On the Evolution Operators of a Class of Time-Delay Systems with Impulsive Parameterizations

Manuel De la Sen (), Asier Ibeas, Aitor J. Garrido and Izaskun Garrido
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Manuel De la Sen: Automatic Control Group–ACG, Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country–UPV/EHU, 48940 Leioa, Bizkaia, Spain
Asier Ibeas: Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, 08193 Barcelona, Spain
Aitor J. Garrido: Automatic Control Group–ACG, Institute of Research and Development of Processes–IIDP, Department of Automatic Control and Systems Engineering, Faculty of Engineering of Bilbao, University of the Basque Country–UPV/EHU, Po Rafael Moreno no3, 48013 Bilbao, Bizkaia, Spain
Izaskun Garrido: Automatic Control Group–ACG, Institute of Research and Development of Processes–IIDP, Department of Automatic Control and Systems Engineering, Faculty of Engineering of Bilbao, University of the Basque Country–UPV/EHU, Po Rafael Moreno no3, 48013 Bilbao, Bizkaia, Spain

Mathematics, 2025, vol. 13, issue 3, 1-32

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.

Keywords: delay differential systems; point delays; evolution operator; impulsive actions; global stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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