is partitioned into two differential equations with respect to ${\lambda }_{\mathrm{u}1}\left(t\right)$ and ${\lambda }_{\mathrm{u}2}\left(t\right)$ in System (26), and the state variable ${\lambda }_{\mathrm{u}2}\left(t\right)$ varies much faster than the state variable ${\lambda }_{\mathrm{u}1}\left(t\right)$. Thus, the aforementioned matrices’ partitioning and the state variables’ representation allow us to convert the boundary-value Problem (17) into the explicit form of a singularly perturbed boundary-value problem, namely, Problems (26) and (27). The state variables $z_1\left(t\right)$, ${\lambda }_{\mathrm{u}1}\left(t\right)$ and ${\lambda }_{\mathrm{v}}\left(t\right)$, ${\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right)$ are slow ones, while the state variables $z_2\left(t\right)$ and ${\lambda }_{\mathrm{u}2}\left(t\right)$ are fast ones. The differential equations with respect to the fast state variables $z_2\left(t\right)$ and ${\lambda }_{\mathrm{u}2}\left(t\right)$ (the second and the fourth equations in (26)) are fast modes of System (26), while the other differential equations of this system are its slow modes.

4.2. First-Order Formal Asymptotic Solution to Problems (26) and (27)

Based on the Boundary Functions Method (see, e.g., [66,67]), we look for the first-order asymptotic solution

$$\mathrm{col}\left(z_1^1(t,\epsilon ),z_2^1(t,\epsilon ),{\lambda }_{\mathrm{u}1}^1(t,\epsilon ),{\lambda }_{\mathrm{u}2}^1(t,\epsilon ),{\lambda }_{\mathrm{v}}^1(t,\epsilon ),{\lambda }_{\mathrm{u},\mathrm{v}}^1(t,\epsilon )\right)$$

of Problems (26) and (27) in the form

$$\begin{array}{c}\hfill z_1^1(t,\epsilon )=\sum _{k=0}^1{\epsilon }^k\left[{\overline{z}}_{1,k}\left(t\right)+z_{1,k}^0\left(\xi \right)+z_{1,k}^f\left(\eta \right)\right],\\ \hfill z_2^1(t,\epsilon )=\sum _{k=0}^1{\epsilon }^k\left[{\overline{z}}_{2,k}\left(t\right)+z_{2,k}^0\left(\xi \right)+z_{2,k}^f\left(\eta \right)\right],\\ \hfill {\lambda }_{\mathrm{u}1}^1(t,\epsilon )=\sum _{k=0}^1{\epsilon }^k\left[{\overline{\lambda }}_{\mathrm{u}1,k}\left(t\right)+{\lambda }_{\mathrm{u}1,k}^0\left(\xi \right)+{\lambda }_{\mathrm{u}1,k}^f\left(\eta \right)\right],\\ \hfill {\lambda }_{\mathrm{u}2}^1(t,\epsilon )=\sum _{k=0}^1{\epsilon }^k\left[{\overline{\lambda }}_{\mathrm{u}2,k}\left(t\right)+{\lambda }_{\mathrm{u}2,k}^0\left(\xi \right)+{\lambda }_{\mathrm{u}2,k}^f\left(\eta \right)\right],\\ \hfill {\lambda }_{\mathrm{v}}^1(t,\epsilon )=\sum _{k=0}^1{\epsilon }^k\left[{\overline{\lambda }}_{\mathrm{v},k}\left(t\right)+{\lambda }_{\mathrm{v},k}^0\left(\xi \right)+{\lambda }_{\mathrm{v},k}^f\left(\eta \right)\right],\\ \hfill {\lambda }_{\mathrm{u},\mathrm{v}}^1(t,\epsilon )=\sum _{k=0}^1{\epsilon }^k\left[{\overline{\lambda }}_{\mathrm{u},\mathrm{v},k}\left(t\right)+{\lambda }_{\mathrm{u},\mathrm{v},k}^0\left(\xi \right)+{\lambda }_{\mathrm{u},\mathrm{v},k}^f\left(\eta \right)\right],\end{array}$$

(28)

where

$$\begin{array}{c}\hfill \xi ={\displaystyle \frac{t}{\epsilon }},\eta ={\displaystyle \frac{t-t_f}{\epsilon }}.\end{array}$$

(29)

Remark 5. 

In Equation (28), the terms with the overbar constitute the so-call outer solution, the terms with the superscript 0 are the boundary corrections in the right-hand neighborhood of $t=0$, the terms with the superscript f are the boundary corrections in the left-hand neighborhood of $t=t_f$. Equations and boundary conditions for the asymptotic solution terms are obtained substituting $z_1^1(t,\epsilon )$, $z_2^1(t,\epsilon )$, ${\lambda }_{\mathrm{u}1}^1(t,\epsilon )$, ${\lambda }_{\mathrm{u}2}^1(t,\epsilon )$, ${\lambda }_{\mathrm{v}}^1(t,\epsilon )$, and ${\lambda }_{\mathrm{u},\mathrm{v}}^1(t,\epsilon )$ from (28) into Problems (26) and (27), instead of $z_1\left(t\right)$, $z_2\left(t\right)$, ${\lambda }_{\mathrm{u}1}\left(t\right)$, ${\lambda }_{\mathrm{u}2}\left(t\right)$, ${\lambda }_{\mathrm{v}}\left(t\right)$, and ${\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right)$, respectively, and equating the coefficients for the same power of ε on both sides of the resulting equations, separately depending on t, ξ, and η. Additionally, the following should be noted: if $\epsilon \to +0$, then, for any $t\in (0,t_f]$, $\xi \to +\infty$, while, for any $t\in [0,t_f)$, $\eta \to -\infty$.

4.2.1. Obtaining the Boundary Corrections $z_{1,0}^0\left(\xi \right)$, ${\lambda }_{\mathrm{u}1,0}^0\left(\xi \right)$, ${\lambda }_{\mathrm{v},0}^0\left(\xi \right)$ and ${\lambda }_{\mathrm{u},\mathrm{v},0}^0\left(\xi \right)$

The equations for obtaining these boundary corrections are

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_{1,0}^0\left(\xi \right)}{d\xi }}=0,\xi \ge 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}1,0}^0\left(\xi \right)}{d\xi }}=0,\xi \ge 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{v},0}^0\left(\xi \right)}{d\xi }}=0,\xi \ge 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u},\mathrm{v},0}^0\left(\xi \right)}{d\xi }}=0,\xi \ge 0.\end{array}$$

(30)

To obtain the unique solutions of these equations, one should give some additional conditions for the unknown functions. By virtue of the Boundary Functions Method, these conditions are

$$\underset{\xi \to +\infty }{lim}{z}_{1,0}^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{\lambda }_{\mathrm{u}1,0}^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{\lambda }_{\mathrm{v},0}^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{\lambda }_{\mathrm{u},\mathrm{v},0}^0\left(\xi \right)=0.$$

(31)

The set of Equations (30) subject to Conditions (31) has the unique solution

$$z_{1,0}^0\left(\xi \right)\equiv 0,{\lambda }_{\mathrm{u}1,0}^0\left(\xi \right)\equiv 0,{\lambda }_{\mathrm{v},0}^0\left(\xi \right)\equiv 0,{\lambda }_{\mathrm{u},\mathrm{v},0}^0\left(\xi \right)\equiv 0,\xi \ge 0.$$

(32)

4.2.2. Obtaining the Boundary Corrections $z_{1,0}^f\left(\eta \right)$, ${\lambda }_{\mathrm{u}1,0}^f\left(\eta \right)$, ${\lambda }_{\mathrm{v},0}^f\left(\eta \right)$ and ${\lambda }_{\mathrm{u},\mathrm{v},0}^f\left(\eta \right)$

For these boundary corrections, we obtain the following equations:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_{1,0}^f\left(\eta \right)}{d\eta }}=0,\eta \le 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}1,0}^f\left(\eta \right)}{d\eta }}=0,\eta \le 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{v},0}^f\left(\eta \right)}{d\eta }}=0,\eta \le 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u},\mathrm{v},0}^f\left(\eta \right)}{d\eta }}=0,\eta \le 0.\end{array}$$

(33)

Due to the Boundary Functions Method, we require that

$$\underset{\eta \to -\infty }{lim}{z}_{1,0}^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{\lambda }_{\mathrm{u}1,0}^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{\lambda }_{\mathrm{v},0}^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{\lambda }_{\mathrm{u},\mathrm{v},0}^f\left(\eta \right)=0.$$

(34)

Thus, the set of Equations (33) subject to Conditions (34) yields the unique solution

$$z_{1,0}^f\left(\eta \right)\equiv 0,{\lambda }_{\mathrm{u}1,0}^f\left(\eta \right)\equiv 0,{\lambda }_{\mathrm{v},0}^f\left(\eta \right)\equiv 0,{\lambda }_{\mathrm{u},\mathrm{v},0}^f\left(\eta \right)\equiv 0,\eta \le 0.$$

(35)

4.2.3. Obtaining the Outer Solution Terms ${\overline{z}}_{1,0}\left(t\right)$, ${\overline{z}}_{2,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}2,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{v},0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)$

Due to Remark 5 and Equations (32) and (35), the outer solution’s terms ${\overline{z}}_{1,0}\left(t\right)$, ${\overline{z}}_{2,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}2,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{v},0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)$ satisfy the following system of equations in the time interval $[0,t_f]$:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\overline{z}}_{1,0}\left(t\right)}{dt}}=A_1\left(t\right){\overline{z}}_{1,0}\left(t\right)+A_2\left(t\right){\overline{z}}_{2,0}\left(t\right)-S_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right),\\ \hfill {\overline{\lambda }}_{\mathrm{u}2,0}\left(t\right)=0,\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)}{dt}}=-D_{\mathrm{u},1}\left(t\right){\overline{z}}_{1,0}\left(t\right)-A_1^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)+D_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right),\\ \hfill 0=-D_{\mathrm{u},2}\left(t\right){\overline{z}}_{2,0}\left(t\right)-A_2^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)+D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{v},0}\left(t\right)}{dt}}=-D_{\mathrm{v}}\left(t\right)\left(\begin{array}{c}{\overline{z}}_{1,0}\left(t\right)\\ {\overline{z}}_{2,0}\left(t\right)\end{array}\right)-A^T\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)}{dt}}=S_{\mathrm{v}}\left(t\right)\left(\begin{array}{c}{\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)\\ 0\end{array}\right)-S_{\mathrm{u},\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right)+A\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right),\end{array}$$

(36)

$$\begin{array}{c}\hfill {\overline{z}}_{1,0}\left(0\right)=z_{01},{\overline{\lambda }}_{\mathrm{u}1,0}\left(t_f\right)=0,{\overline{\lambda }}_{\mathrm{v},0}\left(t_f\right)=0,{\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(0\right)=0.\end{array}$$

(37)

Remark 6. 

System (36) consists of four differential equations with respect to ${\overline{z}}_{1,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{v},0}\left(t\right)$, and ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)$, and two algebraic equations with respect to ${\overline{\lambda }}_{\mathrm{u}2,0}\left(t\right)$ and ${\overline{z}}_{2,0}\left(t\right)$. Therefore, in (37), boundary conditions for ${\overline{\lambda }}_{\mathrm{u}2,0}\left(t\right)$ and ${\overline{z}}_{2,0}\left(t\right)$ are absent.

Resolving the fourth equation in System (36) with respect to ${\overline{z}}_{2,0}\left(t\right)$, and taking into account the invertibility of the matrix $D_{\mathrm{u},2}\left(t\right)$ for all $t\in [0,t_f]$, we obtain

$${\overline{z}}_{2,0}\left(t\right)=D_{\mathrm{u},2}^{-1}\left(t\right)\left(-A_2^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)+D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)\right),t\in [0,t_f].$$

(38)

Substitution of (38) into the first and fifth equations of System (36), use of Equations (20)–(23), and rearrangement of the sixth equation of this system yield after a routine algebra the following set of four differential equations with respect to ${\overline{z}}_{1,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{v},0}\left(t\right)$, and ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)$ in the time interval $[0,t_f]$:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\overline{z}}_{1,0}\left(t\right)}{dt}}=A_1\left(t\right){\overline{z}}_{1,0}\left(t\right)-A_2\left(t\right)D_{\mathrm{u},2}^{-1}\left(t\right)A_2^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)-S_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right)\\ \hfill +A_2\left(t\right)D_{\mathrm{u},2}^{-1}\left(t\right)D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)}{dt}}=-D_{\mathrm{u},1}\left(t\right){\overline{z}}_{1,0}\left(t\right)-A_1^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)+D_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{v},0}\left(t\right)}{dt}}=-D_{\mathrm{v},1}^T\left(t\right){\overline{z}}_{1,0}\left(t\right)+D_{\mathrm{v},2}^T\left(t\right)D_{\mathrm{u},2}^{-1}\left(t\right)A_2^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)-A^T\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right)\\ \hfill -D_{\mathrm{v},2}^T\left(t\right)D_{\mathrm{u},2}^{-1}\left(t\right)D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)}{dt}}=S_{\mathrm{v},1}^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)-S_{\mathrm{u},\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right)+A\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right).\end{array}$$

(39)

The system of differential Equations (39) is subject to boundary Conditions (37).

Remark 7. 

Let us note the following. The dimension of each of the state variables $z\left(t\right)$, ${\lambda }_{\mathrm{u}}\left(t\right)$, ${\lambda }_{\mathrm{v}}\left(t\right)$, and ${\lambda }_{\mathrm{u},\mathrm{v}}\left(t\right)$ of the differential system in (17) is n. The dimension of the state variables ${\overline{z}}_{1,0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)$ of the differential system (39) is $n-r$, while the dimension of the state variables ${\overline{\lambda }}_{\mathrm{v},0}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)$ of this system is n. Hence, differential System (39) is of a lower dimension than differential System (17). Moreover, in contrast with System (17), System (39) is independent of ε.

In what follows, we assume the following.

• (A7) The boundary-value Problems ((39) and (37)) have the unique solution

$\mathrm{col}\left({\overline{z}}_{1,0}\left(t\right),{\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right),{\overline{\lambda }}_{\mathrm{v},0}\left(t\right),{\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)\right)$, $t\in [0,t_f]$.

Thus, the solution to the boundary-value Problems ((39) and (37)), along with ${\overline{\lambda }}_{\mathrm{u}2,0}\left(t\right)=0$ (see the second equation in (36)) and ${\overline{z}}_{2,0}\left(t\right)$ (see Equation (38)), constitute the zero-order (with respect to $\epsilon$) outer solution.

4.2.4. Control-Theoretic Interpretation of the Boundary-Value Problem (39) and (37)

Consider the following system of differential equations in the time interval $[0,t_f]$:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\overline{z}}_1\left(t\right)}{dt}}=A_1\left(t\right){\overline{z}}_1\left(t\right)-S_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{v}}\left(t\right)+A_2\left(t\right)\overline{w}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{v}}\left(t\right)}{dt}}=-D_{\mathrm{v},1}^T\left(t\right){\overline{z}}_1\left(t\right)-A^T\left(t\right){\overline{\lambda }}_{\mathrm{v}}\left(t\right)-D_{\mathrm{v},2}^T\left(t\right)\overline{w}\left(t\right),\end{array}$$

(40)

where ${\overline{z}}_1\left(t\right)\in E^{n-r}$ and ${\overline{\lambda }}_{\mathrm{v}}\left(t\right)\in E^n$ are state variables; $\overline{w}\left(t\right)\in E^r$ is a control variable; and the vector-valued function $\overline{w}\left(t\right)$ is assumed to be continuous in the interval $[0,t_f]$.

