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Controllability for forward stochastic parabolic equations with dynamic boundary conditions without extra forces

Said Boulite, Abdellatif Elgrou, Lahcen Maniar, Omar Oukdach

Abstract

In this paper, we continue the study of some controllability issues for the forward stochastic parabolic equation with dynamic boundary conditions. The main novelty in the present paper consists of considering only one control without extra forces in the noise parts. Utilizing an adequate spectral inequality and the iterative Lebeau-Robiano strategy, we first establish an observability inequality for the corresponding adjoint backward stochastic system. The null controllability result is then established by the classical duality approach. As a consequence of the null controllability property, an approximate controllability result is proved.

Controllability for forward stochastic parabolic equations with dynamic boundary conditions without extra forces

Abstract

In this paper, we continue the study of some controllability issues for the forward stochastic parabolic equation with dynamic boundary conditions. The main novelty in the present paper consists of considering only one control without extra forces in the noise parts. Utilizing an adequate spectral inequality and the iterative Lebeau-Robiano strategy, we first establish an observability inequality for the corresponding adjoint backward stochastic system. The null controllability result is then established by the classical duality approach. As a consequence of the null controllability property, an approximate controllability result is proved.
Paper Structure (5 sections, 10 theorems, 96 equations)

This paper contains 5 sections, 10 theorems, 96 equations.

Table of Contents

  1. Introduction and main results
  2. Well-posedness and preliminary results
  3. Observability problem
  4. Controllability results
  5. Conclusion

Key Result

Theorem 1.1

For any $T>0$, any nonempty open subset $G_0$ of $G$ and for all $E\subset(0,T)$ such that $\textbf{m}(E)>0$, the system 1.1 is null controllable at time $T$ and the control $u$ can be chosen so that where $C$ is a positive constant depending only on $G$, $G_0$, $E$, $T$, $|a|_{L^\infty_\mathcal{F}(0,T)}$ and $|b|_{L^\infty_\mathcal{F}(0,T)}$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Proposition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • ...and 6 more