On the existence, uniqueness and nature of Caratheodory and Filippov solutions for bimodal piecewise affine dynamical systems

TL;DR

This paper investigates the conditions for existence, uniqueness, and the nature of solutions, including Zeno behavior, in bimodal piecewise affine systems within a differential inclusion framework.

Contribution

It provides necessary and sufficient conditions for the uniqueness of Filippov solutions in bimodal piecewise affine systems, addressing well-posedness and solution behavior.

Findings

Uniqueness conditions for Filippov solutions are highly restrictive.

Established necessary and sufficient conditions for solution uniqueness.

Analyzed Zeno behavior in the context of bimodal systems.

Abstract

In this paper, we deal with the well-posedness (in the sense of existence and uniqueness of solutions) and nature of solutions for discontinuous bimodal piecewise affine systems in a differential inclusion setting. First, we show that the conditions guaranteeing uniqueness of Filippov solutions in the context of general differential inclusions are quite restrictive when applied to bimodal piecewise affine systems. Later, we present a set of necessary and sufficient conditions for uniqueness of Filippov solutions for bimodal piecewise affine systems. We also study the so-called Zeno behavior (possibility of infinitely many switchings within a finite time interval) for Filippov solutions.