arXiv subjects

Saad Benjelloun

Publications and source records attributed to Saad Benjelloun.

16 recordsLinked to original sources

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 through diffusion-driven instability. Steering these patterns from one configuration to another can be formulated as an optimal control problem. When the control acts on the entire spatial domain, existence and optimality conditions are well understood. Yet in practice, the control can only act on a part of the domain. Taking the Nodal--Lefty reaction--diffusion system as a case study, we consider the setting where the control is restricted to a subdomain. We derive an explicit upper bound on the optimality loss, defined as the difference between the subdomain optimal cost and the full-domain optimal cost. From this bound, we obtain an explicit formula for the minimum size of the control region needed to reach a target pattern with prescribed accuracy. We also consider the case where the control is distributed over several disjoint regions instead of a single one, with the same total area, and prove that the distributed configuration gives a tighter bound under natural conditions on the spatial structure of the target. Numerical illustrations confirm the theoretical results and show that a control region covering roughly forty percent of the domain is sufficient to drive the system from stripes to spots with high accuracy.

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

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

Boundary controllers have been recently proposed in the literature, via modal decomposition, to achieve $H^1$ stabilization of linear parabolic equations in two and three dimensions. In one dimension ($1$-D), $H^1$ exponential stability is known to imply boundedness and asymptotic convergence of the state to zero in the sense of the max norm. However, in two ($2$-D) and three dimensions ($3$-D), this implication does not systematically hold. In this paper, focusing on the full-state feedback case, our objective is to prove that the modal-decomposition based controller in \cite{Munteanu2017IJC} guarantees, not only $H^1$ exponential stability, but also $H^2$ exponential stability. This implies, in particular, boundedness and asymptotic convergence of the state to zero in the sense of the max norm. Our approach consists in rewriting the Laplacian of the state, required in the $H^2$ norm, as a linear combination of the state and its time derivative. The $L^2$ norm of the state being bounded by the $H^1$ norm, we only analyze the $L^2$ norm of the time derivative of the state.

math.OC↗

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↗

arXiv2026

Global existence of weak solutions to incompressible anisotropic Cahn-Hilliard-Navier-Stokes system

We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain $Ω\subset \mathbb{R}^d$. This work extends previous results on the isotropic case by incorporating anisotropic surface energy, represented by $\mathfrak{F}= \int_Ω \fracε{2}\, Γ^2(\nabla ϕ) $. The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in \cite{anderson2000phase,taylor-cahn98, zaidni2024}. Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two- and three-dimensions $(d=2,3)$. A key ingredient in extending the local existence of approximate solutions to a global one is the application of Bihari's inequality combined with a fixed-point argument.

math.AP↗

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↗

arXiv2025

Homogenization of 2D materials in the Thomas-Fermi-von Weizsacker theory

We study the homogenization of the Thomas-Fermi-von Weizsacker (TFW) model for 2D materials. It consists in considering 2D-periodic nuclear densities with periods going to zero. We study the behavior of the corresponding ground state electronic densities and ground state energies. The main result is that these three dimensional problems converge to a limit model that is one dimensional. We also illustrate this convergence with numerical simulations and estimate the converging rate for the ground state electronic densities and the ground state energies.

math-ph↗

arXiv2024

Thermodynamically consistent Cahn-Hilliard-Navier-Stokes equations using the metriplectic dynamics formalism

Cahn-Hilliard-Navier-Stokes (CHNS) systems describes flows with two-phases, e.g., a liquid with bubbles. Obtaining constitutive relations for general dissipative processes for such a systems, which are thermodynamically consistent, can be a challenge. We show how the metriplectic 4-bracket formalism achieves this in a straightforward, in fact algorithmic, manner. First, from the noncanonical Hamiltonian formulation for the ideal part of a CHNS system we obtain an appropriate Casimir to serve as the entropy in the metriplectic formalism that describes the dissipation (e.g. viscosity, heat conductivity and diffusion effects). General thermodynamics with the thermodynamic conjugates of concentration and chemical potential are included. Having expressions for the Hamiltonian (energy), entropy, and Poisson bracket, we describe a procedure for obtaining a metriplectic 4-bracket that describes thermodynamically consistent dissipative effects. The 4-bracket formalism leads naturally to a general CHNS system that allows for anisotropic surface energy effects. This general CHNS system reduces to cases in the literature, to which we can compare.

math-ph↗

arXiv2023

Stability of the one electron atom Schrödinger model with magnetic field in two dimensions

We study the stability of the one electron atom Schrödinger model with self-generated magnetic field in two dimensions. The magnetic energy is taken of the general form $K\int_{\mathbb{R}^2} |B|^p$ and we study the stability of the model as a function of the power $p$ and the coupling constant $K$. We show that for $p>3/2$, the model is always stable, and for $p<3/2$, the model is always unstable. In the critical case $p=3/2$, there is a critical stability constant $K_c$, that we characterize in terms of zero modes of the Dirac-Weyl operator. The value of $K_c$ is approximated using analytic and numerical methods.

math-ph↗

arXiv2021

On the sound speed in two-fluid mixtures and the implications for CFD model validation

