Lyapunov functions in stability of nonlinear systems on time scales

Abstract

Uniform stability and uniform asymptotic stability of nonlinear finite-dimensional systems on arbitrary time scales are studied. Sufficient as well as necessary conditions of stability are given with the aid of Lyapunov functions. They extend known stability criteria for continuous-time and discrete-time systems. The Massera lemma, needed to construct a Lyapunov function for uniformly asymptotically stable systems, is extended to time scales. The role played by the graininess function associated to the time scale is discussed. Some peculiarities that appear for time scales with an unbounded graininess function are exhibited.

Keywords:

• asymptotic stability
• Lyapunov function
• time scale

AMS Subject Classification::

• 93D30
• 93D20
• 93D05
• 93C15
• 93C55

Acknowledgement

This work was supported by the Bialystok Technical University grant No. W/WI/1/07