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Null Controllability for Backward Stochastic Parabolic Convection-Diffusion Equations with Dynamic Boundary Conditions

Mahmoud Baroun, Said Boulite, Abdellatif Elgrou, Lahcen Maniar

TL;DR

The paper addresses null controllability for backward stochastic parabolic equations with dynamic boundary conditions and convection. It develops a new global Carleman estimate for forward stochastic parabolic equations with weak divergence terms and uses duality to derive an observability inequality for the adjoint problem, enabling null controllability and explicit cost bounds. A penalized Hilbert Uniqueness Method links controllability of the backward system to properties of the forward adjoint equation, and the results extend to dynamic boundary conditions with surface diffusion. The approach yields an $e^{CT^{-1}}$-type observability constant for small times and provides a constructive way to obtain null controls with quantified energy costs, advancing the theory of SPDE controllability under complex boundary interactions.

Abstract

This paper is concerned with the null controllability for linear backward stochastic parabolic equations with dynamic boundary conditions and convection terms. Using the classical duality argument, the null controllability is obtained via an appropriate observability inequality of the corresponding adjoint forward stochastic parabolic equation. To prove this observability inequality, we develop a new global Carleman estimate for forward stochastic parabolic equations that contains some first-order terms in the weak divergence form. Our Carleman estimate is established by applying the duality technique. Moreover, an estimate of the null-control cost is provided.

Null Controllability for Backward Stochastic Parabolic Convection-Diffusion Equations with Dynamic Boundary Conditions

TL;DR

The paper addresses null controllability for backward stochastic parabolic equations with dynamic boundary conditions and convection. It develops a new global Carleman estimate for forward stochastic parabolic equations with weak divergence terms and uses duality to derive an observability inequality for the adjoint problem, enabling null controllability and explicit cost bounds. A penalized Hilbert Uniqueness Method links controllability of the backward system to properties of the forward adjoint equation, and the results extend to dynamic boundary conditions with surface diffusion. The approach yields an -type observability constant for small times and provides a constructive way to obtain null controls with quantified energy costs, advancing the theory of SPDE controllability under complex boundary interactions.

Abstract

This paper is concerned with the null controllability for linear backward stochastic parabolic equations with dynamic boundary conditions and convection terms. Using the classical duality argument, the null controllability is obtained via an appropriate observability inequality of the corresponding adjoint forward stochastic parabolic equation. To prove this observability inequality, we develop a new global Carleman estimate for forward stochastic parabolic equations that contains some first-order terms in the weak divergence form. Our Carleman estimate is established by applying the duality technique. Moreover, an estimate of the null-control cost is provided.
Paper Structure (6 sections, 11 theorems, 111 equations)

This paper contains 6 sections, 11 theorems, 111 equations.

Table of Contents

  1. Introduction and main results
  2. Well-posedness of equations
  3. Global Carleman estimate for forward stochastic parabolic equations and proof of Theorem \ref{['thm1.2']}
  4. Global Carleman estimate
  5. Proof of Theorem \ref{['thm1.2']}
  6. Proof of Theorem \ref{['thm1.1']}

Key Result

Theorem 1.1

For any given $T>0$, $G_0\Subset G$ a nonempty open subset of $G$ and for all $(y_T,y_{\Gamma,T})\in L^2_{\mathcal{F}_T}(\Omega;\mathbb{L}^2)$, there exists a control $\hat{u}\in L^2_\mathcal{F}(0,T;L^2(G_0))$ such that the corresponding solution $(\hat{y},\hat{y}_\Gamma,\hat{Y},\hat{\widetilde{Y}}) Moreover, the control $\hat{u}$ can be chosen such that where $K$ has the following form

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Lemma 1.1
  • Remark 1.4
  • Lemma 1.2
  • Remark 1.5
  • Definition 2.1
  • ...and 14 more