Optimal control problems for a neutral integro-differential system with infinite delay

Abstract

<p style='text-indent:20px;'>This work devotes to the study on problems of optimal control and time optimal control for a neutral integro-differential evolution system with infinite delay. The main technique is the theory of resolvent operators for linear neutral integro-differential evolution systems constructed recently in literature. We first establish the existence and uniqueness of mild solutions and discuss the compactness of the solution operator for the considered control system. Then, we investigate the existence of optimal controls for the both cases of bounded and unbounded admissible control sets under some assumptions. Meanwhile, the existence of time optimal control to a target set is also considered and obtained by limit arguments. An example is given at last to illustrate the applications of the obtained results.</p>

Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order r∈(1,2)

“…In practice, integrodifferential systems are always used to describe a model that has hereditary features, for more details, refer to articles. 10,11,19,20,24,28,32,[39][40][41][42] Currently, the authors of References 43-46 demonstrated fractional differential evolution systems of order 1 < 𝛼 < 2 using various fractional derivatives, sectorial operators of type (M, 𝜃, 𝛼, 𝜇), measures of noncompactness, nonlocal conditions, and various fixed point techniques.…”

“…A general formulation and solution scheme for fractional optimal control problems are discussed in Reference 18. The authors of References 19‐31 used impulsive systems, Clarke subdifferential type, neutral integrodifferential systems, Hilfer fractional derivative, stochastic systems, Riemann–Liouville derivative, and various fixed point theorems to develop optimal control results for functional differential systems with or without delay. The existence and stability of solutions for an optimal control problems derived from an integrodifferential equations with a compact control set in the space studied in Reference 32.…”

“…Furthermore, integrodifferential systems are used in a variety of scientific fields where an aftereffect or delay must be considered, such as biology, control theory, ecology, and medicine. In practice, integrodifferential systems are always used to describe a model that has hereditary features, for more details, refer to articles 10,11,19,20,24,28,32,39‐42 …”

Optimal control results for Sobolev‐type fractional mixed Volterra–Fredholm type integrodifferential equations of order 1 < r < 2 with sectorial operators

“…In Reference 34, the authors present a general formulation and solution strategy for fractional optimal control issues. The authors of References 7, 18, 21, and 35–40 used impulsive systems, Clarke subdifferential type, neutral integrodifferential systems, Hilfer fractional derivative, stochastic systems, Riemann–Liouville derivative, and various fixed point theorems to develop optimal control results for functional differential systems with or without delay. The existence and stability of solutions for an optimal control problems derived from an integrodifferential equations with a compact control set in $$$ {L}^1\left(\left[0,b\right];X\right) $$$ studied in Reference 41.…”

Minimum Energy Problem in the Sense of Caputo for Fractional Neutral Evolution Systems in Banach Spaces

“…with A 2 and A 3 n × n matrices whose elements belong to L 2 (−1, 0); B is a constant n × r matrix; and the control u is an L 2 -function [1]. Nowadays, many researchers have investigated neutral differential equations in Banach spaces [2][3][4]. This interest is explained by the fact that neutral-argument differential equations have interesting applications in real-life problems: they appear, e.g., while modeling networks containing lossless transmission lines or in super-computers.…”