arXiv subjects

Mohamed Amine Ouchdiri

Publications and source records attributed to Mohamed Amine Ouchdiri.

5 recordsLinked to original sources

arXiv2026

$H^2$ Stabilization of the $2$-D and $3$-D Heat Equation via Modal Decomposition

Modal decomposition has been used recently to achieve boundary stabilization of linear parabolic equations in the $H^1$ norm, in two and three dimensions. In one dimension, $H^1$ exponential stability is well established and known to imply boundedness and asymptotic convergence of the state's max norm to zero. However, this implication does not necessarily hold in higher dimensions. In this paper, we prove that the modal-decomposition-based full-state feedback controller in \cite{Munteanu2017IJC} guarantees, not only $H^1$ exponential stability, but also $H^2$ exponential stability, which implies boundedness and asymptotic convergence of the state's max norm to zero. Our approach is based on rewriting the state's Laplacian, required in the expression of the state's $H^2$ norm, as a linear combination of the state and its time derivative. Thereby, the $L^2$ norm of the state's time derivative is the one to be analyzed in terms of boundedness and asymptotic convergence to zero.

math.OC↗

arXiv2026

How Much Spatial Control Is Enough? Subdomain Optimal Control of Reaction-Diffusion Systems in Synthetic Developmental Biology

Reaction--diffusion systems can produce spatial patterns such as stripes and spots by diffusion-driven instability. In synthetic developmental biology, such patterns are engineered in cell populations, and steering them toward a prescribed target pattern can be formulated as an optimal control problem. When the control acts on the entire spatial domain, the existence of optimal controls and the first-order optimality conditions are well established. In experiments, the control is applied with light by optogenetic tools, and the light illuminates a chosen region of the tissue, so the control acts on a subdomain. We consider the setting where the control is restricted to a subdomain fixed in advance, for a synthetic circuit in which the slowly diffusing activator Nodal and the fast-diffusing inhibitor Lefty form the pattern. We study the optimality loss, the difference between the optimal costs of the subdomain problem and of the full-domain problem, and we ask how much coverage keeps it within a desired precision. In simulations with one parameter set and two target patterns, nine squares give a smaller terminal error than one square at the same coverage. Moreover, the error depends on the average distance from the uncontrolled points to the control region. In particular, small errors occur when this distance is comparable to the length scale set by Nodal diffusion and degradation.

math.OC↗

arXiv2026

Certified Reachable Sets for Nonlinear Reaction--Diffusion Systems

Reachability analysis for dynamical systems seeks to compute a set containing all reachable states at a given time. Compared to ordinary differential equations (ODEs), the analysis of nonlinear reaction--diffusion PDEs with parametric uncertainties remains largely underexplored, due to the infinite-dimensional state space and the variety of solutions under different parameters. We address this through a three-step procedure: 1) Finite Element Methods (FEM)s to discretise the space and generate a finite-dimensional FEM-based model, 2) Proper Orthogonal Decomposition (POD) to build a Reduced-Order Model (ROM), and 3) set-based reachability-analysis methods applied to the ROM. We propose a framework that enables us to derive explicit upper bounds on the approximation errors introduced at each stage of the pipeline. In particular, we quantify the discrepancy between trajectories of the original PDE and those of the FEM-based discretization, as well as the error between the FEM-based model and the reduced-order model. Importantly, these bounds are shown to hold uniformly over the considered set of parameters. By combining these error estimates, we obtain an over-approximation of the reachable set of the original PDE. The approach is illustrated on the Allen--Cahn equation and a logistic growth PDE.

math.NA↗

arXiv2026

An optimal-control framework for reaction diffusion systems with application to synthetic developmental biology

Reaction-diffusion systems offer a powerful framework for understanding self-organized patterns in biological systems, yet controlling these patterns remains a significant challenge. As a consequence, we present a rigorous framework of optimal control for a class of coupled reaction-diffusion systems. The couplings are justified by the shared regulatory mechanisms encountered in synthetic biology. Furthermore, we introduce inputs and polynomial input-gain functions to guarantee well-posedness of the control system while maintaining biological relevance. As a result, we formulate an optimal control problem and derive necessary optimality conditions. We demonstrate our framework on an instance of such equations modeling the Nodal-Lefty interactions in mammalian cells. Numerical simulations showcase the effectiveness in directing pattern towards diverse targeted ones.

math.OC↗

arXiv2025

Turing Patterns in a Morphogenetic Model with Single Regulatory Function

Confirming Turing's theory of morphogens in developmental processes is challenging, and synthetic biology has opened new avenues for testing Turing's predictions. Synthetic mammalian pattern formation has been recently achieved through a reaction-diffusion system based on the short-range activator (Nodal) and the long-range inhibitor (Lefty) topology, where a single function regulates both morphogens. In this paper, we investigate the emergence of Turing patterns in the synthetic Nodal-Lefty system. First, we prove the existence of a global solution and derive conditions for Turing instability through linear stability analysis. Subsequently, we examine the behavior of the system near the bifurcation threshold, employing weakly nonlinear analysis, and using multiple time scales, we derive the amplitude equations for supercritical and subcritical cases. The results demonstrate the ability of the system to support various patterns, with the subcritical Turing instability playing a crucial role in the formation of dissipative structures observed experimentally.

nlin.PS↗