arXiv subjects

Hassan Saoud

Publications and source records attributed to Hassan Saoud.

5 recordsLinked to original sources

arXiv2026

Equilibria in Motion: Stability, Tracking, and Convergence

We study the stability, tracking, and convergence of nonautonomous systems with moving nonisolated equilibrium sets. We develop a Lyapunov framework based on coupled dissipation channels to analyze the evolution of trajectories relative to a moving equilibrium family whose motion is quantified by an equilibrium speed through localized Attouch--Wets estimates. Under suitable dissipation and energy--distance comparison conditions, we establish Lyapunov stability, quantitative tracking estimates, asymptotic tracking under integrable equilibrium motion, and input-to-state stability (ISS) estimates relative to the moving equilibrium family. We further show that, under an appropriate nonescape condition, integrable equilibrium speed guarantees the existence of a limiting equilibrium geometry in the Attouch--Wets sense, allowing convergence to the moving equilibrium family to be transferred to convergence relative to the limiting equilibrium set. Explicit convergence-rate estimates are also derived. The theoretical results are illustrated by a dynamic resource allocation model with time-varying demand.

math.DS↗

arXiv2026

Convergence and Stability of a Catching-Up Algorithm for Differential Inclusions with Maximal Monotone Operators

We study a catching-up algorithm for a class of differential inclusions driven by maximal monotone operators with continuous perturbations. Using a decomposition of the monotone operator into the closed convex hull of its single-valued part and the normal cone to a closed convex set, we establish existence of solutions and derive global energy bounds under a mild tangent dissipativity assumption. Under an additional local Lipschitz assumption on the perturbation, we also obtain uniqueness and stability with respect to the initial data. We then analyze a time-discretized catching-up scheme with variable step sizes and approximate projections. On every finite horizon, we prove convergence of the discrete trajectories to solutions of the continuous problem. A discrete velocity decomposition together with a discrete energy inequality yields uniform boundedness of the iterates, quantitative stability estimates, and explicit error bounds. We also establish asymptotic feasibility of the predictor step in an $L^2$ sense, as well as a Ces\`aro-type averaged feasibility property, showing that the constraint violations generated by the free step vanish as the discretization is refined. Finally, we illustrate the theory on explicit examples, including a fully explicit one--dimensional test case and a multidimensional constrained dry-friction system.

math.OC↗

arXiv2025

Composite Lyapunov Criteria for Stability and Convergence with Applications to Optimization Dynamics

We propose a composite Lyapunov framework for nonlinear autonomous systems that ensures strict decay through a pair of differential inequalities. The approach yields integral estimates, quantitative convergence rates, vanishing of dissipation measures, convergence to a critical set, and semistability under mild conditions, without relying on invariance principles or compactness assumptions. The framework unifies convergence to points and sets and is illustrated through applications to inertial gradient systems and Primal--Dual gradient flows.

math.OC↗

arXiv2025

Geometric Stability Analysis for Differential Inclusions Governed by Maximally Monotone Operators

This paper develops a geometric framework for the stability analysis of differential inclusions governed by maximally monotone operators. A key structural decomposition expresses the operator as the sum of a convexified limit mapping and a normal cone. However, the resulting dynamics are often difficult to analyze directly due to the absence of Lipschitz selections and boundedness. To overcome these challenges, we introduce a regularized system based on a fixed Lipschitz approximation of the convexified mapping. From this approximation, we extract a single-valued Lipschitz selection that preserves the essential geometric features of the original system. This framework enables the application of nonsmooth Lyapunov methods and Hamiltonian-based stability criteria. Instead of approximating trajectories, we focus on analyzing a simplified system that faithfully reflects the structure of the original dynamics. Several examples are provided to illustrate the method's practicality and scope.

math.OC↗

arXiv2023

Locating Theorems of Differential Inclusions Governed by Maximally Monotone Operators

In this paper, we are interested in studying the asymptotic behavior of the solutions of differential inclusions governed by maximally monotone operators. In the case where the LaSalle's invariance principle is inconclusive, we provide a refined version of the invariance principle theorem. This result derives from the problem of locating the $ω$-limit set of a bounded solution of the dynamic. In addition, we propose an extension of LaSalle's invariance principle, which allows us to give a sharper location of the $ω$-limit set. The provided results are given in terms of nonsmooth Lyapunov pair-type functions.

math.OC↗