Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation

A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.


Introduction
Stability problems for non-autonomous systems are very popular in theoretical researches and applications and are difficult enough even in the deterministic case (see, for instance, [1-3, 6-13, 17]). In this paper via the general method of the Lyapunov functionals construction [14][15][16] some new multi-condition of stability in probability is obtained for the zero solution of a nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients. It is shown that the obtained multi-condition of stability gives for one considered equation at once a set of different complementary of each other sufficient stability conditions. Note that other approaches to analyzing stability in random systems are presented for example in [4,18].
Consider the stochastic differential equation [5] dx(t) = a 1 (t, x t )dt + a 2 (t, x t )dw(t), Denote by D the set of functionals, for which the function V ϕ (t, x) defined in (1.3) has a continuous derivative with respect to t and second continuous derivative with respect to x. For functionals from D the generator L of Equation (1.2) has the form [5,16] (1.4) 2) has the zero solution and the following theorems hold. Theorem 1.1. Let there exist a functional V (t, ϕ) ∈ D, positive constants c 1 , c 2 and the function µ(t) such that the following conditions hold: Then the zero solution of Equation (1.2) is asymptotically mean square stable. If, in particular, µ(t) = c 1 e λt , λ > 0, then the zero solution of Equation (1.2) is exponentially mean square stable.
Proof. Integrating the last inequality in (1.5), we obtain EV (t, x t ) ≤ EV (0, φ). So, It means that the zero solution of (1.2) is mean square stable. Besides, from the inequality E|x(t)| 2 ≤ µ −1 (t)EV (0, φ) it follows that the zero solution of (1.2) is asymptotically mean square stable or exponentially mean square stable if µ(t) = c 1 e λt . The proof is completed.
Then the zero solution of Equation (1.2) is stable in probability.
Via Theorems 1.1, 1.2 a construction of stability conditions for a given stochastic differential equation is reduced to construction of appropriate Lyapunov functionals. Via the general method of the Lyapunov functionals construction [14][15][16], below some multi-condition of stability in probability for the zero solution of Equation (1.1) is obtained.

Exponential mean square stability of the linear equation
In this section sufficient conditions of exponential mean square stability are obtained for the linear part of Equation (1.1), i.e., for Equation (1.1) with g(t, x t ) ≡ 0.
Let n i , i = 1, 2, be integers such that 0 ≤ n i ≤ n. Put (2.1) By virtue of S(t) and R k (t, s) defined in (2.1) Equation (1.1) can be presented in the form of a neutral type stochastic differential equation [16] dz(t,

4)
and there exists λ > 0 such that F (t, λ) ≤ 0 then the zero solution of Equation (1.1) is exponentially mean square stable.
Proof. Via (2.4), the zero solution of the auxiliary equationẏ(t) = −S(t)y(t) is exponentially stable and the function v(t) = e λt y 2 (t), λ > 0, is a Lyapunov function for this equation. Following the procedure of the Lyapunov functionals construction [14][15][16], we will construct Lyapunov functional V for Equations (2.2), (2.3) in the x t ) and the additional functional V 2 will be chosen below. Using (1.4) and (2.2) with g(t, x t ) = 0, we have Calculating and estimating z 2 (t, x t ) via (2.3), (2.1), one can show that To neutralize the terms with delays in the estimate of LV 1 consider the additional functional Calculating x t ) = 0 is exponentially mean square stable. The proof is completed.
hold then the zero solution of Equation (1.1) is exponentially mean square stable.
For the proof it is enough to note that (2.5) is equivalent to the condition sup t≥0 F (t, 0) < 0 from which it follows that there exists small enough λ > 0 such that the condition F (t, λ) ≤ 0 holds too.

Stability in probability of the nonlinear equation
In this section it is shown that the sufficient conditions for exponential mean square stability of the linear part of Equation (1.1) also are sufficient conditions for stability in probability of the initial nonlinear equation.
Note that for |x(s)| ≤ ε, s ≤ t, via (1.1) and (2.1) we have Using the additional functional  For the proof it is enough to note that (2.5) is equivalent to the condition sup t≥0 F (t, 0) < 0 from which it follows that there exist small enough λ > 0 and ε > 0 such that Condition (3.1) holds.
Remark 3.1. From 0 ≤ n i ≤ n, i = 1, 2, it follows that the couple (n 1 , n 2 ) in Equation (2.2) has (n + 1) 2 different values. Thus, Theorem 3.1 generally speaking gives (n + 1) 2 different stability conditions at once. Some of these conditions can be infeasible, from some of these conditions can follow some other conditions, the remaining conditions will complement each other.
4 Particular cases of stability condition (2.5) Following Remark 3.1 let us consider some of possible values of the couple (n 1 , n 2 ) and obtain appropriate different stability conditions.
If n 1 = n 2 = 0 then via (2.1) m 1 = m 2 = 0, S(t) = a 0 (t), R k (t, s) = 0, If n 1 = 0, n 2 = n then m 1 = 0, m 2 = n, and from Condition (2.5) we obtain If at last n 1 = n 2 = n then m 1 = m 2 = n, and Condition (2.5) takes the form Using different other combinations of n 1 and n 2 , one can get different other stability conditions. Example 4.1. To demonstrate a possible connection between the obtained different stability conditions consider, for the sake of simplicity, the equation with constant coefficients and without a non-delay term For Equation (4.5) n = 1, so, via Remark 3.1 there are 4 possible stability conditions. Since in Equation (4.5) a 0 = 0 Condition (4.1) does not hold.

Conditions (4.3) take the form
Calculating the integrals in (4.4) separately for a ≥ 0 and a < 0, from Condition (4.4) we obtain  In Fig. 1 stability regions for Equation (4.5), given by Conditions (4.6) (the region (1)), (4.7) (the region (2)) and (4.8) (the region (3)) are shown in the space of the parameters (a, b) for h = 0.5 and p = 0.2. Note that the regions (1) and (3) complement of each other but the region (2) is included in the region (3). It means that Condition (4.8) is less conservative than (4.7). Note also that Condition (4.8) coincides with (4.7) for a = 0 only. In Fig. 2 the similar picture is shown for h = 0.5 and p = 0.55.

Conclusions
In this paper, a nonlinear stochastic non-autonomous differential equation with discrete and distributed delays and the order of nonlinearity higher than one is considered. It is shown that investigation of stability in probability of the nonlinear equation of such type can be reduced to investigation of exponential mean square stability of the linear part of the considered equation. A general multi-condition for stability in probability of the zero solution of the considered equation is obtained which allows in applications to get at once a set of different complementary sufficient stability conditions. Some of these conditions can be infeasible, from some of these conditions can follow some other conditions, the remaining conditions will complement each other. The idea of construction of this multi-condition of stability can be used also for systems of nonlinear stochastic differential equations of such type.