Taming Heterogeneity and Complexity of Embedded Control

Exponential Stability of Dynamic Systems on Time Scales

Zbigniew Bartosiewicz^a, Ewa Piotrowska^a1, Alan S. I. Zinober^b

^aInstitute of Mathematics and Physics, Białystok Technical University, Wiejska 45, 15-351 Białystok, Poland

^bDepartment of Applied Mathematics, The University of Sheffield, Sheffield S10 2TN, UK

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ABSTRACT. The problem of exponential stability of linear dynamics systems on time scales is studied. The stability characteristics of an autonomous linear system of differential or difference equations can be characterized completely by the placement of the eigenvalues of the system matrix. The sufficient condition is that the eigenvalues of the system matrix should be contained in the set of exponential stability, which may change for each time scale on which the system is studied. The boundedness of graininess function guarantees that the set of exponential stability is not empty. In general, the placement of eigenvalues of the system does not guarantee the stability of any time varying systems. Generalization of the Lyapunov criteria for uniform exponential stability of discrete and continuous linear systems results in the sufficient condition for uniform exponential stability for regressive time varying linear dynamic systems.

KEYWORDS: dynamic systems, exponential stability, time scales

1. Introduction

The theory of time scales allows to unify the seemingly disparate fields of discrete and continuous dynamical ...