arXiv subjects

Abdellatif Elgrou

Publications and source records attributed to Abdellatif Elgrou.

15 recordsLinked to original sources

arXiv2026

Controllability of Forward Stochastic Reaction--Convection--Diffusion Systems with Cascade Structure

We investigate the null and approximate controllability of coupled linear forward stochastic reaction--convection--diffusion systems under suitable cascade coupling conditions. The model consists of two forward stochastic parabolic equations governed by general second-order differential operators with time-, space-, and random-dependent coefficients. We consider a localized control acting on the drift term of the first equation, together with controls acting on the diffusion terms. By means of a duality argument, the controllability problem is reduced to an observability problem for the associated adjoint backward stochastic parabolic system. We establish a new global Carleman estimate for coupled backward stochastic parabolic systems whose drift terms belong to a negative Sobolev space. As an application of this estimate, we derive the required observability properties and establish the null and approximate controllability of the original forward system.

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arXiv2026

Insensitizing Control Problems for Coupled Stochastic Parabolic Systems with State and Gradient Observations

We study insensitizing control problems for a class of coupled linear stochastic parabolic systems. We establish the existence of controls such that a sentinel functional, involving localized observations of the state variables and their spatial gradients, is insensitive to small perturbations of the null initial data. We first reformulate the insensitizing control problem as a null controllability problem for a coupled forward--backward stochastic parabolic system, in which the observation terms induce both zeroth- and second-order coupling terms. By duality, the analysis is reduced to an observability inequality for the corresponding adjoint system. The main analytical contribution is the derivation of new global Carleman estimates for coupled stochastic parabolic systems with zeroth- and second-order coupling terms, under suitable geometric assumptions on the control and observation regions. These estimates yield the required observability inequalities and, consequently, the existence of insensitizing controls. Furthermore, depending on the value of a weighting parameter $β\in[0,1]$, which determines the relative contributions of the two state components to the sentinel functional, we consider two cases. If $β\in\{0,1\}$, the sentinel functional depends on only one state component, and a single localized control acting in the drift of the first equation is sufficient. In contrast, if $β\in(0,1)$, both state components contribute to the sentinel functional, and two localized controls acting in the drift terms of the two equations are sufficient. Moreover, the control strategy in this paper involves two additional controls acting throughout the diffusion terms.

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arXiv2026

Long-Time Stability Analysis for Stochastic Evolution Equations with Multiplicative Noise

In this paper, we study the long-time stability behavior of a class of linear stochastic evolution equations in a Hilbert space with multiplicative noise. Explicit sufficient conditions for $p$-th moment and almost sure exponential stability are established, highlighting the interplay between the principal eigenvalue of the governing operator, the drift coefficient, and the noise intensity. The relationship between these two notions of stability is also clarified. Applications to several stochastic partial differential equations are presented. In addition, a fully discrete spectral Galerkin method together with the implicit Euler--Maruyama scheme is shown to preserve these stability properties at the discrete level. Finally, numerical simulations are provided to confirm the theoretical results.

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arXiv2026

A Hierarchical Robust Control Strategy for Stochastic Kuramoto--Sivashinsky--Korteweg--de Vries Equations

We investigate the robust Stackelberg null controllability of a one-dimensional forward linear stochastic Kuramoto--Sivashinsky--Korteweg--de Vries (KS--KdV) equation. The control framework is formulated as a hierarchical Stackelberg game involving two leaders, one follower, and worst-case disturbances acting in both the drift and diffusion terms. The first leader acts to drive the system to rest, while the second leader is introduced to overcome analytical difficulties arising from the stochastic setting. The follower, by reducing the effect of the disturbances, addresses a tracking-type control problem aimed at keeping the system state and its first and second spatial derivatives close to prescribed target trajectories. First, the robust control problem is characterized by the existence of a saddle point. Then, the analysis is reduced to the null controllability of a strongly coupled forward--backward stochastic KS--KdV system. The problem is addressed by combining a duality technique with new Carleman estimates for forward and backward stochastic fourth-order parabolic equations.

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arXiv2026

Carleman Estimates for Backward Anisotropic Stochastic Parabolic Equations with General Dynamic Boundary Conditions and Applications

We investigate a backward anisotropic stochastic parabolic equation with general dynamic boundary conditions, where the drift involves both $\mathbb{L}^2$ and $\mathbb{H}^{-1}$ bulk--surface terms. We first establish the well-posedness of this equation. Subsequently, we derive a new Carleman estimate through a two-step approach. In the first step, using a weighted identity method together with a careful treatment of the boundary integral terms arising from the dynamic boundary conditions, we obtain an intermediate Carleman estimate for backward anisotropic stochastic parabolic equations without weak divergence source terms. In the second step, a duality method combined with suitable optimization techniques is employed to incorporate the weak divergence source terms. As applications of the derived Carleman estimate, we address two control problems. First, we establish null controllability for forward anisotropic stochastic parabolic equations with general dynamic boundary conditions. These equations involve both reaction and convection terms, with adapted, bounded stochastic bulk--surface coefficients. Moreover, we provide an explicit estimate of the null controllability cost, i.e., a bound on the minimal norm of controls required to drive the system to zero at the terminal time $T$. Second, we study an insensitizing control problem for this class of equations. The goal is to determine controls for systems with partially unknown initial data such that a given energy functional remains insensitive to small perturbations of these data. In this work, the functional involves the norm of the state over a localized bulk--surface region, together with the norm of its tangential gradient over a localized boundary region.

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arXiv2025

Stackelberg-Nash Controllability for Abstract Stochastic Evolution Equations and Applications

This paper presents the concepts of exact, null, and approximate controllability in the Stackelberg-Nash sense for abstract forward and backward stochastic evolution equations, involving two types of controls: leaders and followers. We begin by proving the existence and uniqueness of the Nash equilibrium, as well as its characterization for fixed leader controls. We then establish a duality between these controllability concepts and the corresponding observability properties. Finally, we apply our theoretical results to the forward and backward stochastic heat equations. The results for the backward heat equation are obtained by deriving a new Carleman estimate.

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arXiv2025

Multi-objective and hierarchical control for coupled stochastic parabolic systems

We study the Stackelberg-Nash null controllability of a coupled system governed by two linear forward stochastic parabolic equations. The system includes one leader control localized in a subset of the domain, two additional leader controls in the diffusion terms, and $ m $ follower controls, where $ m \geq 2 $. We consider two different scenarios for the followers: first, when the followers minimize a functional involving both components of the system's state, and second, when they minimize a functional involving only the second component of the state. For fixed leader controls, we first establish the existence and uniqueness of the Nash equilibrium in both scenarios and provide its characterization. As a byproduct, the problem is reformulated as a classical null controllability issue for the associated coupled forward-backward stochastic parabolic system. To address this, we derive new Carleman estimates for the adjoint stochastic systems. As far as we know, this problem is among the first to be discussed for stochastic coupled systems.

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arXiv2024

Insensitizing controls for stochastic parabolic equations with dynamic boundary conditions

In this paper, we continue the study of some controllability issues for the forward stochastic heat 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. Under a strong measurability condition, and using a spectral inequality, we first establish an appropriate observability inequality for the corresponding adjoint system. Then, by the classical duality approach, the null and approximate controllability results are established.

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arXiv2024

Null Controllability for Cascade systems of Coupled Backward Stochastic Parabolic Equations with One Distributed Control

We prove the null controllability of a cascade system of $n$ coupled backward stochastic parabolic equations involving both reaction and convection terms, as well as general second-order parabolic operators, with $n \geq 2$. To achieve this, we apply a single distributed control to the first equation, while the other equations are controlled through the coupling. To obtain our results, we develop a new global Carleman estimate for the forward stochastic parabolic adjoint system with some terms in the $H^{-1}$-space. Subsequently, we derive the appropriate observability inequality, and by employing the classical duality argument, we establish our null controllability result. Additionally, we provide an estimate for the null control cost with respect to the final time $T$ and the potentials.

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arXiv2024

Inverse initial problem under Nash strategy for stochastic reaction-diffusion equations with dynamic boundary conditions

In this paper, we study a multi-objective inverse initial problem with a Nash strategy constraint for forward stochastic reaction-diffusion equations with dynamic boundary conditions, where both the volume and surface equations are influenced by randomness. The objective is twofold: first, we maintain the state close to prescribed targets in fixed regions using two controls; second, we determine the history of the solution from observations at the final time. To achieve this, we establish new Carleman estimates for forward and backward equations, which are used to prove an interpolation inequality for a coupled forward-backward stochastic system. Consequently, we obtain two results: backward uniqueness and a conditional stability estimate for the initial conditions.

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arXiv2024

Stackelberg-Nash null controllability for stochastic parabolic equations

We study a hierarchical control problem for stochastic parabolic equations involving gradient terms. We employ the Stackelberg-Nash strategy with two leaders and two followers. The leaders are responsible for selecting the policy targeting null controllability, while the followers solve a bi-objective optimal control problem which consists of maintaining the solution process close to prefixed targets. Once the Nash equilibrium is determined, the problem reduces to achieving null controllability of a coupled forward-backward stochastic system. To solve this problem, via Carleman estimates, we establish a suitable observability inequality. Subsequently, we achieve the desired controllability result.

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arXiv2024

Null controllability for stochastic parabolic equations with Robin boundary conditions

We establish the null controllability of forward and backward linear stochastic parabolic equations with linear Robin (or Fourier) boundary conditions. These equations incorporate zero and first order terms with bounded coefficients. To prove our null controllability results, a key tool will be the derivation of two new global Carleman estimates for the weak solutions of the corresponding adjoint equations in negative Sobolev space. These Carleman estimates are established using a duality method.

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arXiv2024

Multi-objective control for stochastic parabolic equations with dynamic boundary conditions

This paper deals with a hierarchical multi-objective control problem for forward stochastic parabolic equations with dynamic boundary conditions. The controls are divided into two classes: leaders and followers. The goal of the leaders is of null controllability type while the followers are in charge of letting the state close to prescribed targets in fixed observation regions. To solve the problem, Nash and Stackelberg strategies are used. To implement these strategies, we combine some appropriate Carleman estimates and the well-known control duality approach.

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arXiv2024

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

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.

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arXiv2024

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

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.

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