On the Boundedness of Solutions of Discontinuous Differential Equations

Iguer Santos

doi:10.5540/tcam.2021.022.03.00413

Abstract

By using the notion of Carathéodory solution to differential equations, the present work studies the boundedness of solutions of discontinuous differential equations. For these discontinuous systems determined by discontinuous differential equations, results are obtained that guarantee sufficient conditions to boundedness of solutions in terms of nonsmooth Lyapunov functions.

Highlights

Ordinary differential equations with the discontinuous right side, called discontinuous differential equations, determine discontinuous systems
The boundedness of solutions to discontinuous systems determined by autonomous differential equations was studied by [2]
Motivated by the study of the boundedness of solutions for ordinary differential equations in terms of smooth Lyapunov functions, this report studies the boundedness of solutions of discontinuous systems determined by nonautonomous differential equations

Summary

INTRODUCTION

Ordinary differential equations with the discontinuous right side, called discontinuous differential equations, determine discontinuous systems. The boundedness of solutions to discontinuous systems determined by autonomous differential equations was studied by [2]. On the other hand, [3], [12] and [7] address Lyapunov stability of equilibria of discontinuous systems by using the notion of Caratheodory solution. Where f : R × Rn → Rn and g : Rn → Rn. For the study of the boundedness of solutions of the system (1.1) and (1.2), it is assumed throughout the work that for every x0 ∈ Rn, each of the differential equations given in (1.1) and (1.2) admit at least one Caratheodory solution with the initial condition x(t0) = x0. In the present study results are established (Theorems 3.1, 3.2, 3.3 and 3.4) that provide sufficient conditions for the boundedness of solutions of (1.1) and (1.2) in terms of nonsmooth Lyapunov functions. Theorems 3.1 and 3.3 establish results for the boundedness of solutions of (1.1), while Theorems 3.2 and 3.4 establish results for the boundedness of solutions of (1.2)

Caratheodory solution

Nonpathological functions

BOUNDEDNESS OF SOLUTIONS

Findings

CONCLUSIONS