Approximate Controllability of Nonlocal Stochastic Integrodifferential System in Hilbert Spaces
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.