Common diagonal Lyapunov function for third order linear switched system

Consider the stability problem for the following linear switched system (differential inclusion) ẋ=Ax,A∈{A1,A2,…,AN}. Here Ai (i=1,2,…,N) are n×n dimensional Hurwitz stable real matrices. In this study for this system we investigate the problem of the existence and construction of a common diagonal Lyapunov function of the form V(x)=xTDx where D is a positive diagonal matrix. In the case of n=3, i.e. third order system, we suggest a simple elimination algorithm which gives a common D in the case of existence.

► We consider n-dimensional linear switched systems. ► Construction of a common diagonal Lyapunov function for the systems is considered. ► A simple elimination algorithm for third order system is given. ► For the general case, we apply Kelly's cutting-plane method to the given problem.