Practical Stability in Terms of Two Measures for Hybrid Dynamic Systems

We study hybrid dynamic systems on time scales. Using Lyapunov-like functions , we obtain sufficient conditions for practical stability and strict practical stability in terms of two measures for hybrid dynamic systems on time scales. 1. Introduction. In this paper, we present some sufficient conditions for practical stability and strict practical stability in terms of two measures for hybrid dynamic systems on time scales by using Lyapunov-like functions. The practical stability problem, introduced by LaSalle and Lefschetz [12], deals with the question of whether the system state evolves within certain subsets of the state-space. This is very useful in estimating the worst-case transient and steady-state responses and in verifying pointwise in time constraints imposed on the state trajectories. Thus, practical stability is concerned with quantitative analysis as opposed to Lyapunov analysis which is qualitative in nature. Lyapunov analysis may not always be useful in an engineering sense. For instance, an equilibrium point may not be stable in the sense of Lyapunov and yet the system response may be acceptable in the vicinity of this equilibrium; an example involving a van der Pol oscillator is given in [12]. On the other hand, an equilibrium point may be stable in the sense of Lyapunov, but the domain of stability (see [5] for its definition) could be so small as to render the system practically unstable. A striking example of this phenomenon is the Reynolds system used to study the flow of water along a tube of circular cross-section at various speeds [1]. Laminar