PERTURBATIONS OF ASYMPTOTICALLY STABLE DIFFERENTIAL SYSTEMS

We consider systems of differential equations of the form (1) x' = f(t,x) , for t e [ a,°° ) , χ in some domain D C 3R and f e c 1 ([ a,°°) x D) (a a fixed real number). We assume that the solution x(t,t 0,x 0) of (1) defined for t a satisfies |x(t,t0,x0)| <_ c | x 0 | h (t)h (t0) 1 (t ^ t 0 a) for x0 small enough, for some constant c > 0 and h a continuous positive function defined in [a,°°) . We give conditions for the perturbed system y' = f(t,y) + g(t,y) (g ε c ( [ a,°°) x D) ) to have the same type of stability as (1) . 1980 Mathematical subject classification. 34 C 11; 34 D 10. 1. Alekseev's formula generalizes the variation of constants formula, and permits the study of a nonlinear perturbation of a system with certain stability properties. Nevertheless, in the study of asymptotic stability it is difficult to work with non exponential types of stability