System (40) is subject to the boundary conditions

$${\overline{z}}_1\left(0\right)=z_{01},{\overline{\lambda }}_{\mathrm{v}}\left(t_f\right)=0.$$

(41)

The control variable $\overline{w}\left(t\right)$ in the boundary-value Problems (40) and (41) is evaluated by the performance index

$$\begin{array}{c}\hfill {\overline{J}}_{\mathrm{u}}\left(\overline{w}\left(t\right)\right)\stackrel{▵}{=}{\displaystyle \frac{1}{2}}{\int }_0^{t_f}[{\overline{z}}_1^T\left(t\right)D_{\mathrm{u},1}\left(t\right){\overline{z}}_1\left(t\right)+{\overline{\lambda }}_{\mathrm{v}}^T\left(t\right)S_{\mathrm{u},\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v}}\left(t\right)\\ \hfill +{\overline{w}}^T\left(t\right)D_{u,2}\left(t\right)\overline{w}\left(t\right)]dt\to \underset{\overline{w}\left(t\right)}{min}.\end{array}$$

(42)

Proposition 3. 

Let Assumptions (A1)–(A7) be valid. Then, the optimal control Problems (40)–(42) have the unique open-loop optimal control

$$\begin{array}{c}\hfill {\overline{w}}^{*}\left(t\right)=-D_{\mathrm{u},2}^{-1}\left(t\right)A_2^T{\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)+D_{\mathrm{u},2}^{-1}\left(t\right)D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right),t\in [0,t_f],\end{array}$$

(43)

where ${\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)$ and ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)$ are the corresponding components of the solution to the boundary-value Problems (39) and (37). The optimal value ${\overline{J}}_{\mathrm{u}}^{*}$ of the functional in the optimal control Problems (40)–(42) has the form

$$\begin{array}{c}\hfill {\overline{J}}_{\mathrm{u}}^{*}={\displaystyle \frac{1}{2}}{\int }_0^{t_f}[{\overline{z}}_{1,0}^T\left(t\right)D_{\mathrm{u},1}\left(t\right){\overline{z}}_{1,0}\left(t\right)+{\overline{\lambda }}_{\mathrm{v},0}^T\left(t\right)S_{\mathrm{u},\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right)\\ \hfill +{\left({\overline{w}}^{*}\left(t\right)\right)}^T{D}_{u,2}\left(t\right){\overline{w}}^{*}\left(t\right)]dt,\end{array}$$

where ${\overline{z}}_{1,0}\left(t\right)$ and ${\overline{\lambda }}_{\mathrm{v},0}$ are the corresponding components of the solution to the boundary-value Problems (39) and (37).

Proof. 

First of all, let us observe the following. Due to Proposition 1, the matrix $D_{\mathrm{u},1}\left(t\right)$ is positive semi-definite for all $t\in [0,t_f]$, while the matrix $D_{\mathrm{u},2}\left(t\right)$ is positive definite in this time interval. Moreover, due to Assumption (A3) and Equation (18), the matrix $S_{\mathrm{u},\mathrm{v}}\left(t\right)$ is positive semi-definite for all $t\in [0,t_f]$. Using this observation, and based on the results of [68] (Section 9.2, the existence of solution to an optimal control problem) and [69] (Section 2.5, the control optimality conditions), we directly obtain the statements of the proposition. □

Remark 8. 

Comparing the expressions for ${\overline{w}}^{*}\left(t\right)$ and ${\overline{z}}_{2,0}\left(t\right)$ (see Equations (43) and (38)), we obtain that ${\overline{w}}^{*}\left(t\right)={\overline{z}}_{2,0}\left(t\right)$, $t\in [0,t_f]$.

4.2.5. Obtaining the Boundary Corrections $z_{2,0}^0\left(\xi \right)$, ${\lambda }_{\mathrm{u}2,0}^0\left(\xi \right)$ and $z_{2,0}^f\left(\eta \right)$, ${\lambda }_{\mathrm{u}2,0}^f\left(\eta \right)$

Due to Remark 5 and Equations (32) and (35), we have the following differential equations for these boundary corrections:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_{2,0}^0\left(\xi \right)}{d\xi }}=-{\lambda }_{\mathrm{u}2,0}^0\left(\xi \right),\xi \ge 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}2,0}^0\left(\xi \right)}{d\xi }}=-D_{\mathrm{u},2}\left(0\right)z_{2,0}^0\left(\xi \right),\xi \ge 0,\end{array}$$

(44)

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_{2,0}^f\left(\eta \right)}{d\eta }}=-{\lambda }_{\mathrm{u}2,0}^f\left(\eta \right),\eta \le 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}2,0}^f\left(\eta \right)}{d\eta }}=-D_{\mathrm{u},2}\left(t_f\right)z_{2,0}^f\left(\eta \right),\eta \le 0.\end{array}$$

(45)

By virtue of the Boundary Functions Method [66,67], we require (similarly to (31) and (34)) that the solutions of Systems (44) and (45) satisfy the conditions

$$\underset{\xi \to +\infty }{lim}{z}_{2,0}^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{\lambda }_{\mathrm{u}2,0}^0\left(\xi \right)=0,$$

(46)

$$\underset{\eta \to -\infty }{lim}{z}_{2,0}^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{\lambda }_{\mathrm{u}2,0}^f\left(\eta \right)=0.$$

(47)

Substituting the expression for $z_2^1(t,\epsilon )$ (see Equation (28)) into the initial condition for $z_2\left(t\right)$ (see Equation (27)), equating the coefficients for $\epsilon$ in the power of 0 on both sides of the resulting equation, and using the first limit equality in (47), we obtain one more condition for $z_{2,0}^0\left(\xi \right)$, namely

$$z_{2,0}^0\left(0\right)=z_{02}-{\overline{z}}_{2,0}\left(0\right).$$

(48)

Similarly, substituting the expression for ${\lambda }_{\mathrm{u}2}^1(t,\epsilon )$ (see Equation (28)) into the terminal condition for ${\lambda }_{\mathrm{u}2}\left(t\right)$ (see Equation (27)), equating the coefficients for $\epsilon$ in the power of 0 on both sides of the resulting equation, and using the second limit equality in (46) and that ${\overline{\lambda }}_{\mathrm{u}2,0}\left(t\right)\equiv 0$, we obtain one more condition for ${\lambda }_{\mathrm{u}2,0}^f\left(\eta \right)$, namely

$${\lambda }_{\mathrm{u}2,0}^f\left(0\right)=0.$$

(49)

Proceed to the solution to System (44) subject to Conditions (46) and (48).

Let $D_{\mathrm{u},2}^{1/2}\left(t\right)$, $t\in [0,t_f]$ denote the unique symmetric positive definite square root of the symmetric positive definite matrix $D_{\mathrm{u},2}\left(t\right)$, and $D_{\mathrm{u},2}^{-1/2}\left(t\right)$ denote the inverse matrix of this square root. Using these matrices, we consider the following block-form matrix:

$$\Theta \left(t\right)=\left(\begin{array}{c}\hfill I_{r}0.5D_{\mathrm{u},2}^{-1/2}\left(t\right)\hfill \\ \hfill -D_{\mathrm{u},2}^{1/2}\left(t\right)0.5I_r\hfill \end{array}\right),t\in [0,t_f].$$

(50)

This matrix is invertible for all $t\in [0,t_f]$, and its inverse matrix is

$${\Theta }^{-1}\left(t\right)=\left(\begin{array}{cc}0.5I_r\hfill & -0.5D_{\mathrm{u},2}^{-1/2}\left(t\right)\hfill \\ D_{\mathrm{u},2}^{1/2}\left(t\right)\hfill & I_r\hfill \end{array}\right),t\in [0,t_f].$$

(51)

Using the matrices $\Theta \left(t\right)$ and ${\Theta }^{-1}\left(t\right)$, we make the following transformation of the state variables in System (44) and Conditions (46) and (48):

$$\left(\begin{array}{c}\hfill z_{2,0}^0\left(\xi \right)\hfill \\ \\ \hfill {\lambda }_{\mathrm{u}2,0}^0\left(\xi \right)\hfill \end{array}\right)=\Theta \left(0\right)\left(\begin{array}{c}\hfill x_0^0\left(\xi \right)\hfill \\ \\ \hfill y_0^0\left(\xi \right)\hfill \end{array}\right),$$

(52)

where $x_0^0\left(\xi \right)\in E^r$ and $y_0^0\left(\xi \right)\in E^r$ are new state variables.

Transformation (52) System (44) and Conditions (46) and (48) to the system

$$\begin{array}{c}\hfill {\displaystyle \frac{d{x}_0^0\left(\xi \right)}{d\xi }}=D_{\mathrm{u},2}^{1/2}\left(0\right)x_0^0\left(\xi \right),\xi \ge 0,\\ \hfill {\displaystyle \frac{d{y}_0^0\left(\xi \right)}{d\xi }}=-D_{\mathrm{u},2}^{1/2}\left(0\right)y_0^0\left(\xi \right),\xi \ge 0,\end{array}$$

(53)

and the conditions

$$\underset{\xi \to +\infty }{lim}{x}_0^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{y}_0^0\left(\xi \right)=0,$$

(54)

$$\begin{array}{c}\hfill x_0^0\left(0\right)=0.5\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right)-0.5D_{\mathrm{u},2}^{-1/2}\left(0\right){\lambda }_{\mathrm{u}2,0}^0\left(0\right),\\ \hfill y_0^0\left(0\right)=D_{\mathrm{u},2}^{1/2}\left(0\right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right)+{\lambda }_{\mathrm{u}2,0}^0\left(0\right).\end{array}$$

(55)

The first equation in (53) yields (subject to the first condition in (55))

$$x_0^0\left(\xi \right)=exp\left(D_{\mathrm{u},2}^{1/2}\left(0\right)\xi \right)x_0^0\left(0\right),\xi \ge 0.$$

(56)

Since the matrix $D_{\mathrm{u},2}^{1/2}\left(0\right)$ is positive definite, then the vector-valued function $x_0^0\left(\xi \right)$, obtained in (56), satisfies the first limit equality in (54) if and only if $x_0^0\left(0\right)=0$. The latter, along with Equation (55), yields

$$\begin{array}{c}\hfill {\lambda }_{\mathrm{u}2,0}^0\left(0\right)=D_{\mathrm{u},2}^{1/2}\left(0\right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right),\\ \hfill y_0^0\left(0\right)=2D_{\mathrm{u},2}^{1/2}\left(0\right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right).\end{array}$$

(57)

Hence, the solution to the first equation in (53), subject to the initial condition in (55), is identically zero, i.e.,

$$x_0^0\left(\xi \right)\equiv 0,\xi \ge 0.$$

(58)

Solving the second equation in (53) subject to the initial condition, obtained in (57), we have

$$y_0^0\left(\xi \right)=2exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)\xi \right)D_{\mathrm{u},2}^{1/2}\left(0\right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right),\xi \ge 0.$$

(59)

Since the matrix $D_{\mathrm{u},2}^{1/2}\left(0\right)$ is positive definite, then the vector-valued function $y_0^0\left(\xi \right)$, obtained in (59), satisfies the second limit equality in (54).

Thus, the vector-valued functions $x_0^0\left(\xi \right)$ and $y_0^0\left(\xi \right)$, obtained in (58) and (59), respectively, constitute the solution to System (53) satisfying Conditions (54) and (55). Substituting this solution into (52) and using (50), we obtain, after a routine rearrangement, the solution to System (44) satisfying Conditions (46) and (48),

$$\begin{array}{c}\hfill z_{2,0}^0\left(\xi \right)=exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)\xi \right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right),\xi \ge 0,\\ \hfill {\lambda }_{\mathrm{u}2,0}^0\left(\xi \right)=D_{\mathrm{u},2}^{1/2}\left(0\right)exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)\xi \right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right),\xi \ge 0.\end{array}$$

(60)

This solution satisfies the inequalities

$$\parallel z_{2,0}^0\left(\xi \right)\parallel \le a^{0}exp(-{\beta }^0\xi ),\parallel {\lambda }_{\mathrm{u}2,0}^0\left(\xi \right)\parallel \le a^{0}exp(-{\beta }^0\xi ),\xi \ge 0,$$

(61)

where $a^0>0$ is some constant, and ${\beta }^0>0$ is the minimal eigenvalue of the matrix $D_{\mathrm{u},2}^{1/2}\left(0\right)$.

For obtaining the solution to System (45) satisfying Conditions (47) and (49), we use the procedure similar to that for the derivation of the solution to Problems (44), (46), and (48). Thus, we have

$$z_{2,0}^f\left(\eta \right)\equiv 0,{\lambda }_{\mathrm{u}2,0}^f\left(\eta \right)\equiv 0,\eta \le 0.$$

(62)

4.2.6. Obtaining the Boundary Corrections $z_{1,1}^0\left(\xi \right)$, ${\lambda }_{\mathrm{u}1,1}^0\left(\xi \right)$, ${\lambda }_{\mathrm{v},1}^0\left(\xi \right)$ and ${\lambda }_{\mathrm{u},\mathrm{v},1}^0\left(\xi \right)$

Using Remark 5 and Equation (32) yields the following equations for obtaining these boundary corrections:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_{1,1}^0\left(\xi \right)}{d\xi }}=A_2\left(0\right)z_{2,0}^0\left(\xi \right),\xi \ge 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}1,1}^0\left(\xi \right)}{d\xi }}=0,\xi \ge 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{v},1}^0\left(\xi \right)}{d\xi }}=-D_{\mathrm{v}}\left(0\right)\left(\begin{array}{c}0\\ z_{2,0}^0\left(\xi \right)\end{array}\right),\xi \ge 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u},\mathrm{v},1}^0\left(\xi \right)}{d\xi }}=0,\xi \ge 0.\end{array}$$

(63)

Similarly to the results of Section 4.2.1 (see Equation (31)), we look for the solution to System (63) subject to the conditions

$$\underset{\xi \to +\infty }{lim}{z}_{1,1}^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{\lambda }_{\mathrm{u}1,1}^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{\lambda }_{\mathrm{v},1}^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{\lambda }_{\mathrm{u},\mathrm{v},1}^0\left(\xi \right)=0.$$

(64)

Thus, using the expression for $z_{2,0}^0\left(\xi \right)$ (see Equation (60)), we directly obtain the unique solution to System (63) subject to Conditions (64)

$$\begin{array}{c}\hfill z_{1,1}^0\left(\xi \right)=-A_2\left(0\right)D_{\mathrm{u},2}^{-1/2}\left(0\right)exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)\xi \right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right),\xi \ge 0,\\ \hfill {\lambda }_{\mathrm{u}1,1}^0\left(\xi \right)\equiv 0,{\lambda }_{\mathrm{u},\mathrm{v},1}^0\left(\xi \right)\equiv 0,\xi \ge 0,\\ \hfill {\lambda }_{\mathrm{v},1}^0\left(\xi \right)=D_{\mathrm{v}}\left(0\right)\left(\begin{array}{c}0\\ D_{\mathrm{u},2}^{-1/2}\left(0\right)exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)\xi \right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right)\end{array}\right),\xi \ge 0.\end{array}$$

(65)

Due to the first inequality in (61), we have the following inequalities for the vector-valued functions $z_{1,1}^0\left(\xi \right)$ and ${\lambda }_{\mathrm{v},1}^0\left(\xi \right)$:

$$\begin{array}{c}\hfill \parallel z_{1,1}^0\left(\xi \right)\parallel \le a^0\parallel A_2\left(0\right)D_{\mathrm{u},2}^{-1/2}\left(0\right)\parallel exp(-{\beta }^0\xi ),\\ \hfill \parallel {\lambda }_{\mathrm{v},1}^0\left(\xi \right)\parallel \le a^0\parallel D_{\mathrm{v}}\left(0\right)\parallel \parallel D_{\mathrm{u},2}^{-1/2}\left(0\right)\parallel exp(-{\beta }^0\xi ),\end{array}$$

(66)

where the positive constants $a^0$ and ${\beta }^0$ were introduced in Equation (61).

