Stability of Switching ODEs

Abstract

The main motivational forces for this chapter are the work of Davis [30] on piecewise deterministic systems, and the work of Kac and Krasovskii [79] on stability of randomly switched systems. In recent years, growing attention has been drawn to deterministic dynamic systems formulated as differential equations modulated by a random switching process. This is because of the increasing demands for modeling large-scale and complex systems, designing optimal controls, and carrying out optimization tasks.In this chapter, we consider stability of such hybrid systems modulated by a random-switching process, which are “equivalent” to a number of ordinary differential equations coupled by a switching or jump process.

In this chapter, for random-switching systems, we first obtain suficient conditions for stability and instability. Our approach leads to a necessary and suficient condition for systems whose continuous component is onedimensional. For multidimensional systems, our conditions involve the use of minimal and maximal eigenvalues of appropriate matrices. The difference of maximal and minimal eigenvalues results in a gap for stability and instability. To close this gap, we introduce a logarithm transformation leading to the continuous component taking values in the unit sphere. This in turn, enables us to obtain necessary and suficient conditions for stability. The essence is the utilization of the so called Liapunov exponent.