Stability of dynamical systems: A constructive approach

A set A of complex matrices is stable if for every neighborhood of the origin , there exists another neighborhood of the origin , such that for each (the set of finite products of matrices in A), . Matrix and Liapunov stability are related. Theorem: A set of matrices is stable if and only if there exists a norm, , such that for all , and all , . The norm is a Liapunov function for the set . It need not be smooth; using smooth norms to prove stability can be inadequate. A novel central result is a constructive algorithm which can determine the stability of based on the following. Theorem: is a set of m distinct complex matrices. Let Wo be a bounded neighborhood of the origin. Define for , Wk =convexhbull ~ Mk\´Wk - I where . Then A isstableifand only if is bounded. is the norm of the first theorem. The constructive algorithm represents a convex set by its extreme points and uses linear programming to construct the successive . Sufficient conditions for the finiteness of constructing from , and for stopping the algorithm when either the set is proved stable or unstable are presented. is generalized to be any convex set of matrices. A dynamical system of differential equations is stable if a corresponding set of matrices --associated with the logarithmic norms of the matrices of the linearized equations--is stable. The concept of the stability of a set of matrices is related to the existence of a matrix norm. Such a norm is an induced matrix norm if and only if the set of matrices is maximally stable (ie., it cannot be enlarged and remain stable).