Two theorems in stability theory

Abstract

In studies concerned with determining the stability of solutions of some systems of linear differential equations, it often is helpful to be able to transform the given system into another whose stability properties are known or are more easily obtainable. It is of interest, therefore, to consider transformations usable for this purpose, and to develop criteria to indicate when one system has the same stability properties as another. In Theorem 1 we give conditions sufficient to identify a Lyapunov transformation if certain properties are possessed by the coefficient matrix in the linear differential equation satisfied by the transformation, and in Theorem 2 we present an extension of Erugin's theorem [1, p. 122]. However, we need a few definitions and some preliminary work first. DEFINITION. A nonsingular matrix Q = Q(t) is said to be a Lyapunov transformation if it is continuously differentiable for all t ?0 and if Q, Q-1, Q', and (Q-1)' are each bounded. We designate the class of all such by the symbol L(0, oo). (Primes indicate differentiation.) DEFINITION. A matrix X(t) is said to be reducible to a matrix Y(t) if both are defined for all t >0 and if there exists a Lyapunov transformation such that X-Q Y for all t ? 0. When this holds, clearly Y is reducible to X also. DEFINITION. Matrices A (t) and B(t) are said to be L-equivalent to each other if both are defined for t ? 0 and if there exists a Lyapunov transformation Q Q(t) such that B = Q-A Q Q-'Q' in which case it is easy to see that we also have A = P-'BP -P-'P' where P=Q-1. In defining the class L(0, oo), we follow Gantmacher [1] and Lyapunov [2] rather than Nemytskii and Stepanov [3]. Although L(0, oo) is a restricted class of transformations it is a very useful one, for Lyapunov (loc. cit.) showed that matrices in this class preserve the stability properties of the Zero solution of a linear system of differential equations. More precisely, if X' =A(t)X and Y'=B(t) Y and if X = Q Y for some Q in L(0, oo ) then the Zero solution for Y is asymptotically stable, stable, or unstable according as the Zero solution for X is. Gantmacher [1, p. 117] points out that for fixed n,