Stability, boundedness and controllability of solutions of measure functional differential equations

Abstract

Generalized ordinary differential equations (we write generalized ODEs), introduced by J. Kurzweil in 1957, are known to encompass several other types of equations as measure functional differential equations, for instance. In this paper, we obtain converse Lyapunov theorems for generalized ODEs and, in particular, for measure functional differential equations which, in turn, encompass impulsive functionals differential equations as well as functional dynamic equations on time scales. We also relate uniform stability to boundedness of solutions. As an application, we establish necessary and sufficient conditions for a system of non–homogeneous nonlinear generalized ODEs defined in a Banach space and for a system of non–homogeneous measure functional differential equations to be asymptotically controllable. We include an example which illustrates the main results.