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Mohamed Ouzahra

Publications and source records attributed to Mohamed Ouzahra.

12 recordsLinked to original sources
arXiv2026

On Asymptotic and Finite-Time Stabilization of Bilinear Systems

This manuscript addresses the analysis and design of feedback laws for the stabilization of bilinear control systems in infinite-dimensional spaces. It first examines weak, strong, and polynomial stabilization within a Hilbert space framework, emphasizing the role of observability conditions. It then studies exponential stabilization in Banach spaces, highlighting the additional challenges arising from the lack of a Hilbertian structure. Finally, it introduces finite-time stabilization, presenting recent results and open problems within the broader context of nonlinear infinite-dimensional control theory. Several applications are also discussed.

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arXiv2025

On Finite- and Fixed-Time Stabilization of Abstract Nonlinear Systems with Well-Posedness Guarantees

This paper addresses the problem of stabilization for infinite-dimensional systems. In particular, we design nonlinear stabilizers for both linear and nonlinear abstract systems. We focus on two classes of systems: the first class comprises linear abstract systems subject to matched perturbations, while the second class encompasses fully nonlinear abstract systems. Our main objective is to synthesize state-feedback controllers that guarantee finite- or fixed-time stability of the closed-loop system, along with possible estimation of the settling time. For the first class, the presence of persistent perturbations introduces significant challenges in the well-posedness analysis, particularly due to the discontinuous nature of the control law. To address this, we employ maximal monotone operator theory to rigorously establish the existence and uniqueness of solutions, extending classical results from continuous abstract systems. For the second class, which includes nonlinearities, we further show that the proposed feedback law ensures fixed-time stability and well-posedness of the closed-loop system, again using maximal monotone theory. The results provide a unified framework for robust, finite /fixed-time stabilization in the presence of discontinuities and nonlinearities in infinite-dimensional settings.

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arXiv2025

Finite-Time Stabilization of a Class of Nonlinear Systems in Hilbert Space

This paper deals with the finite-time stabilization of a class of nonlinear infinite-dimensional systems. First, we consider a bounded matched perturbation in its linear form. It is shown that by using a set-valued function, both the convergence objective (finite-time) and the rejection of perturbations are achieved. Second, we consider a class of nonlinear systems and design a feedback control that ensures the closed-loop system is finite-time stable. All proofs presented in this paper regarding convergence are based on Lyapunov theory. The existence of solutions to the closed-loop system and its well-posedness are established using maximal monotone theory. To illustrate the applicability of the theoretical results, a heat equation is considered as an application of the main results.

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arXiv2025

A Decomposition Method for Finite-Time Stabilization of Bilinear Systems with Applications to Parabolic and Hyperbolic Equations

In this work, we address the problem of finite-time stabilization for a class of bilinear system. We propose a decomposition-based approach in which the nominal system is split into two subsystems, one of which is inherently finite-time stable without control. This allows the stabilization analysis to focus solely on the remaining subsystem. To ensure the well-posedness of the closed-loop system, we establish sufficient conditions on the system and control operators. The stabilization results are then derived using a suitable Lyapunov function and an observation condition. The effectiveness of the proposed approach is demonstrated through examples involving both parabolic and hyperbolic infinite-dimensional systems.

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arXiv2025

Analyticity of Exponential Dirichlet Series and Applications to the Approximate Controllability of Parabolic Equations

In this paper, we investigate the analyticity of a class of exponential Dirichlet series. We then explicitly determine the coefficients of their power series decomposition and provide an estimate for the remainder. As an application, we study the approximate controllability property of linear parabolic equations with locally distributed or lumped controls by employing the moment method, which relies on the exponential Dirichlet series associated with the spectrum of the system's operator.

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arXiv2021

Exponential stability of linear systems under a class of Desch-Schappacher perturbations

In this paper we investigate the uniform exponential stability of the system $\frac{dx(t)}{dt}=Ax(t)-ρBx(t), \; (ρ>0), $ where the unbounded operator $A$ is the infinitesimal generator of a linear $C_0-$semigroup of contractions $S(t)$ in a Hilbert space $X$ and $B$ is a Desch-Schappacher operator. Then we give sufficient conditions for exponential stability of the above system. The obtained stability result is then applied to show the uniform exponential stabilization of bilinear partial differential equations.

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arXiv2021

Problem of optimal control for bilinear systems with endpoint constraint

In this work, we will investigate the question of optimal control for bilinear systems with constrained endpoint. The optimal control will be characterized through a set of unconstrained minimization problems that approximate the former. Then a class of bilinear systems for which the optimal control can be expressed as a time-varying feedback law will be identified. Finally, applications to parabolic and hyperbolic partial differential equations are provided.

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arXiv2020

Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control

In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Moreover, in the case of a globally supported control, we prove the approximate controllability within any time-interval given in advance which does not depend on the initial and target states. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions.

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arXiv2019

Controllability of the semilinear wave equation governed by a multiplicative control

In this paper we establish several results on approximate controllability of a semilinear wave equation by making use of a single multiplicative control. These results are then applied to discuss the exact controllability properties for the one dimensional version of the system at hand. The proof relies on linear semigroup theory and the results on the additive controllability of the semilinear wave equation. The approaches are constructive and provide explicit steering controls. Moreover, in the context of undamped wave equation, the exact controllability is established for a time which is uniform for all initial states.

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