On the Almost Sure Stability of Dynamical Systems with Respect to White Noise
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ON THE ALMOST SURE STABILITY OF DYNAMICAL SYSTEMS WITH RESPECT TO WHITE NOISE O. A. Sultanov
UDC 517.925.51
Abstract. We consider a system of ordinary differential equations possessing an asymptotically stable equilibrium position and examine the stability of this equilibrium under permanent stochastic whitenoise-type perturbations. The perturbed system is considered in the form of Stratonovich stochastic differential equations. We also describe restrictions on the parameters of perturbations that guarantee the trajectory-wise stability of the equilibrium with probability 1. Keywords and phrases: dynamical system, perturbation, white noise, stability. AMS Subject Classification: 93E15, 34D10, 60H10
1.
Statement of the problem. We consider the system of ordinary differential equations dx = f (x, t), x = (x1 , . . . , xn )T ∈ Rn , t ≥ 0, (1) dt which has the trivial solution x(t) ≡ 0. We assume that the vector-valued function f = (f1 , . . . , fn )T is continuous and has bounded partial derivatives: ∂x f (x, t) ≤ M1 for all x ∈ Rn and t ≥ 0, where M1 > 0 is a constant. We assume that for the system considered, there exists a Lyapunov function V (x, t) satisfying the following estimates: n 2 ∂V ∂V x ≤ V (x, t) ≤ α|x|2 , ∂x V ≤ β|x|, dV def + = fk ≤ −γV (2) dt (1) ∂t ∂xk k=1
for |x| ≤ ρ and t ≥ 0, where α, β, γ, ρ = const > 0. The existence of such a function implies the local asymptotic stability of the trivial solution. We examine the question on the preservation of stability of this solution in the presence of permanent random perturbations of the white-noise type. Together with (1), we consider the following system of stochastic differential equations in the Stratonovich form (see [12]): dy(t) = f (y(t), t) dt + μ G(y(t), t) ∗ dW (t),
t > 0,
y(0) = ξ ∈ Rn ,
(3)
trajectories where W (t) ≡ (W1 (t), . . . , Wn (t))T is an n-dimensional Wiener process with continuous defined on the probability space (Ω, F(Ft ), P), G(y, t) = diag y1 σ1 (t), . . . , yn σn (t) is a given diagonal matrix with continuous elements, which is independent of ω ∈ Ω, and 0 < μ 1 is a small control parameter of the intensity of perturbations. These restrictions on the coefficients of the system (3) guarantee the existence and uniqueness of a continuous solution with probability 1 for all t > 0 (see, e.g., [12, Sec. 5.2]). The purpose of this paper is to find conditions for elements of the matrix G(y, t) under which almost all trajectories of the solution y(t) with sufficiently small initial data remain in a small neighborhood of the trivial solution x(t) ≡ 0 of the deterministic system (1). We note that stability problems for dynamical systems under stochastic perturbations were studied in many papers. In particular, the stability of solutions of stochastic differential equations in probability was discussed in [5, 6, 8]. The asymptotic stability at large time of dynamical systems in the probability under the action of a white noise was studied in [4, 13, 15]. Problems on the almost Translated from Itogi Nauki i Tekhn
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