4.2.7. Obtaining the Boundary Corrections $z_{1,1}^f\left(\eta \right)$, ${\lambda }_{\mathrm{u}1,1}^f\left(\eta \right)$, ${\lambda }_{\mathrm{v},1}^f\left(\eta \right)$ and ${\lambda }_{\mathrm{u},\mathrm{v},1}^f\left(\eta \right)$

Using Equations (35) and (62), for these boundary corrections, we obtain the following equations:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_{1,1}^f\left(\eta \right)}{d\eta }}=0,\eta \le 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}1,1}^f\left(\eta \right)}{d\eta }}=0,\eta \le 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{v},1}^f\left(\eta \right)}{d\eta }}=0,\eta \le 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u},\mathrm{v},1}^f\left(\eta \right)}{d\eta }}=0,\eta \le 0.\end{array}$$

(67)

Similarly to the results of Section 4.2.2 (see Equation (34)), we require that

$$\underset{\eta \to -\infty }{lim}{z}_{1,1}^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{\lambda }_{\mathrm{u}1,1}^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{\lambda }_{\mathrm{v},1}^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{\lambda }_{\mathrm{u},\mathrm{v},1}^f\left(\eta \right)=0.$$

(68)

Hence, System (67) subject to Condition (68) has the unique solution

$$z_{1,1}^f\left(\eta \right)\equiv 0,{\lambda }_{\mathrm{u}1,1}^f\left(\eta \right)\equiv 0,{\lambda }_{\mathrm{v},1}^f\left(\eta \right)\equiv 0,{\lambda }_{\mathrm{u},\mathrm{v},1}^f\left(\eta \right)\equiv 0,\eta \le 0.$$

(69)

4.2.8. Obtaining the Outer Solution Terms ${\overline{z}}_{1,1}\left(t\right)$, ${\overline{z}}_{2,1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{v},1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right)$

Using the second equation in (36), and Equations (65) and (69), we obtain (similarly to Equations (36) and (37)) the following system of equation for the outer solution’s terms ${\overline{z}}_{1,1}\left(t\right)$, ${\overline{z}}_{2,1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{v},1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right)$ in the time interval $[0,t_f]$:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\overline{z}}_{1,1}\left(t\right)}{dt}}=A_1\left(t\right){\overline{z}}_{1,1}\left(t\right)+A_2\left(t\right){\overline{z}}_{2,1}\left(t\right)-S_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{v},1}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{z}}_{2,0}\left(t\right)}{dt}}=A_3\left(t\right){\overline{z}}_{1,0}\left(t\right)+A_4\left(t\right){\overline{z}}_{2,0}\left(t\right)-{\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)-S_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)}{dt}}=-D_{\mathrm{u},1}\left(t\right){\overline{z}}_{1,1}\left(t\right)-A_1^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)+D_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right),\\ \hfill 0=-D_{\mathrm{u},2}\left(t\right){\overline{z}}_{2,1}\left(t\right)-A_2^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)+D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{v},1}\left(t\right)}{dt}}=-D_{\mathrm{v}}\left(t\right)\left(\begin{array}{c}{\overline{z}}_{1,1}\left(t\right)\\ {\overline{z}}_{2,1}\left(t\right)\end{array}\right)-A^T\left(t\right){\overline{\lambda }}_{\mathrm{v},1}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right)}{dt}}=S_{\mathrm{v}}\left(t\right)\left(\begin{array}{c}{\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)\\ 0\end{array}\right)-S_{\mathrm{u},\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v},1}\left(t\right)+A\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right),\end{array}$$

(70)

$$\begin{array}{c}\hfill {\overline{z}}_{1,1}\left(0\right)=-z_{1,1}^0\left(0\right)=A_2\left(0\right)D_{\mathrm{u},2}^{-1/2}\left(0\right)\left(z_{02}-{\overline{z}}_{2,0}\left(0\right)\right),\\ \hfill {\overline{\lambda }}_{\mathrm{u}1,1}\left(t_f\right)=0,{\overline{\lambda }}_{\mathrm{v},1}\left(t_f\right)=0,{\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(0\right)=0.\end{array}$$

(71)

Note that, for System (70), the assertion similar to Remark 6 is valid.

From the second equation of System (70), we directly have the expression for ${\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)$

$$\begin{array}{c}\hfill {\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)=-{\displaystyle \frac{d{\overline{z}}_{2,0}\left(t\right)}{dt}}+A_3\left(t\right){\overline{z}}_{1,0}\left(t\right)+A_4\left(t\right){\overline{z}}_{2,0}\left(t\right)-S_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right),t\in [0,t_f].\end{array}$$

(72)

Furthermore, resolving the fourth equation in System (70) with respect to ${\overline{z}}_{2,1}\left(t\right)$ and taking into account the invertibility of the matrix $D_{\mathrm{u},2}\left(t\right)$ for all $t\in [0,t_f]$, we obtain

$${\overline{z}}_{2,1}\left(t\right)=D_{\mathrm{u},2}^{-1}\left(t\right)\left(-A_2^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)+D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right)\right),t\in [0,t_f].$$

(73)

By substitution of (73) into the first and fifth equations of System (70), we obtain (quite similarly to System (39)) the following system of four differential equations with respect to ${\overline{z}}_{1,1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)$, ${\overline{\lambda }}_{\mathrm{v},1}\left(t\right)$, and ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right)$ in the time interval $[0,t_f]$:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\overline{z}}_{1,1}\left(t\right)}{dt}}=A_1\left(t\right){\overline{z}}_{1,1}\left(t\right)-A_2\left(t\right)D_{\mathrm{u},2}^{-1}\left(t\right)A_2^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)-S_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{v},1}\left(t\right)\\ \hfill +A_2\left(t\right)D_{\mathrm{u},2}^{-1}\left(t\right)D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)}{dt}}=-D_{\mathrm{u},1}\left(t\right){\overline{z}}_{1,1}\left(t\right)-A_1^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)+D_{\mathrm{v},1}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{v},1}\left(t\right)}{dt}}=-D_{\mathrm{v},1}^T\left(t\right){\overline{z}}_{1,1}\left(t\right)+D_{\mathrm{v},2}^T\left(t\right)D_{\mathrm{u},2}^{-1}\left(t\right)A_2^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)-A^T\left(t\right){\overline{\lambda }}_{\mathrm{v},1}\left(t\right)-D_{\mathrm{v},2}^T\left(t\right)D_{\mathrm{u},2}^{-1}\left(t\right)D_{\mathrm{v},2}\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right),\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right)}{dt}}=S_{\mathrm{v},1}^T\left(t\right){\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)-S_{\mathrm{u},\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v},1}\left(t\right)+A\left(t\right){\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right).\end{array}$$

(74)

The system of the differential Equations (74) is subject to the boundary conditions (71). Similarly to System (39), System (74) is of the lower dimension than the differential system in (17).

Remark 9. 

Let us note the following. The boundary-value Problems (74) and (71) are of the same form as the boundary-value Problems (39) and (37). Since, due to Assumption (A7), Problems (39) and (37) have the unique solution, then Problems (74) and (71) also have the unique solution $\mathrm{col}\left({\overline{z}}_{1,1}\left(t\right),{\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right),{\overline{\lambda }}_{\mathrm{v},1}\left(t\right),{\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right)\right)$, $t\in [0,t_f]$.

Thus, the solution to the boundary-value Problems (74) and (71), along with ${\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)$ (see Equation (72)) and ${\overline{z}}_{2,1}\left(t\right)$ (see Equation (73)), constitutes the coefficient of the first-order (with respect to $\epsilon$) addend in the outer solution.

4.2.9. Obtaining the Boundary Corrections $z_{2,1}^0\left(\xi \right)$, ${\lambda }_{\mathrm{u}2,1}^0\left(\xi \right)$ and $z_{2,1}^f\left(\eta \right)$, ${\lambda }_{\mathrm{u}2,1}^f\left(\eta \right)$

Following Remark 5, and using Equations (32), (35), (60), and (62), we have the differential equations for these boundary corrections:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_{2,1}^0\left(\xi \right)}{d\xi }}=A_4\left(0\right)z_{20}^0\left(\xi \right)-{\lambda }_{\mathrm{u}2,1}^0\left(\xi \right),\xi \ge 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}2,1}^0\left(\xi \right)}{d\xi }}=-D_{\mathrm{u},2}\left(0\right)z_{2,1}^0\left(\xi \right)-{\displaystyle \frac{d{D}_{\mathrm{u},2}\left(0\right)}{dt}}\xi z_{20}^0\left(\xi \right)-A_4^T\left(0\right){\lambda }_{\mathrm{u}2,0}^0\left(\xi \right),\xi \ge 0,\end{array}$$

(75)

$$\begin{array}{c}\hfill {\displaystyle \frac{d{z}_{2,1}^f\left(\eta \right)}{d\eta }}=-{\lambda }_{\mathrm{u}2,1}^f\left(\eta \right),\eta \le 0,\\ \hfill {\displaystyle \frac{d{\lambda }_{\mathrm{u}2,1}^f\left(\eta \right)}{d\eta }}=-D_{\mathrm{u},2}\left(t_f\right)z_{2,1}^f\left(\eta \right),\eta \le 0.\end{array}$$

(76)

Similarly to (46) and (47), we solve Systems (75) and (76) subject to the conditions

$$\underset{\xi \to +\infty }{lim}{z}_{2,1}^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{\lambda }_{\mathrm{u}2,1}^0\left(\xi \right)=0,$$

(77)

$$\underset{\eta \to -\infty }{lim}{z}_{2,1}^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{\lambda }_{\mathrm{u}2,1}^f\left(\eta \right)=0.$$

(78)

Moreover, similarly to (48) and (49), we obtain two additional conditions for the solutions of Systems (75) and (76),

$$\begin{array}{c}\hfill z_{2,1}^0\left(0\right)=-{\overline{z}}_{2,1}\left(0\right),\end{array}$$

(79)

$$\begin{array}{c}\hfill {\lambda }_{\mathrm{u}2,1}^f\left(0\right)=-{\overline{\lambda }}_{\mathrm{u}2,1}\left(t_f\right).\end{array}$$

(80)

Let us start with the solution to System (75) subject to Conditions (77) and (79).

Using the matrices $\Theta \left(t\right)$ and ${\Theta }^{-1}\left(t\right)$ (see Equations (50) and (51)), we make (similarly to (52)) the following transformation of the state variables in System (75) and Conditions (77) and (79):

$$\left(\begin{array}{c}\hfill z_{2,1}^0\left(\xi \right)\hfill \\ \\ \hfill {\lambda }_{\mathrm{u}2,1}^0\left(\xi \right)\hfill \end{array}\right)=\Theta \left(0\right)\left(\begin{array}{c}\hfill x_1^0\left(\xi \right)\hfill \\ \\ \hfill y_1^0\left(\xi \right)\hfill \end{array}\right),$$

(81)

where $x_1^0\left(\xi \right)\in E^r$ and $y_1^0\left(\xi \right)\in E^r$ are new state variables.

Transformation (81) converts System (75) and Conditions (77) and (79) to the system

$$\begin{array}{c}\hfill {\displaystyle \frac{d{x}_1^0\left(\xi \right)}{d\xi }}=D_{\mathrm{u},2}^{1/2}\left(0\right)x_1^0\left(\xi \right)+f_x\left(\xi \right),\xi \ge 0,\\ \hfill {\displaystyle \frac{d{y}_1^0\left(\xi \right)}{d\xi }}=-D_{\mathrm{u},2}^{1/2}\left(0\right)y_1^0\left(\xi \right)+f_y\left(\xi \right),\xi \ge 0,\end{array}$$

(82)

and the conditions

$$\underset{\xi \to +\infty }{lim}{x}_1^0\left(\xi \right)=0,\underset{\xi \to +\infty }{lim}{y}_1^0\left(\xi \right)=0,$$

(83)

$$\begin{array}{c}\hfill x_1^0\left(0\right)=-0.5{\overline{z}}_{2,1}\left(0\right)-0.5D_{\mathrm{u},2}^{-1/2}\left(0\right){\lambda }_{\mathrm{u}2,1}^0\left(0\right),\\ \hfill y_1^0\left(0\right)=-D_{\mathrm{u},2}^{1/2}\left(0\right){\overline{z}}_{2,1}\left(0\right)+{\lambda }_{\mathrm{u}2,1}^0\left(0\right),\end{array}$$

(84)

where

$$\begin{array}{c}\hfill f_x\left(\xi \right)=0.5\left(A_4\left(0\right)z_{2,0}^0\left(\xi \right)+D_{\mathrm{u},2}^{-1/2}\left(0\right){\displaystyle \frac{d{D}_{\mathrm{u},2}\left(0\right)}{dt}}\xi z_{2,0}^0\left(\xi \right)+D_{\mathrm{u},2}^{-1/2}\left(0\right)A_4^T\left(0\right){\lambda }_{\mathrm{u}2,0}^0\left(\xi \right)\right),\\ \hfill f_y\left(\xi \right)=D_{\mathrm{u},2}^{1/2}\left(0\right)A_4\left(0\right)z_{2,0}^0\left(\xi \right)-{\displaystyle \frac{d{D}_{\mathrm{u},2}\left(0\right)}{dt}}\xi z_{2,0}^0\left(\xi \right)-A_4^T\left(0\right){\lambda }_{\mathrm{u}2,0}^0\left(\xi \right).\end{array}$$

(85)

By virtue of the inequalities in (61), we immediately have the estimates for $f_x\left(\xi \right)$ and $f_y\left(\xi \right)$

$$\parallel f_x\left(\xi \right)\parallel \le {\alpha }_{x}exp\left(-({\beta }^0/2)\xi \right),\parallel f_y\left(\xi \right)\parallel \le {\alpha }_{y}exp\left(-({\beta }^0/2)\xi \right),\xi \ge 0,$$

(86)

where ${\alpha }_x>0$ and ${\alpha }_y>0$ are some constants; the positive constant ${\beta }^0$ was introduced in (61).

