STABILITY OF DELAY DIFFERENTIAL EQUATIONS WITH FADING STOCHASTIC PERTURBATIONS OF THE TYPE OF WHITE NOISE AND POISSON’S JUMPS

. Delay diﬀerential equation is considered under stochastic perturbations of the type of white noise and Poisson’s jumps. It is shown that if stochastic perturbations fade on the inﬁnity quickly enough then suﬃcient conditions for asymptotic stability of the zero solution of the deterministic diﬀerential equation with delay provide also asymptotic mean square stability of the zero solution of the stochastic diﬀerential equation. Stability conditions are obtained via the general method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). Investigation of the situation when stochastic perturbations do not fade on the inﬁnity or fade not enough quickly is proposed as an unsolved problem.


Introduction
Let {Ω, F, P} be a complete probability space, {F t } t≥0 be a nondecreasing family of sub-σ-algebras of F, i.e., F s ⊂ F t for s < t, P{·} be the probability of an event enclosed in the braces, E be the mathematical expectation, H 2 be the space of F 0 -adapted stochastic processes ϕ(s), s ≤ 0, ϕ 2 = sup s≤0 E|ϕ(s)| 2 .
To explain the idea of the proposed investigation note that the second moment y(t) = Ex 2 (t) of the solution x(t) of the scalar stochastic differential equation and can be represented in the form Via (1.4) we obtain the following   shown for In Figure 1.3 50 trajectories (blue) of the equation (1.2) solution are shown for 2) is asymptotically mean square stable and all trajectories converge to zero.
Below Statement 1.1 is generalized on the equation (1.1) and one unsolved problem in this direction is formulated too. Note also that in [5] a similar research was considered for stochastic difference equations.

Auxiliary statements and definitions
Let D be the set of the functionals, for which the function V ϕ (t, x) defined by (2.1) has a continuous derivative with respect to t and two continuous derivatives with respect to x. The generator L of the equation (1.1) is defined on the functionals from D as follows [1,3,4] -mean square stable if for each ε > 0 there exists a δ > 0 such that E|x(t, φ)| 2 < ε, t ≥ 0, provided that φ 2 < δ; -asymptotically mean square stable if it is mean square stable and lim t→∞ E|x(t, φ)| 2 = 0 for each initial function φ.

Then the zero solution of the equation (1.1) is asymptotically mean square stable.
Schur complement [2]. The symmetric matrix

Main result
Note that in the case of an autonomous system (C(t) = C, G(t, u) = G(u)) conditions of asymptotic mean square stability for the equation (1.1) zero solution can be obtained in the form of the matrix Riccati inequality [3] A ′ P + P A + C ′ P C + G ′ (u)P G(u)Π(du) + R + P BR −1 B ′ P < 0, P > 0, R > 0, which via Schur complement can be reformulated also in the form of LMI [4] Below we will consider non-autonomous system and will suppose that stochastic perturbations in the equation (1.1) quickly enough fade on the infinity, i.e., the function ρ(t) is integrable on [0, ∞).
Theorem 3.1. Let there exist positive definite n × n-matrices P , R and the functions σ(t) and γ(t, u) such that the following inequalities hold:

3). Then the zero solution of the equation (1.1) is asymptotically mean square stable.
Proof. Following the general method of Lyapunov functionals construction [3] we will construct the Lyapunov functional V for the equation (1.1) in the form V = V 1 + V 2 , where V 1 (x(t)) = e − t 0 ρ(s)ds x ′ (t)P x(t). Let L be the generator of the equation (1.1). Then via (2.2), two first inequalities (3.1) and (1.3) for V 1 we have Using the additional functional So, the constructed functional V (x t ) satisfies the conditions of Theorem 2.1. Therefore, the zero solution of the equation (1.1) is asymptotically mean square stable. The proof is completed.

Conclusions
The effect of fading stochastic perturbations of the type of white noise and Poisson's jumps on asymptotically stable delay differential equation is investigated. It is shown that if stochastic perturbations fade on the infinity quickly enough then condition for asymptotic stability of the deterministic system provides also asymptotic mean square stability of the considered system under stochastic perturbations. Consideration of the situation with stochastic perturbations which do not fade on the infinity or fade not enough quickly is proposed as an unsolved problem.