Stability of dynamical systems: A constructive approach

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).