Solving the first equation in (82) subject to the first condition in (83), and using the positive definiteness of $D_{\mathrm{u},2}^{1/2}\left(0\right)$ and the estimate for $f_x\left(\xi \right)$ in (86), we obtain, after a routine algebra,

$$\begin{array}{c}\hfill x_1^0\left(\xi \right)=-{\int }_{\xi }^{+\infty }exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)(\tau -\xi )\right)f_x\left(\tau \right)d\tau ,\xi \ge 0.\end{array}$$

(87)

This equation, along with Equation (84), yields

$$\begin{array}{c}\hfill y_1^0\left(0\right)=2D_{\mathrm{u},2}^{1/2}\left(0\right)\left(-{\overline{z}}_{2,1}\left(0\right)+{\int }_0^{+\infty }exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)\tau \right)f_x\left(\tau \right)d\tau \right).\end{array}$$

(88)

Thus, the second equation in (82) subject to the initial condition (88) yields the solution

$$\begin{array}{c}\hfill y_1^0\left(\xi \right)=exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)\xi \right)y_1^0\left(0\right)+{\int }_0^{\xi }exp\left(-D_{\mathrm{u},2}^{1/2}\left(0\right)(\xi -\tau )\right)f_y\left(\tau \right)d\tau ,\xi \ge 0.\end{array}$$

(89)

Due to the positive definiteness of the matrix $D_{\mathrm{u},2}^{1/2}\left(0\right)$ and the estimate for $f_y\left(\xi \right)$ (see Equation (86)), the vector-valued function $y_1^0\left(\xi \right)$, given by (89), satisfies the limit condition in (83).

Now, using Equations (81) and (87) and (89), we obtain the solution to System (75) subject to Conditions (77) and (79)

$$\begin{array}{c}\hfill z_{2,1}^0\left(\xi \right)=x_1^0\left(\xi \right)+0.5D_{\mathrm{u},2}^{-1/2}\left(0\right)y_1^0\left(\xi \right),\xi \ge 0,\\ \hfill {\lambda }_{\mathrm{u}2,1}^0\left(\xi \right)=-D_{\mathrm{u},2}^{1/2}\left(0\right)x_1^0\left(\xi \right)+0.5y_1^0\left(\xi \right),\xi \ge 0.\end{array}$$

(90)

Due to the positive definiteness of the matrix $D_{\mathrm{u},2}^{1/2}\left(0\right)$, the estimates in (86) and Equations (87) and (89), this solution satisfies the inequalities

$$\begin{array}{c}\hfill \parallel z_{2,1}^0\left(\xi \right)\parallel \le {\gamma }_z^{0}exp\left(-({\beta }^0/2)\xi \right),\parallel {\lambda }_{\mathrm{u}2,1}^0\left(\xi \right)\parallel \le {\gamma }_{\lambda }^{0}exp\left(-({\beta }^0/2)\xi \right),\xi \ge 0,\end{array}$$

(91)

where ${\gamma }_z^0>0$ and ${\gamma }_{\lambda }^0>0$ are some constants.

Proceed to the solution to System (76) subject to Conditions (78) and (80). Similarly to (81), we make the following transformation of the state variables in Problems (76), (78), and (80):

$$\left(\begin{array}{c}\hfill z_{2,1}^f\left(\eta \right)\hfill \\ \\ \hfill {\lambda }_{\mathrm{u}2,1}^f\left(\eta \right)\hfill \end{array}\right)=\Theta \left(t_f\right)\left(\begin{array}{c}\hfill x_1^f\left(\eta \right)\hfill \\ \\ \hfill y_1^f\left(\eta \right)\hfill \end{array}\right),$$

(92)

where $x_1^f\left(\eta \right)\in E^r$ and $y_1^f\left(\eta \right)\in E^r$ are new state variables.

Transformation (92) converts Problems (76), (78), and (80) to the problem

$$\begin{array}{c}\hfill {\displaystyle \frac{d{x}_1^f\left(\eta \right)}{d\eta }}=D_{\mathrm{u},2}^{1/2}\left(t_f\right)x_1^f\left(\eta \right),\eta \le 0,\\ \hfill {\displaystyle \frac{d{y}_1^f\left(\eta \right)}{d\eta }}=-D_{\mathrm{u},2}^{1/2}\left(t_f\right)y_1^f\left(\eta \right),\eta \le 0,\end{array}$$

(93)

$$\underset{\eta \to -\infty }{lim}{x}_1^f\left(\eta \right)=0,\underset{\eta \to -\infty }{lim}{y}_1^f\left(\eta \right)=0,$$

(94)

$$\begin{array}{c}\hfill x_1^f\left(0\right)=0.5z_{2,1}^f\left(0\right)+0.5D_{\mathrm{u},2}^{-1/2}\left(t_f\right){\overline{\lambda }}_{\mathrm{u}2,1}\left(t_f\right),\\ \hfill y_1^f\left(0\right)=D_{\mathrm{u},2}^{1/2}\left(t_f\right)z_{2,1}^f\left(0\right)-{\overline{\lambda }}_{\mathrm{u}2,1}\left(t_f\right).\end{array}$$

(95)

The second equation in (93) yields (subject to the second condition in (95))

$$y_1^f\left(\eta \right)=exp\left(-D_{\mathrm{u},2}^{1/2}\left(t_f\right)\eta \right)y_1^f\left(0\right),\eta \le 0.$$

(96)

Since the matrix $D_{\mathrm{u},2}^{1/2}\left(t_f\right)$ is positive definite, then the vector-valued function $y_1^f\left(\eta \right)$, obtained in (96), satisfies the second limit equality in (94) if and only if $y_1^f\left(0\right)=0$. The latter, along with Equation (95), yields

$$\begin{array}{c}\hfill x_1^f\left(0\right)=D_{\mathrm{u},2}^{-1/2}\left(t_f\right){\overline{\lambda }}_{\mathrm{u}2,1}\left(t_f\right).\end{array}$$

(97)

Hence, the solution to the second equation in (93) subject to the aforementioned condition $y_1^f\left(0\right)=0$, is identically zero, i.e.,

$$y_1^f\left(\eta \right)\equiv 0,\eta \le 0.$$

(98)

Solving the first equation in (93) subject to the condition obtained in (97), we have

$$x_1^f\left(\eta \right)=exp\left(D_{\mathrm{u},2}^{1/2}\left(t_f\right)\eta \right)D_{\mathrm{u},2}^{-1/2}\left(t_f\right){\overline{\lambda }}_{\mathrm{u}2,1}\left(t_f\right),\eta \le 0.$$

(99)

Since the matrix $D_{\mathrm{u},2}^{1/2}\left(t_f\right)$ is positive definite, then the vector-valued function $x_1^f\left(\eta \right)$, obtained in (99), satisfies the first limit equality in (94).

Thus, the vector-valued functions $x_1^f\left(\eta \right)$ and $y_1^f\left(\eta \right)$, obtained in (99) and (98), respectively, constitute the solution to Problems (93)–(95). Substitution of this solution into (92), and use of Equation (50), yields, after a routine rearrangement, the solution to System (76) satisfying Conditions (78) and (80)

$$\begin{array}{c}\hfill z_{2,1}^f\left(\eta \right)=exp\left(D_{\mathrm{u},2}^{1/2}\left(t_f\right)\eta \right)D_{\mathrm{u},2}^{-1/2}\left(t_f\right){\overline{\lambda }}_{\mathrm{u}2,1}\left(t_f\right),\eta \le 0,\\ \hfill {\lambda }_{\mathrm{u}2,1}^f\left(\eta \right)=-exp\left(D_{\mathrm{u},2}^{1/2}\left(t_f\right)\eta \right){\overline{\lambda }}_{\mathrm{u}2,1}\left(t_f\right),\eta \le 0.\end{array}$$

(100)

This solution satisfies the inequalities

$$\parallel z_{2,1}^f\left(\eta \right)\parallel \le {\gamma }_z^{f}exp\left({\beta }^f\eta \right),\parallel {\lambda }_{\mathrm{u}2,1}^f\left(\eta \right)\parallel \le {\gamma }_{\lambda }^{f}exp\left({\beta }^f\eta \right),\eta \le 0,$$

(101)

where ${\gamma }_z^f>0$ and ${\gamma }_{\lambda }^f>0$ are some constants, and ${\beta }^f>0$ is the minimal eigenvalue of the matrix $D_{\mathrm{u},2}^{1/2}\left(t_f\right)$.

4.3. Justification of the First-Order Asymptotic Solution to Problems (26) and (27)

Using the results of Section 4.2.1, Section 4.2.2, Section 4.2.3, Section 4.2.5, Section 4.2.6, Section 4.2.7, Section 4.2.8 and Section 4.2.9, we can rewrite the components of the first-order asymptotic solution to Problems (26) and (27) (see Equation (28)) as follows:

$$\begin{array}{c}\hfill z_1^1(t,\epsilon )={\overline{z}}_{1,0}\left(t\right)+\epsilon \left({\overline{z}}_{1,1}\left(t\right)+z_{1,1}^0(t/\epsilon )\right),\\ \hfill z_2^1(t,\epsilon )={\overline{z}}_{2,0}\left(t\right)+z_{2,0}^0(t/\epsilon )+\epsilon \left({\overline{z}}_{2,1}\left(t\right)+z_{2,1}^0(t/\epsilon )+z_{2,1}^f((t-t_f)/\epsilon )\right),\\ \hfill {\lambda }_{\mathrm{u}1}^1(t,\epsilon )={\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)+\epsilon {\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right),\\ \hfill {\lambda }_{\mathrm{u}2}^1(t,\epsilon )={\lambda }_{\mathrm{u}2,0}^0(t/\epsilon )+\epsilon \left({\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)+{\lambda }_{\mathrm{u}2,1}^0(t/\epsilon )+{\lambda }_{\mathrm{u}2,1}^f((t-t_f)/\epsilon )\right),\\ \hfill {\lambda }_{\mathrm{v}}^1(t,\epsilon )={\overline{\lambda }}_{\mathrm{v},0}\left(t\right)+\epsilon \left({\overline{\lambda }}_{\mathrm{v},1}\left(t\right)+{\lambda }_{\mathrm{v},1}^0(t/\epsilon )\right),{\lambda }_{\mathrm{u},\mathrm{v}}^1(t,\epsilon )={\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)+\epsilon {\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right).\end{array}$$

(102)

Lemma 1. 

Let Assumptions (A1)–(A7) be valid. Then, there exists a number ${\epsilon }_0>0$ such that, for all $t\in [0,t_f]$ and $\epsilon \in (0,{\epsilon }_0]$, the following inequalities are satisfied:

$$\begin{array}{c}\hfill \parallel z_1(t,\epsilon )-z_1^1(t,\epsilon )\parallel \le c_0{\epsilon }^2,\parallel z_2(t,\epsilon )-z_2^1(t,\epsilon )\parallel \le c_0{\epsilon }^2,\\ \hfill \parallel {\lambda }_{\mathrm{u}1}(t,\epsilon )-{\lambda }_{\mathrm{u}1}^1(t,\epsilon )\parallel \le c_0{\epsilon }^2,\parallel {\lambda }_{\mathrm{u}2}(t,\epsilon )-{\lambda }_{\mathrm{u}2}^1(t,\epsilon )\parallel \le c_0{\epsilon }^2,\\ \hfill \parallel {\lambda }_{\mathrm{v}}(t,\epsilon )-{\lambda }_{\mathrm{v}}^1(t,\epsilon )\parallel \le c_0{\epsilon }^2,\parallel {\lambda }_{\mathrm{u},\mathrm{v}}(t,\epsilon )-{\lambda }_{\mathrm{u},\mathrm{v}}^1(t,\epsilon )\parallel \le c_0{\epsilon }^2,\end{array}$$

(103)

where $\mathrm{col}\left(z_1(t,\epsilon ),z_2(t,\epsilon ),{\lambda }_{\mathrm{u}1}(t,\epsilon ),{\lambda }_{\mathrm{u}2}(t,\epsilon ),{\lambda }_{\mathrm{v}}(t,\epsilon ),{\lambda }_{\mathrm{u},\mathrm{v}}(t,\epsilon )\right)$ is the unique solution to the singularly perturbed boundary-value Problems (26) and (27), and $c_0>0$ is some constant independent of ε.

Proof. 

First of all, let us observe the following. The differential system (26) is linear homogeneous. Furthermore, due to Proposition 1 and Equations (18) and (20)–(23), all of the matrix-valued coefficients in this system are continuously differentiable in the interval $[0,t_f]$. Moreover, the matrix-valued coefficients $D_{\mathrm{u},1}\left(t\right)$ and $D_{\mathrm{u},2}\left(t\right)$ are twice continuously differentiable in this interval. The boundary Conditions (27) are explicit and linear. Moreover, the boundary conditions for the fast state variables $z_2\left(t\right)$ and ${\lambda }_{\mathrm{u}2}\left(t\right)$ are given at the opposite ends of the interval $[0,t_f]$. Namely, the condition for $z_2\left(t\right)$ is given at $t=0$, while the condition for ${\lambda }_{\mathrm{u}2}\left(t\right)$ is given at $t=t_f$.

Let us consider the matrix of the coefficients for the fast state vector $\mathrm{col}\left(z_2\left(t\right),{\lambda }_{\mathrm{u}2}\left(t\right)\right)$ in the set of the fast modes of System (26), i.e., the matrix

$$\mathsf{\Phi }(t,\epsilon )\stackrel{▵}{=}\left(\begin{array}{cc}\hfill \epsilon A_4\left(t\right)\hfill & \hfill -I_r\hfill \\ \hfill -D_{\mathrm{u},2}\left(t\right)\hfill & \hfill \epsilon A_4^T\left(t\right)\hfill \end{array}\right),t\in [0,t_f],\epsilon >0.$$

(104)

The matrix-valued function $\mathsf{\Phi }(t,\epsilon )$ is of the dimension $2r\times 2r$, and it is continuous with respect to $(t,\epsilon )\in [0,t_f]\times (-\infty ,+\infty )$. By ${\mu }_i(t,\epsilon )$, $(i=1,\dots ,r,r+1,\dots ,2r)$, we denote the eigenvalues of the matrix $\mathsf{\Phi }(t,\epsilon )$. We are going to show that, for all $t\in [0,t_f]$ and all sufficiently small $\epsilon >0$, r eigenvalues ${\mu }_i(t,\epsilon )$, $(i=1,\dots ,r)$ of this matrix satisfy the inequality

$$\mathrm{Re}\left({\mu }_i(t,\epsilon )\right)\ge \alpha ,i=1,\dots ,r,$$

(105)

while the other eigenvalues satisfy the inequality

$$\mathrm{Re}\left({\mu }_j(t,\epsilon )\right)\le -\alpha ,j=r+1,\dots ,2r,$$

(106)

where $\alpha >0$ is some constant independent of $\epsilon$.

