Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions

TL;DR

This paper establishes existence and uniqueness results for Sobolev type fractional dynamic equations with nonlocal conditions and explores an optimal control problem involving multi-integral solutions.

Contribution

It introduces new existence and uniqueness results for Sobolev fractional equations with nonlocal conditions and addresses an optimal control problem with multi-integral solutions.

Findings

Proved existence and uniqueness of mild solutions.

Established the existence of solutions for an optimal control problem.

Provided an illustrative example of the theoretical results.

Abstract

We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given.