Approximate Controllability of Nonlocal Stochastic Integrodifferential System in Hilbert Spaces

TL;DR

This paper establishes the approximate controllability of a class of nonlocal stochastic integrodifferential systems in Hilbert spaces by leveraging resolvent operator properties, fixed point theorems, and approximation techniques.

Contribution

It introduces a novel approach that does not rely on compactness or Lipschitz conditions for nonlocal terms, using resolvent operator theory and fixed point methods.

Findings

Proves controllability of nonlinear stochastic systems without compactness assumptions

Employs Schauder's fixed point theorem and resolvent operator theory

Provides an example validating the theoretical results

Abstract

This project investigates the approximate controllability of a class of stochastic integrodifferential equations in Hilbert space with non-local beginning conditions. In a departure from the conventional concerns expressed in the literature, we will not consider compactness or the Lipschitz criteria concerning the nonlocal term. We use the fact that the resolvent operator is compact. We first prove the controllability of the nonlinear system using Schauder's fixed point theorem, a method known for its robustness; as well, we also use Grimmer's resolvent operator theory. Subsequently, we employ the reliable approximation methods and the powerful diagonal argument to determine the approximate controllability of the stochastic system. To conclude, we present an example that validates our theoretical statement.