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

Javad A. Asadzade, Nazim I. Mahmudov

TL;DR

This work addresses solvability and optimal control of impulsive stochastic evolution equations in a Hilbert space. The dynamics include a generator $A$ with a $C_0$-semigroup $T(t)$, stochastic forcing via a cylindrical Brownian motion, and impulsive jumps at times $t_k$; the authors establish existence and uniqueness of mild solutions and derive optimal control conditions. The main methods are Krasnoselskii fixed-point theory for mild solutions and Balder’s theorem to prove the existence of optimal controls under suitable Lipschitz, growth, and convexity assumptions. A detailed application confirms the theory by constructing an explicit problem with an optimal pair guaranteed by the derived 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.

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

TL;DR

This work addresses solvability and optimal control of impulsive stochastic evolution equations in a Hilbert space. The dynamics include a generator with a -semigroup , stochastic forcing via a cylindrical Brownian motion, and impulsive jumps at times ; the authors establish existence and uniqueness of mild solutions and derive optimal control conditions. The main methods are Krasnoselskii fixed-point theory for mild solutions and Balder’s theorem to prove the existence of optimal controls under suitable Lipschitz, growth, and convexity assumptions. A detailed application confirms the theory by constructing an explicit problem with an optimal pair guaranteed by the derived 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.
Paper Structure (6 sections, 5 theorems, 109 equations)

This paper contains 6 sections, 5 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Existence and uniqueness of mild solution
  4. Existence of optimal controls
  5. Application
  6. Conclusion

Key Result

Lemma 2.1

For any $p \geq 1$ and an arbitrary predictable process $\chi(\cdot)$ valued in $L^2_0$, the following inequality holds: for all $t \in [0, \infty)$.

Theorems & Definitions (9)

  • Lemma 2.1: see 7
  • Lemma 2.2
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof