arXiv subjects

Omar Oukdach

Publications and source records attributed to Omar Oukdach.

9 recordsLinked to original sources

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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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

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

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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arXiv2021

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.

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