Lyapunov Instability for Discontinuous Differential Equations

Iguer Luis Domini Dos Santos

doi:10.22481/intermaths.v2i2.9811

Abstract

The present work studies the Lyapunov instability for discontinuous differential equations through the use of the notion of Carathéodory solution to differential equations. From Lyapunov's first instability theorem and Chetaev's instability theorem, which deal with instability to ordinary differential equations, two Lyapunov instability results for discontinuous differential equations are obtained.

Highlights

Discontinuous differential equations are ordinary differential equations with the discontinuous right side and determine discontinuous systems
The study of the Lyapunov stability to discontinuous differential equations using the notion of Carathéodory solution can be found in [3] and [6]
Based on instability results for ordinary differential equations, the present work studies the instability for discontinuous systems determined by x (t) = f (t, x(t))

Summary

INTRODUCTION

Discontinuous differential equations are ordinary differential equations with the discontinuous right side and determine discontinuous systems. Such differential equations are treated, for example, in [1], [2], [3], [4], [5] and [6]. The study of the Lyapunov stability to discontinuous differential equations using the notion of Carathéodory solution can be found in [3] and [6]. Based on instability results for ordinary differential equations, the present work studies the instability for discontinuous systems determined by x (t) = f (t, x(t)). The notion of Carathéodory solution to Eq (1) is used here. In [4] we can find the study on the continuation of solutions

Absolutely continuous functions

Extension of solutions

LYAPUNOV INSTABILITY

CONCLUSIONS