Аннотация:The authors consider the perturbed system (1) x ˙=F(t,x,μ), where x=(x 1 ,⋯,x n ) and μ is a small nonnegative parameter. The paper assumes that for each fixed μ∈[0,μ * ] the right-hand side F:ℝ + ×D×[0,μ * ]→ℝ n , where ℝ + =[0,+∞) and D is some neighborhood of the origin in ℝ n , satisfies the Carathéodory condition. Along with the perturbed system (1), the authors consider the nonperturbed system (2) x ˙=F 0 (t,x)≡F(t,x,0).
In addition to the conditions cited above they assume that the Cauchy problem for system (2) is uniquely solvable. The following results are obtained:
(1) If system (1) is μ-stable (respectively, uniformly μ-stable) with respect to x=0, then the nonperturbed system (2) has the zero solution which is Lyapunov stable (respectively, uniformly stable).
(2) If the right-hand side of the nonperturbed system satisfies a Lipschitz condition in x uniformly with respect to (t,x)∈G and the zero solution of the system is uniformly asymptotically stable, then system (1) is uniformly asymptotically μ-stable.