Unpredictable Solutions of Quasilinear Systems with Discontinuous Right-Hand Sides

TL;DR

This paper proves the existence and uniqueness of unpredictable solutions in quasilinear systems with discontinuous right-hand sides, using a Gronwall-Coppel inequality and analyzing stability, with examples demonstrating different stability behaviors.

Contribution

It introduces a rigorous framework for unpredictable solutions in systems with discontinuous perturbations defined by unpredictable sequences, extending existing theory.

Findings

Existence and uniqueness of unpredictable solutions established.

Stability analysis of unpredictable solutions conducted.

Examples show both stable and unstable unpredictable solutions.

Abstract

It is rigorously proved under certain assumptions that a quasilinear system with discontinuous right-hand side possesses a unique unpredictable solution. The discontinuous perturbation function on the right-hand side is defined by means of an unpredictable sequence. A Gronwall-Coppel type inequality is utilized to achieve the main result, and the stability of the unpredictable solution is discussed. Examples with exponentially asymptotically stable and unstable unpredictable solutions are provided.