Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations

TL;DR

This paper investigates the relaxation of control problems governed by Sobolev-type nonlinear fractional differential equations with nonlocal conditions, establishing the existence of optimal solutions and their approximation by minimizing sequences.

Contribution

It introduces the relaxation framework for complex fractional control systems with nonconvex constraints and proves the existence of optimal solutions under certain conditions.

Findings

Optimal solutions exist for the relaxation of the control problems.

Optimal solutions are limits of minimizing sequences in suitable topologies.

The results apply to systems with nonlocal control conditions and nonconvex integrands.

Abstract

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multi-controls, and the functional in suitable topologies.