Necessary and sufficient conditions for L₁-strong- weak lower semicontinuity of integral functionals

In this paper, we introduce and study an optimal control problem for a new class of fractional differential quasi-hemivariational inequalities with nonlocal initial conditions and Clarke subdifferentials. The main tools in our study are the theory of fractional calculus, the fixed point argument for condensing multimaps and Balder’s theorem. We prove the solvability of abstract nonlocal fractional differential quasi-hemivariational inequalities and then show the existence of solutions to the associated optimal control problems.

In this paper, we study the existence and optimal control results for second-order Sobolev-type delay systems with Clarke’s subdifferential type. Initially, the existence of mild solution is established for the proposed second-order delay differential system with the novel ideas of Clarke’s subdifferential. The fixed point theorem of condensing multivalued maps, the strongly continuous cosine family, and the properties of Clarke’s subdifferential are used to establish the existence of mild solution. Moreover, the existence of optimal control pair that is governed by the presented system is verified through Balder’s theorem. Finally, an example is provided to illustrate the main results.

The main concern of the manuscript deals with the optimal control problem of conformable fractional neutral stochastic integrodifferential systems with infinite delay. This study is motivated by SAR and RAR systems advantage of providing control over many factors such as power, frequency, phase, polarization, incidence angle, spatial resolution and swath width, all of which are important when designing and operating a radar system. Initially, we investigate the existence of mild solutions for the conformable fractional stochastic integrodifferential equations with infinite delay using stochastic analysis techniques and Banach fixed point theorem. In the later part we establish the existence of mild solutions of the conformable fractional neutral stochastic integrodifferential system with infinite time delay. Furthermore, the existence of optimal control of the corresponding Lagrange optimal control problem is investigated. An example is provided to illustrate the applications of the obtained results. We explain the limitation we have with the existence software, and developed a numerical scheme to justify the theory.

This paper demonstrates the existence, uniqueness, and optimal control of the time-fractional stochastic Burgers’ equation with a deviated argument governed by mixed fractional Brownian motion. The nonlinear deterministic controlled time-fractional Burgers’ equation is subjected to stochastic perturbations to model the memory effects and disturbances that arise in the turbulent flow of liquids. The solvability of the presented system is investigated using stochastic analysis, fractional calculus, and the Nussbaum fixed point theorem. Balder’s theorem is also used to prove the existence of optimal control. At last, an example is presented to demonstrate the developed theory.