Existence and uniqueness of mild solutions and evolution operators for a class of non-autonomous conformable fractional semi-linear systems and Their Exact Null Controllability

TL;DR

This paper establishes the existence, uniqueness, and controllability of solutions for non-autonomous conformable fractional semi-linear systems using fixed-point theorems and evolution operators, with practical examples.

Contribution

It introduces new sufficient conditions for the existence and uniqueness of mild solutions and analyzes their exact null controllability for fractional systems.

Findings

Existence and uniqueness of evolution operators for fractional systems

New criteria for mild solution existence and uniqueness

Demonstration of exact null controllability with examples

Abstract

This paper investigates the controllability of systems governed by conformable fractional order derivatives. It first establishes the existence and uniqueness of evolution operators for non-autonomous fractional-order homogeneous systems, using a suitable initial time defined as the intersection of two specific time intervals. Using the theory of linear evolution operators, Schauder's fixed-point theorem, and the Banach contraction principle, the study derives a new set of sufficient conditions for the existence and uniqueness of a mild solution to non-autonomous conformable fractional semi-linear systems. Additionally, the paper examines the exact null controllability of abstract systems based on the mild solution. We provide a comprehensive example to demonstrate the applicability of the established theoretical results.