Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay
Abstract
:1. Introduction
Motivation and Novelties
2. Preliminaries
- Let , is a continuous function on and , then, for every , the following hold.
- ;
- There ∃ a positive constant , .
- There ∃ two functions independent of U with continuous and bounded and locally bounded where:
- For function U in , is a D-valued continuous function on .
- The space D is complete.
- .
- is strongly continuous in χ on for each fixed .
- .
- (i)
- , this map measurable ∀ and for all .
- (ii)
- is measurable ∀ and for all .
- (iii)
- is measurable ∀ , and almost .
- (a)
- is compact.
- (b)
- .
- (c)
- .
- (d)
- .
- (e)
- .
- (f)
- .
3. Results of Controllability for the Steady Delay Case
- (H1)
- is compact for ,
- (H2)
- The function is random Caratheodory.
- (H3)
- There ∃ functions and for each is continuous nondecreasing and integrable with:
- (H4)
- There ∃ a random function where:
- (H5)
- The linear given by
- (H6)
- for each is continuous and and are measurable.
- (a)
- I maps bounded sets into equicontinuous sets in .Assume that with are a bounded set, as in Claim 2, and . Now,
- (b)
- Assume that is, fixed and : by assumption ; since is compact, the set
4. Results for State-Dependent Delay Case Controllability
- is compact for in .
- The function is random Caratheodory.
- There ∃ a function and , such that is a continuous nondecreasing function and integrable with:
- There ∃ a random function with for each such that for any bounded .
- There ∃ a random function where:
- The linear LO defined by:
- For each is continuous and, for each , is measurable, and, for each , is measurable.
- (a)
- In , I transforms bounded sets into equicontinuous sets.Let with be a bounded set as in Claim 2, and . Then,
- (b)
- Suppose that is a subset of where . is bounded and equicontinuous, and function is continuous on . Via , and by considering the characteristics of the measure , we have :
5. Applications
6. Example
- (i)
- is an orthonormal basis of ,
- (ii)
- If , then ,
- (iii)
- For , and the associated cosine family isConsequently, is compact for all and
- (iv)
- Let the group of translation be denoted by :
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Abuasbeh, K.; Shafqat, R.; Alsinai, A.; Awadalla, M. Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay. Symmetry 2023, 15, 290. https://doi.org/10.3390/sym15020290
Abuasbeh K, Shafqat R, Alsinai A, Awadalla M. Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay. Symmetry. 2023; 15(2):290. https://doi.org/10.3390/sym15020290
Chicago/Turabian StyleAbuasbeh, Kinda, Ramsha Shafqat, Ammar Alsinai, and Muath Awadalla. 2023. "Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay" Symmetry 15, no. 2: 290. https://doi.org/10.3390/sym15020290
APA StyleAbuasbeh, K., Shafqat, R., Alsinai, A., & Awadalla, M. (2023). Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay. Symmetry, 15(2), 290. https://doi.org/10.3390/sym15020290