Browsing by Author "Marx, Swann"

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    Contributions to the singular perturbation theory of infinite-dimensional coupled systems
    (2025) Arias Neira, Gonzalo Andrés; Cerpa, Eduardo; Marx, Swann; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    Singular perturbation and separation of time scales methods have been used to study the stability and control design for coupled ODE systems with different time scales for many years. This important literature was motivated by the fact that systems with significantly different time scales appear in several applications, in which the constituents of a coupled system may model different physical phenomena taking place in different time scales. The singular perturbation method (SPM), roughly speaking, aims to decouple a full system into two approximated subsystems based on a suitable time-scale separation. This thesis addresses problems concerning the stability, Tikhonov's approximation, stabilization, and control of singularly perturbed coupled infinite-dimensional systems.
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    Singular Perturbation Analysis for a Coupled KdV-ODE System
    (2024) Marx, Swann; Cerpa, Eduardo
    Asymptotic stability is with no doubts an essential property to be studied for any system. This analysis often becomes very difficult for coupled systems and even harder when different time-scales appear. The singular perturbation method allows to decouple a full system into what are called the reduced-order system and the boundary layer system to get simpler stability conditions for the original system. In the infinite-dimensional setting, we do not have a general result making sure this strategy works. This article is devoted to this analysis for some systems coupling the Korteweg-de Vries equation and an ordinary differential equation with different time scales. More precisely, we obtain stability results and Tikhonov-type theorems.
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    Stability analysis of a linear system coupling wave and heat equations with different time scales
    (2025) Arias, Gonzalo; Cerpa, Eduardo; Marx, Swann
    In this paper we consider a system coupling a wave equation with a heat equation through its boundary conditions. The existence of a small parameter in the heat equation, as a factor multiplying the time derivative, implies the existence of different time scales between the constituents of the system. This suggests the idea of applying a singular perturbation method to study stability properties. In fact, we prove that this method works for the system under study. Using this strategy, we get the stability of the system and a Tikhonov theorem, which allows us to approximate the solution of the coupled system using some appropriate uncoupled subsystems. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.