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Null controllability for stochastic semi-discrete parabolic equations

Qingmei Zhao

Abstract

In this paper, we present a null controllability result for a class of stochastic semi-discrete parabolic equations. For this purpose, an observability estimate is established for backward stochastic semi-discrete parabolic equations, with an explicit observability constant that depends on the discretization parameter. This estimate is obtained by a new Carleman estimate for backward stochastic semi-discrete parabolic operators.

Null controllability for stochastic semi-discrete parabolic equations

Abstract

In this paper, we present a null controllability result for a class of stochastic semi-discrete parabolic equations. For this purpose, an observability estimate is established for backward stochastic semi-discrete parabolic equations, with an explicit observability constant that depends on the discretization parameter. This estimate is obtained by a new Carleman estimate for backward stochastic semi-discrete parabolic operators.
Paper Structure (4 sections, 7 theorems, 108 equations)

This paper contains 4 sections, 7 theorems, 108 equations.

Table of Contents

  1. Introduction and main result
  2. Preliminaries
  3. Semi-discrete Carleman estimate for uniform meshes
  4. The observability for the backward stochastic semi-discrete parabolic equations

Key Result

Theorem 1.1

There exist $C$ and $h_0$, for all $h\le h_0$, there exist $(u, v) \in L^2_{\mathbb{F}}(0, T; L^2_h(\omega\cap\mathcal{M}))\times L^2_{\mathbb{F}}(0, T; L^2_h(\mathcal{M}))$ such that the solution to $(e*03a)$ satisfies and

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.1
  • Theorem 3.1
  • Theorem 4.1