Stability with Regime Switching

Abstract

The purpose of this paper is to analyze stability of a system of piecewise continuous differential equations, and its application to disequilibrium economic models. A unique solution in the sense of Filippov for such a system is defined and claimed to exist. This problem frequently appears in disequilibrium models, since the so-called " shortside" rule assigns either demand or supply to the transaction amount which is a state variable of an economic system. The concept of Filippov solution makes it possible to analyze a dynamic evolution of such a model. This paper demonstrates that (i) stability conditions for each piecewise system of differential equations are neither necessary nor sufficient for the overall stability with regime switching, except special cases such as a system of linear differential equations in R2, with two regimes separated by a linear boundary; (ii) several sufficient conditions for an overall stability with many regimes are available, making use of a Lyapunov function common to all regimes; (iii) stability theorems with regime switching are useful for disequilibrium economic models with several regimes.