Unpredictability in Perturbed Quasilinear Systems with Regular Moments of Impulses

TL;DR

This paper proves that quasilinear impulsive systems with regular impulse moments can have unpredictable solutions when perturbed by unpredictable sequences, establishing their existence, uniqueness, and stability.

Contribution

It introduces a new definition for unpredictable functions with regular discontinuities and demonstrates the existence of unpredictable solutions using a Gronwall type inequality.

Findings

Unpredictable solutions exist in perturbed quasilinear impulsive systems.

The solutions are unique and asymptotically stable.

A novel approach using Gronwall inequality supports the theoretical results.

Abstract

It is rigorously proved that quasilinear impulsive systems possess unpredictable solutions when a perturbation generated by an unpredictable sequence is applied. The existence, uniqueness, as well as asymptotic stability of such solutions are demonstrated. The system under consideration is with regular moments of impulses, and for that reason a novel definition for unpredictable functions with regular discontinuity moments is provided. To show the existence of an unpredictable solution a Gronwall type inequality for piecewise continuous functions is utilized. The theoretical results are supported with an illustrative example.