Existence, stability and controllability of piecewise impulsive dynamic systems on arbitrary time domain

•We formulated a new class of piecewise impulsive dynamic systems on arbitrary time domain by using the time scales theory.•Study the existence of solutions via fixed point theorems. Also establish the stability results of the considered problem.•Provide some necessary and sufficient conditions for...

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Bibliographic Details
Published in:Applied mathematical modelling 2023-05, Vol.117, p.529-548
Main Authors: Kumar, Vipin, Djemai, Mohamed
Format: Article
Language:English
Subjects:
Controllability
Existence
Impulses
Stability
Time scales
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Summary:•We formulated a new class of piecewise impulsive dynamic systems on arbitrary time domain by using the time scales theory.•Study the existence of solutions via fixed point theorems. Also establish the stability results of the considered problem.•Provide some necessary and sufficient conditions for the controllability of the piecewise linear system.•Develop a new class of control function and establish sufficient conditions for the controllability of nonlinear systems.•Some numerical simulations for different time domains are given to verify the proposed theoretical results. The main objective of this article is to establish the existence of solutions, stability, and controllability results for piecewise nonlinear impulsive dynamic systems on an arbitrary time domain. Using the Banach fixed point theorem, we prove the existence of a unique solution while by applying Schauder’s fixed point theorem, we prove the existence of at least one solution. Further, we establish the controllability results by converting the controllability problem into a fixed point problem of an operator equation in some suitable function space. Mainly, we used the fixed point theorems, Gramian type matrices, functional analysis, and time scales theory to establish these results. Since the problem is formulated by using the theory of time scales, therefore, the obtained results are true for the continuous-time domain, discrete-time domain, as well as any combination of these two, which shows that the obtained results are non-trivial and generalize the existing ones. In the last, we have given a simulated example for two different time domains to verify the obtained analytical results.
ISSN:0307-904X
DOI:10.1016/j.apm.2022.12.027