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

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 con-trollability 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, func-tional analysis, and time scales theory to establish these results. Since the problem is for-mulated 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.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

The main objective of this article is to establish the existence, stability, and controllability results for piecewise nonlinear impulsive dynamic systems on an arbitrary time domain. The existence of a unique solution is proved using the Banach fixed point theorem, while the existence of at least one solution is proved using Schauder's fixed point theorem. Further, the controllability results are established by converting the controllability problem into a fixed point problem of an operator equation in some suitable function space. The obtained results are true for continuous-time domain, discrete-time domain, as well as any combination of these two, and generalize the existing ones.