Along with $\mathsf{\Phi }(t,\epsilon )$, let us consider the matrix-valued function

$$\mathsf{\Psi }(t,\epsilon )\stackrel{▵}{=}{\Theta }^{-1}\left(t\right)\mathsf{\Phi }(t,\epsilon )\Theta \left(t\right),$$

(107)

where the matrix-valued functions $\Theta \left(t\right)$ and ${\Theta }^{-1}\left(t\right)$ are given by (50) and (51), respectively.

Like the matrix-valued function $\mathsf{\Phi }(t,\epsilon )$, the matrix-valued function $\mathsf{\Psi }(t,\epsilon )$ is also continuous with respect to $(t,\epsilon )\in [0,t_f]\times (-\infty ,+\infty )$. Moreover, for any pair $(t,\epsilon )\in [0,t_f]\times (-\infty ,+\infty )$, both matrices $\mathsf{\Phi }(t,\epsilon )$ and $\mathsf{\Psi }(t,\epsilon )$ have the same set of eigenvalues ${\mu }_i(t,\epsilon )$, $(i=1,\dots ,r,r+1,\dots ,2r)$..

Using Equations (50), (51), and (104), we can represent the matrix $\mathsf{\Psi }(t,\epsilon )$ in the following block form:

$$\begin{array}{c}\hfill \mathsf{\Psi }(t,\epsilon )=\left(\begin{array}{c}\hfill {\mathsf{\Psi }}_1(t,\epsilon ){\mathsf{\Psi }}_2(t,\epsilon )\hfill \\ \hfill {\mathsf{\Psi }}_3(t,\epsilon ){\mathsf{\Psi }}_4(t,\epsilon )\hfill \end{array}\right),(t,\epsilon )\in [0,t_f]\times (-\infty ,+\infty ),\\ \hfill {\mathsf{\Psi }}_1(t,\epsilon )=D_{\mathrm{u},2}^{1/2}\left(t\right)+0.5\epsilon \left(A_4\left(t\right)+D_{\mathrm{u},2}^{-1/2}\left(t\right)A_4^T\left(t\right)D_{\mathrm{u},2}^{1/2}\left(t\right)\right),\\ \hfill {\mathsf{\Psi }}_2(t,\epsilon )=0.25\epsilon \left(A_4\left(t\right)D_{\mathrm{u},2}^{-1/2}\left(t\right)-D_{\mathrm{u},2}^{-1/2}\left(t\right)A_4^T\left(t\right)\right),\\ \hfill {\mathsf{\Psi }}_3(t,\epsilon )=\epsilon \left(D_{\mathrm{u},2}^{1/2}\left(t\right)A_4\left(t\right)-A_4^T\left(t\right)D_{\mathrm{u},2}^{1/2}\left(t\right)\right),\\ \hfill {\mathsf{\Psi }}_4(t,\epsilon )=-D_{\mathrm{u},2}^{1/2}\left(t\right)+0.5\epsilon \left(D_{\mathrm{u},2}^{1/2}\left(t\right)A_4\left(t\right)D_{\mathrm{u},2}^{-1/2}\left(t\right)+A_4^T\left(t\right)\right).\end{array}$$

(108)

Setting $\epsilon =0$ in (108), we obtain

$$\mathsf{\Psi }(t,0)=\left(\begin{array}{cc}{D}_{\mathrm{u},2}^{1/2}\left(t\right)\hfill & 0\hfill \\ 0\hfill & -D_{\mathrm{u},2}^{1/2}\left(t\right)\hfill \end{array}\right),t\in [0,t_f].$$

(109)

For any $t\in [0,t_f]$, the eigenvalues of $\mathsf{\Psi }(t,0)$ are ${\nu }_i\left(t\right)$, $(i=1,\dots ,r)$ and $-{\nu }_i\left(t\right)$, $(i=1,\dots ,r)$, where ${\nu }_i\left(t\right)$, $(i=1,\dots ,r)$ are the eigenvalues of the matrix $D_{\mathrm{u},2}^{1/2}\left(t\right)$. The latter matrix is symmetric and positive definite for all $t\in [0,t_f]$. Therefore, ${\nu }_i\left(t\right)$, $(i=1,\dots ,r)$ are real and positive for all $t\in [0,t_f]$. Moreover, since the matrix-valued function $D_{\mathrm{u},2}^{1/2}\left(t\right)$ is continuous in the interval $[0,t_f]$, then, due to the results of [70], the functions ${\nu }_i\left(t\right)$, $(i=1,\dots ,r)$ also are continuous for $t\in [0,t_f]$. This feature, along with the aforementioned positiveness of these functions directly yields the existence of a positive number $\alpha$ such that the following inequality is satisfied:

$${\nu }_i\left(t\right)\ge 2\alpha ,t\in [0,t_f],i=1,\dots ,r.$$

(110)

Furthermore, since, for any $t\in [0,t_f]$, the matrix-valued function $\mathsf{\Psi }(t,\epsilon )$ is continuous with respect to $\epsilon \in (-\infty ,+\infty )$, then

$$\begin{array}{c}\hfill {\mu }_i(t,0)={\nu }_i\left(t\right),{\mu }_{i+r}(t,0)=-{\nu }_i\left(t\right),t\in [0,t_f],i=1,\dots ,r,\end{array}$$

meaning, along with (110), that

$${\mu }_i(t,0)\ge 2\alpha ,{\mu }_{i+r}(t,0)\le -2\alpha ,t\in [0,t_f],i=1,\dots ,r.$$

(111)

The matrix-valued function $\mathsf{\Psi }(t,\epsilon )$, being continuous with respect to $(t,\epsilon )\in [0,t_f]\times (-\infty ,+\infty )$, is continuous in $\epsilon \in (-\infty ,+\infty )$ uniformly with respect to $t\in [0,t_f]$. Therefore, by virtue of [70], the eigenvalues ${\mu }_i(t,\epsilon )$, $(i=1,\dots ,r,r+1,\dots ,2r)$ are continuous in $\epsilon \in (-\infty ,+\infty )$ uniformly with respect to $t\in [0,t_f]$. This observation, along with the inequalities in (111), yields the existence of a number $\overline{\epsilon }>0$ such that, for all $t\in [0,t_f]$ and all $\epsilon \in (0,\overline{\epsilon }]$, Inequalities (105) and (106) are satisfied. The validity of these inequalities means that the singularly perturbed boundary-value Problems (26) and (27) are of the conditionally stable type [66,67]. Taking into account this feature of the considered boundary-value problem, as well as the aforementioned properties of System (26) and boundary Conditions (27), and using the results of the work [67], we immediately obtain the statements of the lemma. □

5. Main Results

5.1. Asymptotic Expansion of the Stackelberg Solution to the Game (9)–(11)

Theorem 1. 

Let Assumptions (A1)–(A7) be valid. Then, for all $t\in [0,t_f]$ and all $\epsilon \in (0,{\epsilon }_0]$, the open-loop Stackelberg solution $\left(u^{*}(t,\epsilon ),v^{*}(t,\epsilon )\right)$ to the game (9)–(11) satisfies the inequalities

$$\begin{array}{c}\hfill \parallel \epsilon u^{*}(t,\epsilon )+{\lambda }_{\mathrm{u}2}^1(t,\epsilon )\parallel \le c_0{\epsilon }^2,\\ \hfill \parallel v^{*}(t,\epsilon )+B_{\mathrm{v}}^T\left(t\right){\lambda }_{\mathrm{v}}^1(t,\epsilon )\parallel \le c_1{\epsilon }^2,\end{array}$$

(112)

where ${\lambda }_{\mathrm{u}2}^1(t,\epsilon )$ and ${\lambda }_{\mathrm{v}}^1(t,\epsilon )$ are given in Equation (102), and the numbers ${\epsilon }_0>0$ and $c_0>0$ are introduced in Lemma 1,

$$\begin{array}{c}\hfill c_1=c_0\left(\underset{t\in [0,t_f]}{max}\parallel B_{\mathrm{v}}\left(t\right)\parallel \right).\end{array}$$

Proof. 

Let us start with the proof of the first inequality in (112).

Using Equations (13), (19), and (25), Proposition 2, and the equivalence of the boundary-value Problems (17), (26), and (27), we obtain, after a routine algebra,

$$\epsilon u^{*}(t,\epsilon )=-{\lambda }_{\mathrm{u}2}(t,\epsilon ),t\in [0,t_f],\epsilon \in (0,{\epsilon }_0],$$

(113)

where ${\lambda }_{\mathrm{u}2}(t,\epsilon )$ is the component of the solution to the boundary-value Problems (26) and (27) (see Lemma 1).

Now, using Equation (113) and the inequality for ${\lambda }_{\mathrm{u}2}(t,\epsilon )$ (see Equation (103)), we directly have the first inequality in (112).

Proceed to the proof of the second inequality in (112). This inequality directly follows from Equation (19), Proposition 2, the equivalence of the boundary-value Problems (17), (26), and (27), and the inequality for ${\lambda }_{\mathrm{v}}(t,\epsilon )$ in (103).

Thus, the theorem is proven. □

5.2. Asymptotic Expansion of the Stackelberg Optimal Values of the Cost Functionals in the Game (9)–(11)

Consider the values

$$\begin{array}{c}\hfill {\overline{J}}_{\mathrm{u},0}={\displaystyle \frac{1}{2}}{\int }_0^{t_f}\left[{\overline{z}}_0^T\left(t\right)D_{\mathrm{u}}\left(t\right){\overline{z}}_0\left(t\right)+{\overline{\lambda }}_{\mathrm{v},0}^T\left(t\right)S_{\mathrm{u},\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right)\right]dt,\\ \hfill {\overline{J}}_{\mathrm{v},0}={\displaystyle \frac{1}{2}}{\int }_0^{t_f}[{\overline{z}}_0^T\left(t\right)D_{\mathrm{v}}\left(t\right){\overline{z}}_0\left(t\right)+{\overline{\lambda }}_{\mathrm{v},0}^T\left(t\right)S_{\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right)]dt,\end{array}$$

(114)

where ${\overline{z}}_0\left(t\right)=\left(\begin{array}{c}{\overline{z}}_{1,0}\left(t\right)\\ {\overline{z}}_{2,0}\left(t\right)\end{array}\right)$, ${\overline{z}}_{1,0}\left(t\right)$, and ${\overline{\lambda }}_{\mathrm{v},0}\left(t\right)$ are the corresponding components of the solution to the boundary-value Problems (39) and (37), and ${\overline{z}}_{2,0}\left(t\right)$ is given by Equation (38).

Theorem 2. 

Let Assumptions (A1)–(A7) be valid. Then, for all $\epsilon \in (0,{\epsilon }_0]$, the Stackelberg optimal values of the cost functionals $J_{\mathrm{u}}^{*}\left(\epsilon \right)$ and $J_{\mathrm{v}}^{*}\left(\epsilon \right)$ in the game (9)–(11) satisfy the inequalities

$$\begin{array}{c}\hfill |J_{\mathrm{u}}^{*}\left(\epsilon \right)-{\overline{J}}_{\mathrm{u},0}|\le b\epsilon ,\\ \hfill |J_{\mathrm{v}}^{*}\left(\epsilon \right)-{\overline{J}}_{\mathrm{v},0}|\le b\epsilon ,\end{array}$$

(115)

where $b>0$ is some constant independent of ε.

Proof. 

Let us start with the proof of the first inequality in (115).

Substituting the expressions for the $u^{*}(t,\epsilon )$, $v^{*}(t,\epsilon )$ (see Equations (19) and (113)) into Equation (10) and using Equations (14), (18), and (25), Proposition 2 and the equivalence of the boundary-value Problems (17), (26), and (27), we obtain, after a routine rearrangement, the following expression for $J_{\mathrm{u}}^{*}\left(\epsilon \right)$:

$$\begin{array}{c}\hfill J_{\mathrm{u}}^{*}\left(\epsilon \right)=J_{\mathrm{u}}\left(u^{*}(t,\epsilon ),v^{*}(t,\epsilon )\right)\\ \hfill ={\displaystyle \frac{1}{2}}{\int }_0^{t_f}[z_1^T(t,\epsilon )D_{\mathrm{u},1}\left(t\right)z_1(t,\epsilon )+z_2^T(t,\epsilon )D_{\mathrm{u},2}\left(t\right)z_2(t,\epsilon )\\ \hfill +{\lambda }_{\mathrm{u}2}^T(t,\epsilon ){\lambda }_{\mathrm{u}2}(t,\epsilon )+{\lambda }_{\mathrm{v}}^T(t,\epsilon )S_{\mathrm{u},\mathrm{v}}\left(t\right){\lambda }_{\mathrm{v}}(t,\epsilon )]dt,\epsilon \in (0,{\epsilon }_0],\end{array}$$

(116)

where $z_1(t,\epsilon )$, $z_2(t,\epsilon )$, ${\lambda }_{\mathrm{u}2}(t,\epsilon )$, and ${\lambda }_{\mathrm{v}}(t,\epsilon )$ are the corresponding components of the solution to boundary-value Problems (26) and (27).

Now, using Equation (116), as well as Lemma 1, the inequalities in (61), (66), (91), and (101), and the first equation in (114), immediately yields the first inequality in (115).

The second inequality in (115) is proven quite similarly to the proof of the first inequality. This completes the proof of the theorem. □

Remark 10. 

Using Proposition 3, Remark 8, and the first equation in (114), we can conclude that ${\overline{J}}_{\mathrm{u},0}={\overline{J}}_{\mathrm{u}}^{*}$.

5.3. Asymptotically Suboptimal Stackelberg Solution to the Game (9)–(11)

Consider the following controls for the leader and the follower, respectively,

$$\begin{array}{c}\hfill \tilde{u}(t,\epsilon )=-{\displaystyle \frac{1}{\epsilon }}{\lambda }_{\mathrm{u}2,0}^0(t/\epsilon )-{\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right),t\in [0,t_f],\epsilon \in (0,{\epsilon }_0]\\ \hfill \mathrm{and}\\ \hfill \tilde{v}\left(t\right)=-B_{\mathrm{v}}^T\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right),t\in [0,t_f].\end{array}$$

(117)

Let ${\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)$ and ${\tilde{J}}_{\mathrm{v}}\left(\epsilon \right)$ be the values of the functionals $J_{\mathrm{u}}(u,v)$ and $J_{\mathrm{v}}(u,v)$, respectively, generated by the pair of the controls $\left(\tilde{u}(t,\epsilon ),\tilde{v}\left(t\right)\right)$ in the game (9)–(11).

Theorem 3. 

Let Assumptions (A1)–(A7) be valid. Then, for all $\epsilon \in (0,{\epsilon }_0]$, the following inequalities are satisfied:

$$\begin{array}{c}\hfill |J_{\mathrm{u}}^{*}\left(\epsilon \right)-{\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)|\le \tilde{b}\epsilon ,\\ \hfill |J_{\mathrm{v}}^{*}\left(\epsilon \right)-{\tilde{J}}_{\mathrm{v}}\left(\epsilon \right)|\le \tilde{b}\epsilon ,\end{array}$$

(118)

where $\tilde{b}>0$ is some constant independent of ε.

Proof. 

Let us start with the proof of the first inequality in (118).

