Searcharxiv⌕ Search

arXiv subjects

Walid Zouhair

Publications and source records attributed to Walid Zouhair.

7 recordsLinked to original sources
arXiv2026

Finite-time stabilization via impulse control of degenerate singular parabolic equations

This paper examines the impulse controllability of degenerate singular parabolic equations through a modern framework focused on finite-time stabilization. Furthermore, we provide an explicit estimate for the exponential decay of the solution. The proof of our main result combines a logarithmic convexity estimate with specific spectral properties. Finally, we establish the existence and uniqueness of the minimal norm impulse control associated with the system.

math.AP↗
arXiv2024

Internal null controllability for the one-dimensional heat equation with dynamic boundary conditions

The primary focus of this paper is to establish the internal null controllability for the one-dimensional heat equation featuring dynamic boundary conditions. This achievement is realized by introducing a new Carleman estimate and an observability inequality for the corresponding backward system. In conclusion, the paper includes a set of numerical experiments that serve to confirm the validity of the theoretical findings and underscore the effectiveness of the designed control with a minimal L2-norm.

math.OC↗
arXiv2022

Controllability of suspension bridge model proposed by Lazer and Mckenna under the influence of impulses, delays, and non-local conditions

The main purpose of this paper is to prove the controllability of the model proposed by Lazer and Mckenna under the influence of impulses, delay, and non-local conditions. First, we study approximate controllability by employing a technique that pulls back the control solution to a fixed curve in a short time interval. Subsequently, based on Banach Fixed Point Theorem we investigate the exact controllability.

math.OC↗
arXiv2022

Finite-time stabilization and impulse control of heat equation with dynamic boundary conditions

In this paper, we study the impulse controllability of a multi-dimensional heat equation with dynamic boundary conditions in a bounded smooth domain. Using a recent approach based on finite-time stabilization, we show that the system is impulse null controllable at any positive time via impulse controls supported in a nonempty open subset of the physical domain. Furthermore, we infer an explicit estimate for the exponential decay of the solution. The proof of the main result combines a logarithmic convexity estimate and some spectral properties associated to dynamic boundary conditions. In our setting, the nature of the equations, which couple intern-boundary phenomena, makes it necessary to go into quite sophisticated estimates incorporating several boundary terms.

math.OC↗
arXiv2022

Approximate controllability of semi-linear heat equation with Non-instantaneous impulses, memory and delay

The semilinear heat equation with non-instantaneous impulses \textbf{(NII)}, memory, and delay is considered and its approximate controllability is obtained. This is done by employing a technique that avoids fixed point theorems and pulls back the control solution to a fixed curve in a short time interval. We demonstrate, once again, that the controllability of the system is robust under the influence of non-instantaneous impulses, memory, and delays. Finally, we present some open problems and a possible general framework to study the controllability of non-instantaneous impulses semilinear systems.

math.DS↗
arXiv2022

On the existence and uniqueness of solutions for non-autonomous semi-linear systems with non-instantaneous impulses, delay, and non-local conditions

A non-autonomous evolution semi-linear differential system under non-instantaneous impulses, delays, and perturbed by non-local conditions is studied. Its piece-wise continuous solutions belong to a finite-dimensional Banach space. The existence and uniqueness of solutions on the interval $[-r,τ]$ are obtained by applying Karakostas fixed-point theorem. Further results concerning solution prolongation are developed. An example is presented, and several remarks on the infinite-dimensional case are included.

math.OC↗