Solvability and Optimal Controls of Impulsive Stochastic Evolution Equations in Hilbert Spaces

TL;DR

This paper studies the existence, uniqueness, and optimal control of impulsive stochastic differential equations in Hilbert spaces, providing theoretical results and a practical example to demonstrate their applicability.

Contribution

It establishes the solvability and optimal control conditions for impulsive stochastic equations in infinite-dimensional spaces, extending existing theory.

Findings

Proved existence and uniqueness of mild solutions

Derived necessary optimal control conditions

Provided a practical example illustrating the results

Abstract

This paper investigates the solvability and optimal control of a class of impulsive stochastic differential equations (SDEs) within a Hilbert space setting. First, we establish the existence and uniqueness of mild solutions for the proposed impulsive stochastic system, leveraging fixed-point theorems and appropriate analytical techniques. Next, we identify and derive the necessary conditions for the existence of optimal control pairs, ensuring the feasibility and effectiveness of the control solutions. Finally, to validate and demonstrate the practical applicability of our theoretical findings, we provide a detailed example showcasing the utility of the results in real-world scenarios.