Substituting the open-loop Stackelberg solution $\left(u^{*}(t,\epsilon ),v^{*}(t,\epsilon )\right)$ into System (9) instead of $\left(u\left(t\right),v\left(t\right)\right)$, we obtain, for all $\epsilon \in (0,{\epsilon }_0]$,

$$\begin{array}{c}\hfill {\displaystyle \frac{dz\left(t\right)}{dt}}=A\left(t\right)z\left(t\right)+B_{\mathrm{u}}\left(t\right)u^{*}(t,\epsilon )+B_{\mathrm{v}}\left(t\right)v^{*}(t,\epsilon ),t\in [0,t_f],z\left(0\right)=z_0.\end{array}$$

(119)

The solution $z^{*}(t,\epsilon )$ of this system is the Stackelberg optimal trajectory of the game (9)–(11). Due to Proposition 2 and the equivalence of the boundary-value Problems (17), (26), and (27), we directly have

$$\begin{array}{c}\hfill z^{*}(t,\epsilon )=z(t,\epsilon ),t\in [0,t_f],\epsilon \in (0,{\epsilon }_0],\end{array}$$

(120)

where $z(t,\epsilon )$ is the corresponding component of the solution to the boundary-value Problems (26) and (27).

Now, let us substitute the pair of the controls $\left(\tilde{u}(t,\epsilon ),\tilde{v}\left(t\right)\right)$ into System (9) instead of $\left(u\left(t\right),v\left(t\right)\right)$. This substitution yields, for all $\epsilon \in (0,{\epsilon }_0]$,

$$\begin{array}{c}\hfill {\displaystyle \frac{dz\left(t\right)}{dt}}=A\left(t\right)z\left(t\right)+B_{\mathrm{u}}\left(t\right)\tilde{u}(t,\epsilon )+B_{\mathrm{v}}\left(t\right)\tilde{v}\left(t\right),t\in [0,t_f],z\left(0\right)=z_0.\end{array}$$

(121)

Let $z=\tilde{z}(t,\epsilon )$, and $t\in [0,t_f]$ denotes the solution to this system.

Substituting $\tilde{z}(t,\epsilon )$, $\tilde{u}(t,\epsilon )$ and $\tilde{v}\left(t\right)$ into Functional (10) instead of $z\left(t\right)$, $u\left(t\right)$ and $v\left(t\right)$, respectively, yields, after a routine rearrangement, the following expression for ${\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)$:

$$\begin{array}{c}\hfill {\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)=J_{\mathrm{u}}\left(\tilde{u}(t,\epsilon ),\tilde{v}\left(t\right)\right)={\displaystyle \frac{1}{2}}{\int }_0^{t_f}[{\tilde{z}}^T(t,\epsilon )D_{\mathrm{u}}\left(t\right)\tilde{z}(t,\epsilon )\\ \hfill +{\left({\lambda }_{\mathrm{u}2,0}^0(t/\epsilon )+\epsilon {\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)\right)}^T\left({\lambda }_{\mathrm{u}2,0}^0(t/\epsilon )+\epsilon {\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)\right)+{\overline{\lambda }}_{\mathrm{v},0}^T\left(t\right)S_{\mathrm{u},\mathrm{v}}\left(t\right){\overline{\lambda }}_{\mathrm{v},0}\left(t\right)]dt,\epsilon \in (0,{\epsilon }_0],\end{array}$$

(122)

where $S_{\mathrm{u},\mathrm{v}}\left(t\right)$ is given in (18).

Let us denote

$$\begin{array}{c}\hfill {\delta }_z(t,\epsilon )\stackrel{▵}{=}{z}^{*}(t,\epsilon )-\tilde{z}(t,\epsilon ),t\in [0,t_f],\epsilon \in (0,{\epsilon }_0].\end{array}$$

(123)

Using Equations (119) and (121), we directly obtain that ${\delta }_z(t,\epsilon )$ is the unique solution to the system

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\delta }_z(t,\epsilon )}{dt}}=A\left(t\right){\delta }_z(t,\epsilon )+B_{\mathrm{u}}\left(t\right)\left(u^{*}(t,\epsilon )-\tilde{u}(t,\epsilon )\right)+B_{\mathrm{v}}\left(t\right)\left(v^{*}(t,\epsilon )-\tilde{v}\left(t\right)\right),\\ \hfill t\in [0,t_f],{\delta }_z(0,\epsilon )=0.\end{array}$$

(124)

Let us estimate the differences $\left(u^{*}(t,\epsilon )-\tilde{u}(t,\epsilon )\right)$ and $\left(v^{*}(t,\epsilon )-\tilde{v}\left(t\right)\right)$. Due to Lemma 1 and Equations (91) and (101), we have

$$\begin{array}{c}\hfill \parallel {\lambda }_{\mathrm{u}2}(t,\epsilon )-{\lambda }_{\mathrm{u}2,0}^0(t/\epsilon )-\epsilon {\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)\parallel \le c_0{\epsilon }^2\\ \hfill +{\gamma }_{\lambda }^0\epsilon exp\left(-({\beta }^0/2)t/\epsilon \right)+{\gamma }_{\lambda }^f\epsilon exp\left({\beta }^f(t-t_f)/\epsilon \right),t\in [0,t_f],\epsilon \in (0,{\epsilon }_0].\end{array}$$

(125)

Using this inequality and the expressions for $u^{*}(t,\epsilon )$ and $\tilde{u}(t,\epsilon )$ (see Equations (113) and (117)) directly yields

$$\begin{array}{c}\hfill \parallel u^{*}(t,\epsilon )-\tilde{u}(t,\epsilon )\parallel \le c_0\epsilon +{\gamma }_{\lambda }^{0}exp\left(-({\beta }^0/2)t/\epsilon \right)+{\gamma }_{\lambda }^{f}exp\left({\beta }^f(t-t_f)/\epsilon \right),\\ \hfill t\in [0,t_f],\epsilon \in (0,{\epsilon }_0].\end{array}$$

(126)

Similarly, using the expressions for ${\lambda }_{\mathrm{v}}^1(t,\epsilon )$ and $\tilde{v}\left(t\right)$ (see Equations (102) and (117)), as well as the second inequalities in (66) and (112), we have

$$\begin{array}{c}\hfill \parallel v^{*}(t,\epsilon )-\tilde{v}(t,\epsilon )\parallel \le c_2\epsilon ,t\in [0,t_f],\epsilon \in (0,{\epsilon }_0],\end{array}$$

(127)

where $c_2>0$ is some constant independent of $\epsilon$.

Let $\Gamma (t,\sigma )$, $0\le \sigma \le t\le t_f$ be the fundamental matrix solution to the homogeneous system corresponding the differential equation in (124). Hence, we can express ${\delta }_z(t,\epsilon )$ as follows:

$$\begin{array}{c}\hfill {\delta }_z(t,\epsilon )={\int }_0^t\Gamma (t,\sigma )[B_{\mathrm{u}}\left(\sigma \right)\left(u^{*}(\sigma ,\epsilon )-\tilde{u}(\sigma ,\epsilon )\right)+B_{\mathrm{v}}\left(\sigma \right)\left(v^{*}(\sigma ,\epsilon )-\tilde{v}\left(\sigma \right)\right)]d\sigma ,\\ \hfill t\in [0,t_f],\epsilon \in (0,{\epsilon }_0].\end{array}$$

This equation, along with Inequalities (126) and (127), immediately yields

$$\begin{array}{c}\hfill \parallel {\delta }_z(t,\epsilon )\parallel \le c_3\epsilon ,t\in [0,t_f],\epsilon \in (0,{\epsilon }_0],\end{array}$$

(128)

where $c_3>0$ is some constant independent of $\epsilon$.

Furthermore, by virtue of the equation for ${\lambda }_{\mathrm{v}}^1(t,\epsilon )$ and the inequality for ${\lambda }_{\mathrm{v}}(t,\epsilon )$ (see Equations (102) and (103)), we have the inequality

$$\begin{array}{c}\hfill \parallel {\lambda }_{\mathrm{v}}(t,\epsilon )-{\overline{\lambda }}_{\mathrm{v},0}\left(t\right)\parallel \le c_4\epsilon ,t\in [0,t_f],\epsilon \in (0,{\epsilon }_0],\end{array}$$

(129)

where $c_4>0$ is some constant independent of $\epsilon$.

Now, Equations (116), (122), and (123), and Inequalities (125), (128), and (129), yield the first inequality in (118). The second inequality in (118) is proven similarly. This completes the proof of the theorem. □

Remark 11. 

Due to Theorem 3 and Equation (16), the pair of the controls $\left(\tilde{u}(t,\epsilon ),\tilde{v}\left(t\right)\right)$ is an asymptotically suboptimal Stackelberg solution in the game (9)–(11). To obtain this solution, one has to first solve the boundary-value Problems (39) and (37). Then, using the explicit formulas in (38), (60), and (72), one calculates ${\lambda }_{\mathrm{u}2,0}^0(t/\epsilon )$ and ${\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)$.

As a direct consequence of Theorems 2 and 3, we obtain the following assertion.

Corollary 1. 

Let Assumptions (A1)–(A7) be valid. Then, for all $\epsilon \in (0,{\epsilon }_0]$, the values ${\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)$ and ${\tilde{J}}_{\mathrm{v}}\left(\epsilon \right)$ satisfy the inequalities

$$\begin{array}{c}\hfill |{\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)-{\overline{J}}_{\mathrm{u},0}|\le (b+\tilde{b})\epsilon ,\\ \hfill |{\tilde{J}}_{\mathrm{v}}\left(\epsilon \right)-{\overline{J}}_{\mathrm{v},0}|\le (b+\tilde{b})\epsilon .\end{array}$$

(130)

6. Example

Consider a particular case of the Stackelberg game (9)–(11) with the following data:

$$\begin{array}{c}\hfill n=2,r=1,s=1,t_f=1,\\ \hfill A\left(t\right)=A=\left(\begin{array}{c}00\\ 00\end{array}\right),B_{\mathrm{u}}\left(t\right)=B_{\mathrm{u}}=\left(\begin{array}{c}0\\ 1\end{array}\right),B_{\mathrm{v}}\left(t\right)=B_{\mathrm{v}}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right),\\ \hfill D_{\mathrm{u}}\left(t\right)=D_{\mathrm{u}}=\left(\begin{array}{c}\hfill 20\hfill \\ \hfill 00.5\hfill \end{array}\right),D_{\mathrm{v}}\left(t\right)=D_{\mathrm{v}}=\left(\begin{array}{c}\hfill 00\hfill \\ \hfill 01\hfill \end{array}\right),z_0=\left(\begin{array}{c}\sqrt{2}\\ 1\end{array}\right),\\ \hfill G_{\mathrm{u},\mathrm{v}}\left(t\right)=G_{\mathrm{u},\mathrm{v}}=2,G_{\mathrm{v},\mathrm{u}}\left(t\right)=G_{\mathrm{v},\mathrm{u}}=1.\end{array}$$

(131)

The example of the game (9)–(11) with Data (131) will allow us to clearly illustrate the theoretical results of the paper (Lemma 1 and Theorems 1–3), while avoiding too complicated analytical/numerical calculations.

6.1. Illustration of Lemma 1

To illustrate this lemma, we should construct the functions in (102) subject to Data (131).

By ${\overline{\lambda }}_{\mathrm{v}1,0}\left(t\right)$ and ${\overline{\lambda }}_{\mathrm{v}2,0}\left(t\right)$, we denote the upper and lower components of the vector ${\overline{\lambda }}_{\mathrm{v},0}\left(t\right)$. Similarly, by ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},10}\left(t\right)$ and ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},20}\left(t\right)$, we denote the upper and lower components of the vector ${\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)$. Thus,

$$\begin{array}{c}\hfill {\overline{\lambda }}_{\mathrm{v},0}\left(t\right)=\left(\begin{array}{c}{\overline{\lambda }}_{\mathrm{v},10}\left(t\right)\\ {\overline{\lambda }}_{\mathrm{v},20}\left(t\right)\end{array}\right),{\overline{\lambda }}_{\mathrm{u},\mathrm{v},0}\left(t\right)=\left(\begin{array}{c}{\overline{\lambda }}_{\mathrm{u},\mathrm{v},10}\left(t\right)\\ {\overline{\lambda }}_{\mathrm{u},\mathrm{v},20}\left(t\right)\end{array}\right).\end{array}$$

(132)

Using these notations and Data (131), we can rewrite Problems (39) and (37) as

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\overline{z}}_{1,0}\left(t\right)}{dt}}=-{\overline{\lambda }}_{\mathrm{v},10}\left(t\right)+{\overline{\lambda }}_{\mathrm{v},20}\left(t\right),{\overline{z}}_{1,0}\left(0\right)=\sqrt{2},\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)}{dt}}=-2{\overline{z}}_{1,0}\left(t\right),{\overline{\lambda }}_{\mathrm{u}1,0}\left(1\right)=0,\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{v},10}\left(t\right)}{dt}}=0,{\overline{\lambda }}_{\mathrm{v},10}\left(1\right)=0,\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{v},20}\left(t\right)}{dt}}=-2{\overline{\lambda }}_{\mathrm{u},\mathrm{v}20}\left(t\right),{\overline{\lambda }}_{\mathrm{v},20}\left(1\right)=0,\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u},\mathrm{v},10}\left(t\right)}{dt}}={\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)-2{\overline{\lambda }}_{\mathrm{v},10}\left(t\right)+2{\overline{\lambda }}_{\mathrm{v},20}\left(t\right),{\overline{\lambda }}_{\mathrm{u},\mathrm{v},10}\left(0\right)=0,\\ \hfill {\displaystyle \frac{d{\overline{\lambda }}_{\mathrm{u},\mathrm{v},20}\left(t\right)}{dt}}=-{\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)+2{\overline{\lambda }}_{\mathrm{v},10}\left(t\right)-2{\overline{\lambda }}_{\mathrm{v},20}\left(t\right),{\overline{\lambda }}_{\mathrm{u},\mathrm{v},20}\left(0\right)=0.\end{array}$$

(133)

Problem (133) yields a unique solution in the time interval $[0,1]$

$$\begin{array}{c}\hfill {\overline{z}}_{1,0}\left(t\right)=\left(C_1+C_2(t+\sqrt{2})\right)exp\left(\sqrt{2}t\right)+\left(C_3+C_4(t-\sqrt{2})\right)exp(-\sqrt{2}t),\\ \hfill {\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right)=-\left(C_1\sqrt{2}+C_2(\sqrt{2}t+1)\right)exp\left(\sqrt{2}t\right)+\left(C_3\sqrt{2}+C_4(\sqrt{2}t-1)\right)exp(-\sqrt{2}t),\\ \hfill {\overline{\lambda }}_{\mathrm{v},10}\left(t\right)=0,\\ \hfill {\overline{\lambda }}_{\mathrm{v},20}\left(t\right)=\left(C_1\sqrt{2}+C_2(\sqrt{2}t+3)\right)exp\left(\sqrt{2}t\right)-\left(C_3\sqrt{2}+C_4(\sqrt{2}t-3)\right)exp(-\sqrt{2}t),\\ \hfill {\overline{\lambda }}_{\mathrm{u},\mathrm{v},10}\left(t\right)=\left(C_1+C_2(t+2\sqrt{2})\right)exp\left(\sqrt{2}t\right)+\left(C_3+C_4(t-2\sqrt{2})\right)exp(-\sqrt{2}t),\\ \hfill {\overline{\lambda }}_{\mathrm{u},\mathrm{v},20}\left(t\right)=-{\overline{\lambda }}_{\mathrm{u},\mathrm{v},10}\left(t\right),\end{array}$$