Study of the propagation of sound in a single non-ideal fluid originates with Stokes in 1845 and Kirchhoff in 1868. The situation is much more complex in the case of two-fluid flow, both from the physical point of view, as the configuration of the flow matters greatly, and from the analytical point of view. The principle two-fluid models currently in use for CFD are the focus of this article. It is shown that analytical expressions for the speed of sound depend heavily on the chosen model. These sound speed expressions are compared with experimental values. The consequences for CFD models are discussed in the final section of this paper. It is found that numerical models with inaccurate wave speeds lead to incorrect numerical solutions, despite the accuracy of the numerical scheme.

math.AP↗

arXiv2021

On the sound dispersion and attenuation in fluids due to thermal and viscous effects

In this paper, we derive a dispersion relation for sound waves in viscous and heat conducting fluids. In particular this dispersion (i.e. variation of speed of sound with frequency) is shown to be of second order of magnitude, w.r.t. Knudsen numbers, as in the Stokes [2] case, corresponding to non-conductive fluid (Prandtl number P r = $\infty$). This formula completes the classical attenuation relation called Stokes-Kirchhoff. We represent in a simplified manner the Kirchhoff approach to derive this attenuation [1], starting from the 3D compressible Navier-stokes system. The classical Stokes-Kirchhoff formula has been questioned recently in [3] and a different (and incorrect) formula was proposed. We point out the non-trivial assumptions that are violated in the new derivation in [3] to reestablish the classical Stokes-Kirchhoff formula. Finally, we give an explanation to differences in dispersion and attenuation formulae that one may find in the literature through analysing the form of the considered attenuated solutions.

physics.class-ph↗

arXiv2021

On the dispersion relation for compressible Navier-Stokes Equations

In this paper we revisit the classical sound dispersion and attenuation theory due to Stokes [5], 1845, and Kirchhoff [3], 1868, for the propagation of sound in non-ideal fluids. In particular we reformulate the analysis due to Fletcher[2], 1974, showing conditions for which the sound propagates at the isothermal speed of sound. Also we presents asymptotic developments making precise the physical conditions under which the different dispersion and attenuation formulas apply.The more complex case of two-fluid flow is addressed by Benjelloun and Ghidaglia [1] to which the reader is referred.

math.AP↗

arXiv2021

Thermodynamic identities and thermodynamic consistency of Equation of States

We present a systematic approach to construct complete equations of state (EOSs), or to ensure thermodynamic consistency of complete and incomplete forms of EOSs using a minimal and sufficient set of relations. We apply the proposed approach to some common classical equations of state for pure materials. In fact, classical equations of state come generally in an incomplete form that hides essencial properties necessary for thermodynamic consistency. If not aware of such constraints one may generalize the EOS, or fit its thermodynamic parameters from emprirical data in an inconsistent way.

physics.class-ph↗

arXiv2021

Open data for Moroccan license plates for OCR applications : data collection, labeling, and model construction

Significant number of researches have been developed recently around intelligent system for traffic management, especially, OCR based license plate recognition, as it is considered as a main step for any automatic traffic management system. Good quality data sets are increasingly needed and produced by the research community to improve the performance of those algorithms. Furthermore, a special need of data is noted for countries having special characters on their licence plates, like Morocco, where Arabic Alphabet is used. In this work, we present a labeled open data set of circulation plates taken in Morocco, for different type of vehicles, namely cars, trucks and motorcycles. This data was collected manually and consists of 705 unique and different images. Furthermore this data was labeled for plate segmentation and for matriculation number OCR. Also, As we show in this paper, the data can be enriched using data augmentation techniques to create training sets with few thousands of images for different machine leaning and AI applications. We present and compare a set of models built on this data. Also, we publish this data as an open access data to encourage innovation and applications in the field of OCR and image processing for traffic control and other applications for transportation and heterogeneous vehicle management.

cs.CV↗

arXiv2021

An open access NLP dataset for Arabic dialects : Data collection, labeling, and model construction

Natural Language Processing (NLP) is today a very active field of research and innovation. Many applications need however big sets of data for supervised learning, suitably labelled for the training purpose. This includes applications for the Arabic language and its national dialects. However, such open access labeled data sets in Arabic and its dialects are lacking in the Data Science ecosystem and this lack can be a burden to innovation and research in this field. In this work, we present an open data set of social data content in several Arabic dialects. This data was collected from the Twitter social network and consists on +50K twits in five (5) national dialects. Furthermore, this data was labeled for several applications, namely dialect detection, topic detection and sentiment analysis. We publish this data as an open access data to encourage innovation and encourage other works in the field of NLP for Arabic dialects and social media. A selection of models were built using this data set and are presented in this paper along with their performances.

cs.CL↗

arXiv2013

Existence theory for a kinetic-fluid coupling when small droplets are treated as part of the fluid

We consider in this paper a spray constituted of an incompressible viscous gas and of small droplets which can breakup. This spray is modeled by the coupling (through a drag force term) of the incom- pressible Navier-Stokes equation and of the Vlasov-Boltzmann equation, together with a fragmentation kernel. We first show at the formal level that if the droplets are very small after the breakup, then the solutions of this system converge towards the solution of a simplified system in which the small droplets produced by the breakup are treated as part of the fluid. Then, existence of global weak solutions for this last system is shown to hold, thanks to the use of the DiPerna-Lions theory for singular transport equations.

math.AP↗