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Impulse approximate controllability for stochastic evolution equations and its applications

Yuanhang Liu

Abstract

This paper is concerned with impulse approximate controllability for stochastic evolution equations with impulse controls. As direct applications, we formulate captivating minimal norm and time optimal control problems; The minimal norm problem seeks to identify an optimal impulse control characterized by the minimum norm among all feasible controls, guiding the system's solutions from an initial state within a fixed time interval toward a predetermined target while the minimal time problem is to find an optimal impulse control (among certain control constraint set), which steers the solution of the stochastic equation from a given initial state to a given target set as soon as possible. These problems, to the best of our knowledge, are among the first to discuss in the stochastic case.

Impulse approximate controllability for stochastic evolution equations and its applications

Abstract

This paper is concerned with impulse approximate controllability for stochastic evolution equations with impulse controls. As direct applications, we formulate captivating minimal norm and time optimal control problems; The minimal norm problem seeks to identify an optimal impulse control characterized by the minimum norm among all feasible controls, guiding the system's solutions from an initial state within a fixed time interval toward a predetermined target while the minimal time problem is to find an optimal impulse control (among certain control constraint set), which steers the solution of the stochastic equation from a given initial state to a given target set as soon as possible. These problems, to the best of our knowledge, are among the first to discuss in the stochastic case.
Paper Structure (8 sections, 10 theorems, 127 equations)

This paper contains 8 sections, 10 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and main results
  3. Auxiliary conclusions
  4. Impulse approximate controllability
  5. Norm optimal control problem
  6. Time optimal control problem
  7. Examples
  8. Declarations

Key Result

Theorem 2.1

Let $0<\tilde{T}<T\leq2\tilde{T}$. Suppose the assumptions ${\bf(H)}$ and ${\bf(B)}$ hold. Then system eq:main-for is impulse approximate controllability. That is to say, for any $y_0 \in L^2_{\mathcal{F}_0}(\Omega;H)$, $\epsilon>0$, the solution $y$ of eq:main-for satisfies the following inequality Moreover, there exist two constants $C_3>0,C_4>0$ such that the control $u\in L^2_{\mathcal{F}_{\ti

Theorems & Definitions (24)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 3.1
  • Lemma 3.2
  • ...and 14 more