(134)

where

$$\begin{array}{c}\hfill C_1=0.26323,C_2=-0.05581,C_3=2.56519,C_4=0.94419.\end{array}$$

Using Equations (38), (131), and (134), we obtain for $t\in [0,1]$

$$\begin{array}{c}\hfill {\overline{z}}_{2,0}\left(t\right)=-2\left(C_1+C_2(t+2\sqrt{2})\right)exp\left(\sqrt{2}t\right)-2\left(C_3+C_4(t-2\sqrt{2})\right)exp(-\sqrt{2}t),\end{array}$$

(135)

yielding

$$\begin{array}{c}\hfill {\overline{z}}_{2,0}\left(0\right)=0.\end{array}$$

(136)

Using Equations (60), (65), (131), and (136), we directly obtain

$$\begin{array}{c}\hfill z_{2,0}^0\left(\xi \right)=exp\left(-{\displaystyle \frac{\xi }{\sqrt{2}}}\right),{\lambda }_{\mathrm{u}2,0}^0\left(\xi \right)={\displaystyle \frac{1}{\sqrt{2}}}exp\left(-{\displaystyle \frac{\xi }{\sqrt{2}}}\right),\xi \ge 0,\\ \hfill z_{1,1}^0\left(\xi \right)\equiv 0,{\lambda }_{\mathrm{u}1,1}^0\left(\xi \right)\equiv 0,{\lambda }_{\mathrm{u},\mathrm{v},1}^0\left(\xi \right)\equiv \left(\begin{array}{c}0\\ 0\end{array}\right),\xi \ge 0,\\ \hfill {\lambda }_{\mathrm{v},1}^0\left(\xi \right)=\left(\begin{array}{c}0\\ \sqrt{2}exp\left(-{\displaystyle \frac{\xi }{\sqrt{2}}}\right)\end{array}\right),\xi \ge 0.\end{array}$$

(137)

Taking into account that $z_{1,1}^0\left(\xi \right)\equiv 0$ and $\xi \ge 0$, and solving the boundary-value Problems (74) and (71), we immediately have

$$\begin{array}{c}\hfill {\overline{z}}_{1,1}\left(t\right)\equiv 0,{\overline{\lambda }}_{\mathrm{u}1,1}\left(t\right)\equiv 0,{\overline{\lambda }}_{\mathrm{v},1}\left(t\right)\equiv \left(\begin{array}{c}0\\ 0\end{array}\right),{\overline{\lambda }}_{\mathrm{u},\mathrm{v},1}\left(t\right)\equiv \left(\begin{array}{c}0\\ 0\end{array}\right),t\in [0,1],\end{array}$$

(138)

which, along with (73), yields

$$\begin{array}{c}\hfill {\overline{z}}_{2,1}\left(t\right)\equiv 0,t\in [0,1].\end{array}$$

(139)

Furthermore, due to Equations (72), (131), (134), and (135),

$$\begin{array}{c}\hfill {\overline{\lambda }}_{\mathrm{u}2,1}\left(t\right)=-{\displaystyle \frac{d{\overline{z}}_{2,0}\left(t\right)}{dt}}-{\overline{\lambda }}_{\mathrm{v},20}\left(t\right),t\in [0,1].\end{array}$$

(140)

To complete the construction of the functions $z_2^1(t,\epsilon )$ and ${\lambda }_{\mathrm{u}2}^1(t,\epsilon )$ (see Equation (102)), we need to obtain $z_{2,1}^0\left(\xi \right)$, ${\lambda }_{\mathrm{u}2,1}^0\left(\xi \right)$, $\xi \ge 0$ and $z_{2,1}^f\left(\eta \right)$, ${\lambda }_{\mathrm{u}2,1}^f\left(\eta \right)$, $\eta \le 0$.

Due to the data of Example (131), the functions $f_x\left(\xi \right)$ and $f_y\left(\xi \right)$ (see Equation (85)) are identically zero. This circumstance, along with the equality ${\overline{z}}_{2,1}\left(0\right)=0$ and Equations (87)–(90), yields

$$\begin{array}{c}\hfill z_{2,1}^0\left(\xi \right)\equiv 0,{\lambda }_{\mathrm{u}2,1}^0\left(\xi \right)\equiv 0,\xi \ge 0.\end{array}$$

(141)

Furthermore, using Equations (100), (131), and (140), and with the fact that ${\overline{\lambda }}_{\mathrm{v},20}\left(1\right)=0$, we obtain

$$\begin{array}{c}\hfill z_{2,1}^f\left(\eta \right)=-\sqrt{2}{\displaystyle \frac{d{\overline{z}}_{2,0}\left(t_f\right)}{dt}}exp\left({\displaystyle \frac{\eta }{\sqrt{2}}}\right),{\lambda }_{\mathrm{u}2,1}^f\left(\eta \right)={\displaystyle \frac{d{\overline{z}}_{2,0}\left(t_f\right)}{dt}}exp\left({\displaystyle \frac{\eta }{\sqrt{2}}}\right),\eta \le 0.\end{array}$$

(142)

Thus, using the Equations (134), (135), and (137)–(142), the functions of Equation (102) become

$$\begin{array}{c}\hfill z_1^1(t,\epsilon )={\overline{z}}_{1,0}\left(t\right),t\in \left[0.1\right],\\ \hfill z_2^1(t,\epsilon )={\overline{z}}_{2,0}\left(t\right)+exp\left(-{\displaystyle \frac{t}{\epsilon \sqrt{2}}}\right)-\epsilon \sqrt{2}{\displaystyle \frac{d{\overline{z}}_{2,0}\left(t_f\right)}{dt}}exp\left({\displaystyle \frac{t-1}{\epsilon \sqrt{2}}}\right),t\in [0,1],\\ \hfill {\lambda }_{\mathrm{u}1}^1(t,\epsilon )={\overline{\lambda }}_{\mathrm{u}1,0}\left(t\right),t\in \left[0.1\right],\\ \hfill {\lambda }_{\mathrm{u}2}^1(t,\epsilon )={\displaystyle \frac{1}{\sqrt{2}}}exp\left(-{\displaystyle \frac{t}{\epsilon \sqrt{2}}}\right)\\ \hfill +\epsilon \left(-{\displaystyle \frac{d{\overline{z}}_{2,0}\left(t\right)}{dt}}-{\overline{\lambda }}_{\mathrm{v},20}\left(t\right)+{\displaystyle \frac{d{\overline{z}}_{2,0}\left(t_f\right)}{dt}}exp\left({\displaystyle \frac{t-1}{\epsilon \sqrt{2}}}\right)\right),t\in [0,1],\\ \hfill {\lambda }_{\mathrm{v}}^1(t,\epsilon )=\left(\begin{array}{c}\hfill {\lambda }_{\mathrm{v}1}^1(t,\epsilon )\hfill \\ \hfill {\lambda }_{\mathrm{v}2}^1(t,\epsilon )\hfill \end{array}\right)\\ \hfill =\left(\begin{array}{c}\hfill {\overline{\lambda }}_{\mathrm{v},10}\left(t\right)\hfill \\ \hfill {\overline{\lambda }}_{\mathrm{v},20}\left(t\right)+\epsilon \sqrt{2}exp\left(-{\displaystyle \frac{t}{\epsilon \sqrt{2}}}\right)\hfill \end{array}\right),t\in [0,1],\\ \hfill {\lambda }_{\mathrm{u},\mathrm{v}}^1(t,\epsilon )=\left(\begin{array}{c}\hfill {\lambda }_{\mathrm{u},\mathrm{v}1}^1(t,\epsilon )\hfill \\ \hfill {\lambda }_{\mathrm{u},\mathrm{v}2}^1(t,\epsilon )\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\overline{\lambda }}_{\mathrm{u},\mathrm{v},10}\left(t\right)\hfill \\ \hfill {\overline{\lambda }}_{\mathrm{u},\mathrm{v},20}\left(t\right)\hfill \end{array}\right),t\in [0,1].\end{array}$$

(143)

Consider the following values:

$$\begin{array}{c}\hfill \Delta z_1\left(\epsilon \right)\stackrel{▵}{=}\underset{t\in [0,1]}{max}|z_1(t,\epsilon )-z_1^1(t,\epsilon )|,\Delta z_2\left(\epsilon \right)\stackrel{▵}{=}\underset{t\in [0,1]}{max}|z_2(t,\epsilon )-z_2^1(t,\epsilon )|,\\ \hfill \Delta {\lambda }_{\mathrm{u}1}\left(\epsilon \right)\stackrel{▵}{=}\underset{t\in [0,1]}{max}\parallel {\lambda }_{\mathrm{u}1}(t,\epsilon )-{\lambda }_{\mathrm{u}1}^1(t,\epsilon )|,\Delta {\lambda }_{\mathrm{u}2}\left(\epsilon \right)\stackrel{▵}{=}\underset{t\in [0,1]}{max}\parallel {\lambda }_{\mathrm{u}2}(t,\epsilon )-{\lambda }_{\mathrm{u}2}^1(t,\epsilon )|,\\ \hfill \Delta {\lambda }_{\mathrm{v}2}\left(\epsilon \right)\stackrel{▵}{=}|{\lambda }_{\mathrm{v}2}(t,\epsilon )-{\lambda }_{\mathrm{v}2}^1(t,\epsilon )|,\Delta {\lambda }_{\mathrm{u},\mathrm{v}1}\left(\epsilon \right)\stackrel{▵}{=}|{\lambda }_{\mathrm{u},\mathrm{v}1}(t,\epsilon )-{\lambda }_{\mathrm{u},\mathrm{v}1}^1(t,\epsilon )|,\end{array}$$

(144)

where $z_1(t,\epsilon )$, $z_2(t,\epsilon )$, ${\lambda }_{\mathrm{u}1}(t,\epsilon )$, ${\lambda }_{\mathrm{u}2}(t,\epsilon )$, ${\lambda }_{\mathrm{v}2}(t,\epsilon )$, ${\lambda }_{\mathrm{u},\mathrm{v}1}(t,\epsilon )$ are the corresponding components of the solution to the boundary-value Problems (26) and (27) subject to Data (131).

Remark 12. 

Note that, in (144), the components ${\lambda }_{\mathrm{v}1}(t,\epsilon )$ and ${\lambda }_{\mathrm{u},\mathrm{v}2}(t,\epsilon )$ of the solution to Problems (26) and (27) are absent. Similarly, the functions ${\lambda }_{\mathrm{v}1}^1(t,\epsilon )$ and ${\lambda }_{\mathrm{u},\mathrm{v}2}^1(t,\epsilon )$ also are absent. This happens because ${\lambda }_{\mathrm{v}1}(t,\epsilon )\equiv 0$ and ${\lambda }_{\mathrm{v}1}^1(t,\epsilon )\equiv 0$, while ${\lambda }_{\mathrm{u},\mathrm{v}2}(t,\epsilon )=-{\lambda }_{\mathrm{u},\mathrm{v}1}(t,\epsilon )$ and ${\lambda }_{\mathrm{u},\mathrm{v}2}^1(t,\epsilon )=-{\lambda }_{\mathrm{u},\mathrm{v}1}^1(t,\epsilon )$.

In Figure 1, the graphs of the functions $z_1(t,\epsilon )$ and $z_1^1(t,\epsilon )$ are depicted for different values of $\epsilon >0$. Note that for $z_1^1(t,\epsilon )$, we have only one curve (the dashed one) because this function is independent of $\epsilon$ (see Equation (143)). It is seen from the figure that, for decreasing $\epsilon$, the graph of $z_1(t,\epsilon )$ approaches the graph of $z_1^1(t,\epsilon )$. In Figure 2, the graph of $\Delta z_1\left(\epsilon \right)$ (as a function of $\epsilon$) is depicted. Along with this graph, the graph of the function $2{\epsilon }^2$ is presented, showing that $\Delta z_1\left(\epsilon \right)$ is estimated by $2{\epsilon }^2$, i.e., $\Delta z_1\left(\epsilon \right)\le 2{\epsilon }^2$ for all $\epsilon \in (0,0.2]$. Similar results with respect to $z_2(t,\epsilon )$, $z_2^1(t,\epsilon )$, and $\Delta z_2\left(\epsilon \right)$ are shown in Figure 3 and Figure 4, the results with respect to ${\lambda }_{\mathrm{u}2}(t,\epsilon )$, ${\lambda }_{\mathrm{u}2}^1(t,\epsilon )$, $\Delta {\lambda }_{\mathrm{u}2}\left(\epsilon \right)$ are shown in Figure 5 and Figure 6, and the results with respect to ${\lambda }_{\mathrm{v}2}(t,\epsilon )$, ${\lambda }_{\mathrm{v}2}^1(t,\epsilon )$, $\Delta {\lambda }_{\mathrm{v}2}\left(\epsilon \right)$ are shown in Figure 7 and Figure 8.

Remark 13. 

Note that the functions ${\lambda }_{\mathrm{u}1}(t,\epsilon )$ and ${\lambda }_{\mathrm{u},\mathrm{v}}(t,\epsilon )$ do not appear in the optimal controls and in the optimal values of the cost functionals of the Stackelberg game (9)–(11) with Data (131). Therefore, in order not to overload the paper with additional figures, we do not present the figures with respect to ${\lambda }_{\mathrm{u}1}(t,\epsilon )$, ${\lambda }_{\mathrm{u}1}^1(t,\epsilon )$, $\Delta {\lambda }_{\mathrm{u}1}\left(\epsilon \right)$ and ${\lambda }_{\mathrm{u},\mathrm{v}1}(t,\epsilon )$, ${\lambda }_{\mathrm{u},\mathrm{v}1}^1(t,\epsilon )$, $\Delta {\lambda }_{\mathrm{u},\mathrm{v}1}\left(\epsilon \right)$. However, for the sake of the results’ completeness, in the table below, the values $\Delta {\lambda }_{\mathrm{u}1}\left(\epsilon \right)$ and $\Delta {\lambda }_{\mathrm{u},\mathrm{v}1}\left(\epsilon \right)$ are presented. The solution to the boundary-value Problems (26) and (27) subject to Data (131) are obtained by using the Matlab R2022b function bvp4c (boundary-value problem solver of the fourth order), which is based on a finite difference method. Another finite difference methods and their applications to approximate solution to boundary-value problems associated with various optimization problems of dynamic systems can be found in [58,71] and in the references therein.

In

Table 1. Values of $\Delta z_1\left(\epsilon \right)$, $\Delta z_2\left(\epsilon \right)$, $\Delta {\lambda }_{\mathrm{u}1}\left(\epsilon \right)$, $\Delta {\lambda }_{\mathrm{u}2}\left(\epsilon \right)$, $\Delta {\lambda }_{\mathrm{v}2}\left(\epsilon \right)$, and $\Delta {\lambda }_{\mathrm{u},\mathrm{v}2}\left(\epsilon \right)$.

