We consider a mathematical model describing the initial stage of a capture into autoresonance in nonlinear oscillating systems with a dissipation. Solutions whose amplitude increases unboundedly in time correspond to a resonance. An asymptotic expansion for such solutions is constructed as a power series with constant coefficients. The stability of autoresonance with respect to persistent perturbations is studied by means of Lapunov's second method. We describe the classes of perturbations for which a capture into autoresonance occurs.