Stability theory

Abstract

This chapter presents the study of the stability theory for generalized ordinary differential equations (ODEs). The results on the stability of the trivial solution in the framework of the generalized ODE are inspired in the theory, developed by Aleksandr M. Lyapunov on the stability of solutions for classic ODEs. Converse Lyapunov theorems confirm the effectiveness of the Direct Method of Lyapunov. The chapter presents a concept of a Lyapunov functional for generalized ODEs. The concept of variational stability for ordinary differential equations was introduced by H. Okamura in 1943, who called it strong stability. The chapter presents direct methods of Lyapunov for uniform stability and for uniform asymptotic stability of the trivial solution of the measure differential equations. It examines the concepts of Lyapunov stability in the framework of dynamic equations on time scales. The chapter provides the results for measure differential equations and for dynamic equations on time scales.