$\mathit{\epsilon }$	$\mathbf{\Delta }{\mathit{z}}_1$	$\mathbf{\Delta }{\mathit{z}}_2$	$\mathbf{\Delta }{\mathit{\lambda }}_{\mathbf{u}1}$	$\mathbf{\Delta }{\mathit{\lambda }}_{\mathbf{u}2}$	$\mathbf{\Delta }{\mathit{\lambda }}_{\mathbf{v}2}$	$\mathbf{\Delta }{\mathit{\lambda }}_{\mathbf{u},\mathbf{v}2}$
$0.2$	$0.0662$	$0.0755$	$0.1058$	$0.1479$	$0.0164$	$0.1102$
$0.1$	$0.0175$	$0.0285$	$0.0305$	$0.0501$	$0.0061$	$0.0284$
$0.05$	$0.0047$	$0.0075$	$0.0082$	$0.0142$	$0.0017$	$0.0074$
$0.01$	$1.995·{10}^{-4}$	$4.597·{10}^{-4}$	$3.493·{10}^{-4}$	$6.385·{10}^{-4}$	$0.703·{10}^{-4}$	$3.062·{10}^{-4}$
$0.005$	$5.006·{10}^{-5}$	$1.233·{10}^{-4}$	$8.801·{10}^{-5}$	$1.621·{10}^{-4}$	$1.760·{10}^{-5}$	$7.656·{10}^{-5}$

, the values $\Delta z_1\left(\epsilon \right)$, $\Delta z_2\left(\epsilon \right)$, $\Delta {\lambda }_{\mathrm{u}1}\left(\epsilon \right)$, $\Delta {\lambda }_{\mathrm{u}2}\left(\epsilon \right)$, $\Delta {\lambda }_{\mathrm{v}2}\left(\epsilon \right)$ and $\Delta {\lambda }_{\mathrm{u},\mathrm{v}1}\left(\epsilon \right)$ are presented for various values of $\epsilon >0$. The results of the table show that these values are estimated by ${\epsilon }^2$ with the coefficients which do not exceed $6.5$.

6.2. Illustration of Theorem 1

Using Equations (19), (113), and (131), and that ${\lambda }_{\mathrm{v}1}(t,\epsilon )\equiv 0$, ${\lambda }_{\mathrm{v}1}^1(t,\epsilon )\equiv 0$, we can rewrite the left-hand sides of the inequalities in (112) as:

$$\begin{array}{c}\hfill |{\lambda }_{\mathrm{u}2}^1(t,\epsilon )-{\lambda }_{\mathrm{u}2}(t,\epsilon )|,|{\lambda }_{\mathrm{v}2}^1(t,\epsilon )-{\lambda }_{\mathrm{v}2}(t,\epsilon )|,t\in [0,1],\end{array}$$

which, along with Equation (144) and Figure 6 and Figure 8, yields the inequalities

$$\begin{array}{c}\hfill |\epsilon u^{*}(t,\epsilon )+{\lambda }_{\mathrm{u}2}^1(t,\epsilon )|\le 6.5{\epsilon }^2,t\in [0,1],\epsilon \in (0,0.2],\\ \hfill |v^{*}(t,\epsilon )+B_{\mathrm{v}}^T\left(t\right){\lambda }_{\mathrm{v}}^1(t,\epsilon )|\le 0.7{\epsilon }^2,t\in [0,1],\epsilon \in (0,0.2].\end{array}$$

6.3. Illustration of Theorem 2

To illustrate this theorem, first of all let us calculate the values ${\overline{J}}_{\mathrm{u}0}$ and ${\overline{J}}_{\mathrm{v}0}$. Using Equations (114), (131), (132), and (134), we obtain after a routine calculation of the integrals

$$\begin{array}{c}\hfill {\overline{J}}_{\mathrm{u},0}=1.6738,{\overline{J}}_{\mathrm{v},0}=0.2809.\end{array}$$

Consider the values

$$\begin{array}{c}\hfill \Delta J_{\mathrm{u},0}\left(\epsilon \right)\stackrel{▵}{=}\left|J_{\mathrm{u}}^{*}\left(\epsilon \right)-{\overline{J}}_{\mathrm{u},0}\right|,\Delta J_{v,0}\left(\epsilon \right)\stackrel{▵}{=}\left|J_{\mathrm{v}}^{*}\left(\epsilon \right)-{\overline{J}}_{\mathrm{v},0}\right|,\end{array}$$

where $J_{\mathrm{u}}^{*}\left(\epsilon \right)$ and $J_{\mathrm{v}}^{*}\left(\epsilon \right)$ are the optimal values of the cost functionals of the leader and the follower, respectively, in the Stackelberg game (9)–(11) with Data (131).

In Figure 9, the graph of $\Delta J_{\mathrm{u},0}\left(\epsilon \right)$ (as a function of $\epsilon$) is depicted. Along with this graph, the graph of the function $\epsilon$ is presented, showing that $\Delta J_{\mathrm{u},0}\left(\epsilon \right)$ is estimated by $\epsilon$, i.e., $\Delta J_{\mathrm{u},0}\left(\epsilon \right)\le \epsilon$ for all $\epsilon \in (0,0.2]$. A similar result with respect to $\Delta J_{\mathrm{v},0}\left(\epsilon \right)$ is shown in Figure 10.

In

Table 2. Values of $J_{\mathrm{u}}^{*}\left(\epsilon \right)$ and $J_{\mathrm{v}}^{*}\left(\epsilon \right)$.

$\mathit{\epsilon }$	${\mathit{J}}_{\mathbf{u}}^{*}\left(\mathit{\epsilon }\right)$	${\mathit{J}}_{\mathbf{v}}^{*}\left(\mathit{\epsilon }\right)$
$0.2$	$1.8622$	$0.3619$
$0.1$	$1.7432$	$0.3260$
$0.05$	$1.7009$	$0.3053$
$0.01$	$1.6788$	$0.2882$
$0.005$	$1.6778$	$0.2878$

, the values of $J_{\mathrm{u}}^{*}\left(\epsilon \right)$ and $J_{\mathrm{v}}^{*}\left(\epsilon \right)$ are presented for various values of $\epsilon >0$, while in

Table 3. Values of $\Delta J_{\mathrm{u},0}\left(\epsilon \right)$ and $\Delta J_{\mathrm{v},0}\left(\epsilon \right)$.

$\mathit{\epsilon }$	$\mathbf{\Delta }{\mathit{J}}_{\mathbf{u},0}\left(\mathit{\epsilon }\right)$	$\mathbf{\Delta }{\mathit{J}}_{\mathbf{v},0}\left(\mathit{\epsilon }\right)$
$0.2$	$0.1884$	$0.0810$
$0.1$	$0.0694$	$0.0452$
$0.05$	$0.0271$	$0.0244$
$0.01$	$0.005$	$0.0073$
$0.005$	$0.004$	$0.0069$

, the values of $\Delta J_{\mathrm{u},0}\left(\epsilon \right)$ and $\Delta J_{\mathrm{v},0}\left(\epsilon \right)$ are presented for the same values of $\epsilon >0$. It is seen that both $\Delta J_{\mathrm{u},0}\left(\epsilon \right)$ and $\Delta J_{\mathrm{v},0}\left(\epsilon \right)$ are estimated by $\epsilon$.

6.4. Illustration of Theorem 3

Using Remark 11 and Equations (117), (131), (137), and (140), we design the asymptotically suboptimal Stackelberg solution $\left(\tilde{u}(t,\epsilon ),\tilde{v}\left(t\right)\right)$ in the game (9)–(11) with Data (131). The components of this solution are

$$\begin{array}{c}\hfill \tilde{u}(t,\epsilon )=-{\displaystyle \frac{1}{\epsilon \sqrt{2}}}exp\left(-{\displaystyle \frac{t}{\epsilon \sqrt{2}}}\right)+{\displaystyle \frac{d{\overline{z}}_{2,0}\left(t\right)}{dt}}+{\overline{\lambda }}_{\mathrm{v},20}\left(t\right),t\in [0,1],\\ \hfill \tilde{v}\left(t\right)={\overline{\lambda }}_{\mathrm{v},20}\left(t\right),t\in [0,1],\end{array}$$

where ${\overline{\lambda }}_{\mathrm{v},20}\left(t\right)$ is given in (134), and ${\overline{z}}_{2,0}\left(t\right)$ is given by (135).

In

Table 4. Values of ${\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)$ and ${\tilde{J}}_{\mathrm{v}}\left(\epsilon \right)$.

$\mathit{\epsilon }$	${\tilde{\mathit{J}}}_{\mathbf{u}}\left(\mathit{\epsilon }\right)$	${\tilde{\mathit{J}}}_{\mathbf{v}}\left(\mathit{\epsilon }\right)$
$0.2$	$1.8085$	$0.3992$
$0.1$	$1.7252$	$0.3325$
$0.05$	$1.6958$	$0.3068$
$0.01$	$1.6786$	$0.2883$
$0.005$	$1.6778$	$0.2878$

, the values ${\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)$ and ${\tilde{J}}_{\mathrm{v}}\left(\epsilon \right)$ of the functionals $J_{\mathrm{u}}(u,v)$ and $J_{\mathrm{v}}(u,v)$, respectively, generated by the asymptotically suboptimal Stackelberg solution $\left(\tilde{u}(t,\epsilon ),\tilde{v}\left(t\right)\right)$ in the game (9)–(11) and (131), are presented for various values of $\epsilon >0$.

Let us compare the values ${\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)$ and ${\tilde{J}}_{\mathrm{v}}\left(\epsilon \right)$ with the optimal values $J_{\mathrm{u}}^{*}\left(\epsilon \right)$ and $J_{\mathrm{v}}^{*}\left(\epsilon \right)$, respectively. For this purpose, we consider the values

$$\begin{array}{c}\hfill \Delta {\tilde{J}}_{\mathrm{u}}^{*}\left(\epsilon \right)\stackrel{▵}{=}\left|J_{\mathrm{u}}^{*}\left(\epsilon \right)-{\tilde{J}}_{\mathrm{u}}\left(\epsilon \right)\right|,\Delta {\tilde{J}}_{\mathrm{v}}^{*}\left(\epsilon \right)\stackrel{▵}{=}\left|J_{\mathrm{v}}^{*}\left(\epsilon \right)-{\tilde{J}}_{\mathrm{v}}\left(\epsilon \right)\right|.\end{array}$$

In Figure 11, the graph of $\Delta {\tilde{J}}_{\mathrm{u}}^{*}\left(\epsilon \right)$ (as a function of $\epsilon$) is depicted. Along with this graph, the graph of the function $0.26\epsilon$ is presented, showing that $\Delta {\tilde{J}}_{\mathrm{u}}^{*}\left(\epsilon \right)$ is estimated by $0.26\epsilon$, i.e., $\Delta {\tilde{J}}_{\mathrm{u}}^{*}\left(\epsilon \right)\le 0.26\epsilon$ for all $\epsilon \in (0,0.2]$. Similar result with respect to $\Delta {\tilde{J}}_{\mathrm{v}}^{*}\left(\epsilon \right)$ is shown in Figure 12.

In

Table 5. Values of $\Delta {\tilde{J}}_{\mathrm{u}}^{*}\left(\epsilon \right)$ and $\Delta {\tilde{J}}_{\mathrm{v}}^{*}\left(\epsilon \right)$.

$\mathit{\epsilon }$	$\mathbf{\Delta }{\tilde{\mathit{J}}}_{\mathbf{u}}^{*}\left(\mathit{\epsilon }\right)$	$\mathbf{\Delta }{\tilde{\mathit{J}}}_{\mathbf{v}}^{*}\left(\mathit{\epsilon }\right)$
$0.2$	$0.0537$	$0.0374$
$0.1$	$0.0180$	$0.0065$
$0.05$	$0.0051$	$0.0016$
$0.01$	$2.302·{10}^{-4}$	$0.679·{10}^{-4}$
$0.005$	$5.844·{10}^{-5}$	$1.771·{10}^{-5}$

, the values of $\Delta {\tilde{J}}_{\mathrm{u}}^{*}\left(\epsilon \right)$ and $\Delta {\tilde{J}}_{\mathrm{v}}^{*}\left(\epsilon \right)$ are presented for the same values of $\epsilon >0$ as in Table 4. It is seen that $\Delta {\tilde{J}}_{\mathrm{u}}^{*}\left(\epsilon \right)$ and $\Delta {\tilde{J}}_{\mathrm{v}}^{*}\left(\epsilon \right)$ are estimated by $\epsilon$ with the coefficients which do not exceed $0.26$ and $0.19$, respectively.

7. Conclusions

In this paper, a two-player finite horizon linear–quadratic Stackelberg differential game was studied in the case where the control cost of the leader in the cost functionals of both players is much smaller than the state cost and the cost of the follower’s control. Due to this feature of the leader’s control cost, the considered game is a cheap control game. For this game, an open-loop solution was sought. By the proper change of the state variable, the initially formulated game was transformed equivalently to a simpler cheap control Stackelberg game. The dynamics equation of the latter consists of two modes. The first mode is controlled directly only by the follower, while the second mode is controlled directly by both players. Moreover, the dimension of the second mode equals to the dimension of the leader’s control and the gain matrix for the leader’s control in this mode is the identity matrix. The transformed game also is a cheap control game. Based on the Stackelberg game’s solvability conditions, the solution to the transformed game was converted to the solution to the linear singularly perturbed boundary-value problem which dimension is four times larger than the dimension of the game’s dynamics. This boundary-value problem is of the conditionally stable type. Based on the Boundary Functions Method, the first-order asymptotic solution to this singularly perturbed boundary-value problem was derived. Using this asymptotic solution, the asymptotic expansions of the optimal open-loop controls of the leader and the follower, as well as the asymptotic expansions of the optimal values of their cost functionals, were obtained. Based on these results, asymptotically suboptimal open-loop controls of the players were designed.

It should be noted that the cost functionals of the leader and the follower in the considered game do not contain terminal parts. The case of the presence of these parts requires to develop another approach to the asymptotic analysis and solution to the boundary-value problem appearing in the Stackelberg game’s open-loop solvability conditions. In addition, it should be noted that the assumption on the positive definiteness of the matrix-coefficient for the fast state variable in the leader’s cost functional is essential for the asymptotic analysis and solution to the considered cheap control Stackelberg game. A weaker assumption on this matrix-coefficient (namely, its positive semi-definiteness) requires to develop another method of the asymptotic analysis and solution to the corresponding boundary-value problem.

Completing this section, we would like to mention several issues connected with the topic of the paper, which are interesting ones for future investigations. These issues are the following: (a) asymptotic analysis of the open-loop solution in the Stackelberg linear–quadratic differential game with cheap control of a follower; (b) asymptotic analysis of the feedback solution in the Stackelberg linear–quadratic differential game with cheap control of a leader/follower; (c) the open-loop solution and the feedback solution of a singular Stackelberg linear–quadratic differential game; and (d) real-life applications of the cheap control/singular Stackelberg linear–quadratic differential games.