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On the Controllability of a Fully Nonlocal Coupled Stochastic Reaction--Convection--Diffusion System

Abdellatif Elgrou, Federica Gregorio, Abdelaziz Rhandi

Abstract

In this paper, we study the null and approximate controllability of a class of fully nonlocal coupled stochastic reaction--convection--diffusion systems. The system consists of two forward stochastic parabolic equations driven by general second-order differential operators and incorporates four nonlocal zero-order integral terms. The nonlocality arises from integral kernel terms present in both equations, defined over a bounded domain $G \subset \mathbb{R}^N$ ($N \geq 1$). Since the coefficients depend on time, space, and random variables, we introduce three controls: a spatially localized control acting on the drift term of the first equation, and two additional controls acting on the diffusion terms of both equations. These additional controls are necessary to overcome difficulties due to the stochastic nature of the associated adjoint backward system. Using a standard duality argument, the controllability problem for the forward system is reduced to an observability problem for the corresponding adjoint nonlocal backward system. To establish this observability, we derive a new global Carleman estimate for the adjoint system, in which the drift terms belong to a negative Sobolev space and the equations include nonlocal integral terms. Our results are obtained under suitable cascade structure conditions on the coupling zero-order, nonlocal, and first-order terms of the system.

On the Controllability of a Fully Nonlocal Coupled Stochastic Reaction--Convection--Diffusion System

Abstract

In this paper, we study the null and approximate controllability of a class of fully nonlocal coupled stochastic reaction--convection--diffusion systems. The system consists of two forward stochastic parabolic equations driven by general second-order differential operators and incorporates four nonlocal zero-order integral terms. The nonlocality arises from integral kernel terms present in both equations, defined over a bounded domain (). Since the coefficients depend on time, space, and random variables, we introduce three controls: a spatially localized control acting on the drift term of the first equation, and two additional controls acting on the diffusion terms of both equations. These additional controls are necessary to overcome difficulties due to the stochastic nature of the associated adjoint backward system. Using a standard duality argument, the controllability problem for the forward system is reduced to an observability problem for the corresponding adjoint nonlocal backward system. To establish this observability, we derive a new global Carleman estimate for the adjoint system, in which the drift terms belong to a negative Sobolev space and the equations include nonlocal integral terms. Our results are obtained under suitable cascade structure conditions on the coupling zero-order, nonlocal, and first-order terms of the system.

Paper Structure

This paper contains 6 sections, 10 theorems, 142 equations.

Table of Contents

  1. Introduction and Main Results
  2. Global Carleman Estimates
  3. Preliminary Tools
  4. Main Carleman Estimate
  5. Observability Results
  6. Proofs of the Main Controllability Results

Key Result

Theorem 1.1

Assume that assumK12, assump1.3, assump1.3K21 and assump1.3B21 hold. Then, for any initial states $y_0, z_0 \in L^2_{\mathcal{F}_0}(\Omega; L^2(G))$ and all source terms $\xi_1, \xi_2 \in L^2_\mathcal{F}(0,T; L^2(G))$ that satisfy Then, there exists a control triple such that the corresponding solution $(y,z)$ of ass15cont satisfies that Moreover, there exists a constant $C>0$ such that where

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • proof
  • ...and 7 more