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Axioms
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Advances in Mathematical Methods in Optimal Control and Applications
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Advances in Mathematical Methods in Optimal Control and Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 28 February 2025 | Viewed by 5269

Special Issue Editor

Dr. Cristiana J. Silva

E-Mail Website
Guest Editor
Department of Mathematics, ISTA, Instituto Universitário de Lisboa (ISCTE-IUL), Av. das Forças Armadas, 1649-026 Lisbon, Portugal
Interests: optimal control theory; mathematical modeling; optimization methods; applications to biology and epidemiology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Optimal control theory is a mathematical research area of great development with a wide range of applications across various fields. Optimal control problems aim to find the optimal control strategy that minimizes an objective (also called cost) function, while satisfying control system dynamics and a set of constraints (e.g., boundary, state, and control). The mathematical framework of the control system, cost function, and constraints can be very different, and for each of them, distinct mathematical methods must be applied.

This Special Issue focuses on recent advances in mathematical methods for the optimal control of different types of control systems, state and control constraints, and boundary conditions. We welcome papers on new mathematical methods for optimal control problems with applications in biology, ecology, population dynamics, epidemiology, economy, etc. 

Prof. Dr. Cristiana J. Silva
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • optimal control problems
  • optimality conditions
  • control system
  • mathematical methods
  • applications

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Published Papers (7 papers)

