CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 3, Number 4, Fall 1995

STABILITY ANALYSIS AND APPLICATIONS TO LARGE SCALE IMPULSIVE SYSTEMS:

A NEW APPROACH

XINZHI LIU AND ALLAN WILLMS

ABSTRACT. Stability criteria employing a combined es-timate of continuous and discrete portions of an impulsive system are established. To show stability for an impulsive system, it is not necessary to find a Lyapunov function whose derivative along the trajectories is negative definite; however, the function must not be allowed to grow too quickly. Herein we have defined and characterized the necessary growth con- dition on the Lyapunov function. These stability results are then extended to large scale impulsive systems. This exten- sion, although analogous to extensions for purely discrete or purely continuous systems, is not straightforward due to the hybrid nature of the impulsive system and provides some ad-ditional difficulties not associated with continuous or discrete systems. Examples are also worked through which show that an impulsive system may exhibit asymptotic stability behav- ior even when both the corresponding continuous system and the discrete system are unstable.

1. Introduction. While most physical systems are modelled either by differential equations or difference equations, the theory of impulsive systems has been emerging as an important area of investigation in recent years [2] . An impulsive system is, in essence, a combination of a differential system and a difference system, which describes both continuous evolutions and discrete events occurring in the model of a physical system. Stability theory employing discontinuous Lyapunov functions has recently been developed for such systems [3,4, 51. These stability results do not require a Lyapunov function, v, to have a negative definite derivative along trajectories of the system, but allow for cases where, for example, v increases during the continuous portion of the trajectory and then experiences a jump decrease a t the impulses. Nonetheless, a condition on v is required to ensure that its growth is not too rapid. Such conditions are essentially placed in [5] on the

Received by the editors in revised form on June 6, 1995. Key words and phmses. Im ulsive s stems, large scale, Lyapunov functions. AMS (1991) Mathematics &b'ect dass,jication. 34A37, 93D05, 93D30. Research supported by N S E R ~Canada.

Copyright 01995 Rocky Mountain Mathematics Consortium

420 X. LIU AND A. WILLMS

discrete and continuous portions separately. In this paper we shall seek conditions that are expressed in terms of the combined continuous and discrete portions of the system. The price to pay for this type of estimate is the necessity of placing a condition on the generalized second derivative of the Lyapunov function, v. However, this new approach is advantageous for the stability analysis of large scale impulsive systems.

This paper is organized as follows. In Section 2 we give some basic notions and introduce the concepts of bounded growth of a function over an interval and at a point of discontinuity, respectively, which characterize the necessary growth conditions on the Lyapunov function. We establish, in Section 3, some general stability criteria for impulsive systems. To illustrate our stability results, an example is worked out which shows that an impulsive system can exhibit asymptotic stability behavior even when both the corresponding continuous system and the discrete system are unstable. In Section 4 we shall utilize our results obtained in the previous section to investigate large scale impulsive systems where we will attempt to characterize the stability properties of the large scale system in terms of the stability properties of its subsystems and bounds on the subsystem interconnections. As will shortly become apparent, this extension, although analogous to the extension for purely continuous or purely discrete large-scale systems, is not straightforward due to the hybrid nature of the impulsive system, and provides some additional difficulties not associated with continuous or discrete systems. For a good exposition on stability theory of continuous or discrete large-scale systems, see [I, 6, 71.

2. Preliminaries. We shall consider the impulsive system

where Ax(&) = x( t t ) - x(tk), k = 1,2,. . . ,I : R+ x Rn I+ Rn, and the impulse times satisfy tl < t2 < ... < tk -+ oo as k + oo. For technical reasons, we shall restrict ourselves to initial conditions of the form X(T;) = xo SO that we are not concerned with an impulse occurring immediately at the start of a trajectory. We remark that this condition could be removed, but initial impulses would have to be dealt with as a special case. We shall also assume that there are an infinite number of impulses; otherwise, after some finite time, the system will be