Approximate Controllability of Nonlocal Stochastic Integrodifferential System in Hilbert Spaces

TL;DR

This work addresses approximate controllability for a class of nonlinear stochastic integrodifferential equations in Hilbert spaces with nonlocal initial conditions driven by a cylindrical Wiener process. By dropping compactness and Lipschitz requirements on the nonlocal term and employing Grimmer's resolvent operator $ ext{Re}( au)$ together with Schauder fixed-point arguments, the authors first establish the existence of mild solutions and then derive approximate controllability via approximation and a diagonal argument. The main contributions include relaxing standard nonlocal-term assumptions, clarifying the role of the resolvent and the controllability operator, and validating the theory with a concrete example. The results extend approximate controllability to broader nonlocal stochastic systems and suggest directions toward trajectory controllability under weakened compactness assumptions.

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.

Theorems & Definitions (22)

• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Definition 2.5
• Theorem 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 2.9
• Definition 2.10