Conditions for the Uniform Asymptotic Stability of Delay Differential Equations

Abstract

_x(t )= f (t;xt) ;f (t; 0) 0; (1) where, as usual, x2 R n , t2 R+, f (t; ): C ([ r(t); 0];R n )! R n , xt( )= x(t + ), 2 [ r(t); 0], and C ([ r(t); 0];R n ) is the space of continuous functions on the interval [ r(t); 0]. We suppose that f is a function providing the existence of solutions x (t;t0;’ )o f Eq. (1) at least on some interval (t0;t0 +)] for arbitrary initial data (t0;’ )w itht02 R+ andk’kt0 H. In addition, we assume that r 0 (t) t 0 and "> 0, there exists a (t0) > 0 such that jx (t;t0’)j" wheneverk’kt0 <(t0), (ii) uniformly stable if it is stable for every t02 R+ and (t0) can be chosen to be independent of t0,