مرکز منطقه ای اطلاع رساني علوم و فناوري - Stability of dynamical systems: A constructive approach

Abstract :

A set A of $n \\times n$ complex matrices is stable if for every neighborhood of the origin $U \\subset C^{n}$ , there exists another neighborhood of the origin $V$ , such that for each $M \\in A\´$ (the set of finite products of matrices in A), $MV\\subset U$ . Matrix and Liapunov stability are related. Theorem: A set of matrices $A$ is stable if and only if there exists a norm, $|\\cdot |$ , such that for all $M \\in A$ , and all $z \\in C^{n}$ , $|Mz| < |z|$ . The norm is a Liapunov function for the set $A$ . 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 $A$ based on the following. Theorem: $A,={M0,Mj,. .,Mmi)$ is a set of m distinct complex matrices. Let Wo be a bounded neighborhood of the origin. Define for $k > 0$ , Wk =convexhbull ~ Mk\´Wk - I where $k\´=(k- 1) mod m$ . Then A isstableifand only if $V-U Q,_$ is bounded. $W*$ 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 $W_k$ . Sufficient conditions for the finiteness of constructing $W_k$ from $W_{k-1}$ , and for stopping the algorithm when either the set is proved stable or unstable are presented. $A$ 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).