Approximate Controllability of Fractional Evolution Equations with Nonlocal Conditions via Operator Theory

TL;DR

This paper studies the approximate controllability of fractional evolution equations with nonlocal conditions in Hilbert spaces, providing new theoretical conditions and a constructive control approach using operator theory.

Contribution

It introduces a novel method for establishing approximate controllability of fractional evolution equations with nonlocal conditions via operator theory and Green's functions.

Findings

Established sufficient conditions for approximate controllability.

Developed a control function based on the Gramian controllability operator.

Provided an illustrative example demonstrating the theoretical results.

Abstract

This paper investigates the existence and uniqueness of mild solutions, as well as the approximate controllability, of a class of fractional evolution equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for approximate controllability are established through a novel approach to the approximate solvability of semilinear operator equations. The methodology utilizes Green's function and constructs a control function based on the Gramian controllability operator. The analysis is based on Schauder's fixed point theorem and the theory of fractional order solution operators and resolvent operators. To demonstrate the feasibility of the proposed theoretical results, an illustrative example is provided.