On the Evolution Operators of a Class of Time-Delay Systems with Impulsive Parameterizations
De la Sen, Manuel (Universidad del País Vasco. Departamento de Electricidad y Electrónica)
Ibeas, Asier (Universitat Autònoma de Barcelona. Departament de Telecomunicació i Enginyeria de Sistemes)
Garrido, Aitor J. (Universidad del País Vasco. Departamento de Ingeniería de Sistemas y Control Automático)
Garrido, Izaskun (Universidad del País Vasco. Departamento de Ingeniería de Sistemas y Control Automático)

Date:	2025
Abstract:	This paper formalizes the analytic expressions and some properties of the evolution operator that generates the state-trajectory of dynamical systems combining delay-free dynamics with a set of discrete, or point, constant (and not necessarily commensurate) delays, where the parameterizations of both the delay-free and the delayed parts can undergo impulsive changes. Also, particular evolution operators are defined explicitly for the non-impulsive and impulsive time-varying delay-free case, and also for the case of impulsive delayed time-varying systems. In the impulsive cases, in general, the evolution operators are non-unique. The delays are assumed to be a finite number of constant delays that are not necessarily commensurate, that is, all of them being integer multiples of a minimum delay. On the other hand, the impulsive actions through time are assumed to be state-dependent and to take place at certain isolated time instants on the matrix functions that define the delay-free and the delayed dynamics. Some variants are also proposed for the cases when the impulsive actions are state-independent or state- and dynamics-independent. The intervals in-between consecutive impulses can be, in general, time-varying while subject to a minimum threshold. The boundedness of the state-trajectory solutions, which imply the system's global stability, is investigated in the most general case for any given piecewise-continuous bounded function of initial conditions defined on the initial maximum delay interval. Such a solution boundedness property can be achieved, even if the delay-free dynamics is unstable, by an appropriate distribution of the impulsive actions. An illustrative first-order example is developed in detail to illustrate the impulsive stabilization results.
Grants:	Agencia Estatal de Investigación PID2021-123543OB-C21 Agencia Estatal de Investigación PID2021-123543OB-C22
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Delay differential systems ; Point delays ; Evolution operator ; Impulsive actions ; Global stability
Published in:	Mathematics, Vol. 13, Issue 3 (February 2025) , art. 365, ISSN 2227-7390

DOI: 10.3390/math13030365

29 p, 401.3 KB

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