An iterative method of Lyapunov function construction for differential inclusions (Q1093069)

The authors are concerned with sufficient conditions for the exponential stability of the zero solution of a differential inclusion of the form \(\dot x\in F(x)\), where \(x\in R^ n\), and F(x) denotes a set in \(R^ n\) of the form Ax where A is an \(n\times n\) matrix; this set F(x) is assumed to be the convex hull of a finite set of \(n\times n\) matrices \(A_ s\), \(s=1,...,M\). The conditions are in terms of positive definite quadratic Lyapunov functions which will exist with the required properties if a certain function of the matrix elements defining these Lyapunov functions and vectors in \(R^{nM}\) has a certain type of saddle point. A search algorithm for such saddle points is developed, as is an iterative algorithm for the construction of the Lyapunov function. An example and a generalization are also discussed.