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Research

38 pages, 2057 KiB  
Article
One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution
by Valery Y. Glizer and Vladimir Turetsky
Axioms 2024, 13(11), 801; https://doi.org/10.3390/axioms13110801 - 18 Nov 2024
Viewed by 258
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)
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14 pages, 1040 KiB  
Article
Optimal Control of Microcephaly Under Vertical Transmission of Zika
by Dilara Yapışkan, Cristiana J. Silva and Delfim F. M. Torres
Axioms 2024, 13(11), 772; https://doi.org/10.3390/axioms13110772 - 6 Nov 2024
Viewed by 425
Abstract
The Zika virus, known for its potential to induce neurological conditions such as microcephaly when transmitted vertically from infected mothers to infants, has sparked widespread concerns globally. Motivated by this, we propose an optimal control problem for the prevention of vertical Zika transmission. [...] Read more.
The Zika virus, known for its potential to induce neurological conditions such as microcephaly when transmitted vertically from infected mothers to infants, has sparked widespread concerns globally. Motivated by this, we propose an optimal control problem for the prevention of vertical Zika transmission. The novelty of this study lies in its consideration of time-dependent control functions, namely, insecticide spraying and personal protective measures taken to safeguard pregnant women from infected mosquitoes. New results provide a way to minimize the number of infected pregnant women through the implementation of control strategies while simultaneously reducing both the associated costs of control measures and the mosquito population, resulting in a decline in microcephaly cases. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)
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15 pages, 326 KiB  
Article
Non-Fragile Sampled Control Design for an Interconnected Large-Scale System via Wirtinger Inequality
by Volodymyr Lynnyk and Branislav Rehák
Axioms 2024, 13(10), 702; https://doi.org/10.3390/axioms13100702 - 10 Oct 2024
Viewed by 447
Abstract
A control design for a linear large-scale interconnected system composed of identical subsystems is presented in this paper. The control signal of all subsystems is sampled. For different subsystems, the sampling times are not identical. Nonetheless, it is assumed that a bound exists [...] Read more.
A control design for a linear large-scale interconnected system composed of identical subsystems is presented in this paper. The control signal of all subsystems is sampled. For different subsystems, the sampling times are not identical. Nonetheless, it is assumed that a bound exists for the maximal sampling time. The control algorithm is designed using the Wirtinger inequality, and the non-fragile control law is proposed. The size of the linear matrix inequalities to be solved by the proposed control algorithm is independent of the number of subsystems composing the overall system. Hence, the algorithm is computationally effective. The results are illustrated by two examples. The first example graphically illustrates the function of the proposed algorithm while the second one compares with a method for stabilizing a large-scale system obtained earlier, thus illustrating the improved capabilities of the presented algorithm. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)
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13 pages, 1975 KiB  
Article
A Second-Order Numerical Method for a Class of Optimal Control Problems
by Kamil Aida-zade, Alexander Handzel and Efthimios Providas
Axioms 2024, 13(10), 679; https://doi.org/10.3390/axioms13100679 - 1 Oct 2024
Viewed by 478
Abstract
The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is [...] Read more.
The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)
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32 pages, 4152 KiB  
Article
Enhanced Kepler Optimization Method for Nonlinear Multi-Dimensional Optimal Power Flow
by Mohammed H. Alqahtani, Sulaiman Z. Almutairi, Abdullah M. Shaheen and Ahmed R. Ginidi
Axioms 2024, 13(7), 419; https://doi.org/10.3390/axioms13070419 - 21 Jun 2024
Cited by 2 | Viewed by 723
Abstract
Multi-Dimensional Optimal Power Flow (MDOPF) is a fundamental task in power systems engineering aimed at optimizing the operation of electrical networks while considering various constraints such as power generation, transmission, and distribution. The mathematical model of MDOPF involves formulating it as a non-linear, [...] Read more.
Multi-Dimensional Optimal Power Flow (MDOPF) is a fundamental task in power systems engineering aimed at optimizing the operation of electrical networks while considering various constraints such as power generation, transmission, and distribution. The mathematical model of MDOPF involves formulating it as a non-linear, non-convex optimization problem aimed at minimizing specific objective functions while adhering to equality and inequality constraints. The objective function typically includes terms representing the Fuel Cost (FC), Entire Network Losses (ENL), and Entire Emissions (EE), while the constraints encompass power balance equations, generator operating limits, and network constraints, such as line flow limits and voltage limits. This paper presents an innovative Improved Kepler Optimization Technique (IKOT) for solving MDOPF problems. The IKOT builds upon the traditional KOT and incorporates enhanced local escaping mechanisms to overcome local optima traps and improve convergence speed. The mathematical model of the IKOT algorithm involves defining a population of candidate solutions (individuals) represented as vectors in a high-dimensional search space. Each individual corresponds to a potential solution to the MDOPF problem, and the algorithm iteratively refines these solutions to converge towards the optimal solution. The key innovation of the IKOT lies in its enhanced local escaping mechanisms, which enable it to explore the search space more effectively and avoid premature convergence to suboptimal solutions. Experimental results on standard IEEE test systems demonstrate the effectiveness of the proposed IKOT in solving MDOPF problems. The proposed IKOT obtained the FC, EE, and ENL of USD 41,666.963/h, 1.039 Ton/h, and 9.087 MW, respectively, in comparison with the KOT, which achieved USD 41,677.349/h, 1.048 Ton/h, 11.277 MW, respectively. In comparison to the base scenario, the IKOT achieved a reduction percentage of 18.85%, 58.89%, and 64.13%, respectively, for the three scenarios. The IKOT consistently outperformed the original KOT and other state-of-the-art metaheuristic optimization algorithms in terms of solution quality, convergence speed, and robustness. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)
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17 pages, 2691 KiB  
Article
Combined Observer-Based State Feedback and Optimized P/PI Control for a Robust Operation of Quadrotors
by Oussama Benzinane and Andreas Rauh
Axioms 2024, 13(5), 285; https://doi.org/10.3390/axioms13050285 - 23 Apr 2024
Viewed by 948
Abstract
This paper deals with a discrete-time observer-based state feedback control design by taking into consideration bounded parameter uncertainty, actuator faults, and stochastic noise in an inner control loop which is extended in a cascaded manner by outer PI- and P-control loops for velocity [...] Read more.
This paper deals with a discrete-time observer-based state feedback control design by taking into consideration bounded parameter uncertainty, actuator faults, and stochastic noise in an inner control loop which is extended in a cascaded manner by outer PI- and P-control loops for velocity and position regulation. The aim of the corresponding subdivision of the quadrotor model is the treatment of the control design in a systematic manner. In the inner loop, linear matrix inequality techniques are employed for the placement of poles into a desired area within the complex z-plane. A robustification of the design towards noise is achieved by optimizing both control and observer gains simultaneously guaranteeing stability in a predefined bounded state domain. This procedure helps to reduce the sensitivity of the inner control loop towards changes induced by the outer one. Finally, a model-based optimization process is employed to tune the parameters of the outer P/PI controllers. To allow for the validation of accurate trajectory tracking, a comparison of the novel approach with the use of a standard extended Kalman filter-based linear-quadratic regulator synthesis is presented to demonstrate the superiority of the new design. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)
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16 pages, 830 KiB  
Article
Robustness Analysis for Sundry Disturbed Open Loop Dynamics Using Robust Right Coprime Factorization
by Yuanhong Xu and Mingcong Deng
Axioms 2024, 13(2), 116; https://doi.org/10.3390/axioms13020116 - 9 Feb 2024
Viewed by 1225
Abstract
In this paper, the robustness of a system with sundry disturbed open loop dynamics is investigated by employing robust right coprime factorization (RRCF). These sundry disturbed open loop dynamics are present not only in the feed forward path, but also within the feedback [...] Read more.
In this paper, the robustness of a system with sundry disturbed open loop dynamics is investigated by employing robust right coprime factorization (RRCF). These sundry disturbed open loop dynamics are present not only in the feed forward path, but also within the feedback loop. In such a control framework, the nominal plant is firstly right coprime factorized and a feed forward and a feedback controllers are designed based on Bezout identity to ensure the overall stability. Subsequently, considering the sundry disturbed open loop dynamics, a new condition formulated as a disturbed Bezout identity is put forward to achieve the closed loop stability of the system, even in the presence of disturbances existing in sundry open loops, where in the feedback loop a disturbed identity operator is defined. This approach guarantees the system robustness if a specific inequality condition is satisfied. And, it should be noted that the proposed approach is applicable to both linear and nonlinear systems with sundry disturbed open loop dynamics. Simulations demonstrate the effectiveness of our methodology. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)
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