Fixed Points of Self-Mappings with Jumping Effects: Application to Stability of a Class of Impulsive Dynamic Systems

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, 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, Department of Automatic Control and Systems Engineering, Faculty of Engineering of Bilbao, Institute of Research and Development of Processes–IIDP, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain
Izaskun Garrido: Automatic Control Group–ACG, Institute of Research and Development of Processes, Department of Automatic Control and Systems Engineering, Faculty of Engineering of Bilbao, Institute of Research and Development of Processes–IIDP, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain

Mathematics, 2025, vol. 13, issue 7, 1-42

Abstract: This paper studies the boundedness and convergence properties of the sequences generated by strict and weak contractions in metric spaces, as well as their fixed points, in the event that finite jumps can take place from the left to the right limits of the successive values of the generated sequences. An application is devoted to the stabilization and the asymptotic stabilization of impulsive linear time-varying dynamic systems of the n -th order. The impulses are formalized based on the theory of Dirac distributions. Several results are stated and proved, namely, (a) for the case when the time derivative of the differential system is impulsive at isolated time instants; (b) for the case when the matrix function of dynamics is almost everywhere differentiable with impulsive effects at isolated time instants; and (c) for the case of combinations of the two above effects, which can either jointly take place at the same time instants or at distinct time instants. In the first case, finite discontinuities of the first order in the solution are generated; that is, equivalently, finite jumps take place between the corresponding left and right limits of the solution at the impulsive time instants. The second case generates, equivalently, finite jumps in the first derivative of the solution with respect to time from their left to their right limits at the corresponding impulsive time instants. Finally, the third case exhibits both of the above effects in a combined way.

Keywords: impulsive actions; discontinuities of the first kind; dynamic systems; impulsive dynamic systems; global stability; global asymptotic stability; Dirac distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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