About Stability of Nonlinear Stochastic Differential Equations with State-Dependent Delay

: A nonlinear stage-structured population model with a state-dependent delay under stochastic perturbations is investigated. Delay-independent and delay-dependent conditions of stability in probability for two equilibria of the considered system are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The model under consideration is not the aim of the work and was chosen only to demonstrate the proposed research method, which can be used for the study of other types of nonlinear systems with a state-dependent delay.


Introduction
Among different types of delay differential equations, in particular, stochastic delay differential equations, equations with a delay, that depends on the state of the system under consideration, play a special role and are very popular in research (see, for instance, [1][2][3][4][5][6][7][8][9] and the references therein). Here, the method of stability investigation described in [10,11] for nonlinear stochastic differential equations with usual delay is used for investigation of the following stage-structured single population model with a state-dependent delay [8].
Note that although the results obtained here are new, the stability investigation of the considered model here (1) is not the main aim of this paper. This model was chosen to demonstrate the proposed research method, which can be used for the study of many other types of nonlinear systems with a state-dependent delay under stochastic perturbations.

Stochastic Perturbations, Centering and Linearization
In this section, the necessary preliminary steps of the method under consideration are presented.

Stochastic Perturbations
Summing both Equation (1), we havė Substituting (5) into (1), we obtaiṅ where y(t) is a value of the process in the time moment t, and y t is a trajectory of this process until the time moment t. Let us assume that system (6) is exposed to stochastic perturbations that are of the white noise type, are directly proportional to the deviation of the system state (x(t), y(t)) from the equilibrium E(x * , y * ) and influence (ẋ(t),ẏ(t)) immediately. Then, system (6) transforms to the following system of Ito's stochastic delay differential equations [15].
where σ 1 and σ 2 are constants and w 1 (t) and w 2 (t) are mutually independent standard Wiener processes.

Centering
Let E(x * , y * ) be one of the two equilibria of system (8). From (2), it follows that Consider the new variables x 1 (t) and y 1 (t), such that Using (9) and (10), rewrite (7) as follows Substituting (11) into system (8) and using (9), we obtain the following system of nonlinear Ito's stochastic differential equations where Remark 4. It is clear that the stability of the equilibrium E(x * , y * ) of system (7) and (8) is equivalent to the stability of the zero solution of system (12) and (13).

Stability
Following Remark 4, below we consider stability or instability of the zero solution of system (12) and (13) for each of the two equilibria of the initial system (1).

Some Necessary Definitions
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 t 1 ⊂ F t 2 for t 1 < t 2 , and E be the mathematical expectation with respect to the measure P. Definition 1. The zero solution of system (12) is called stable in probability if for any ε 1 > 0 and ε 2 > 0 there exists δ > 0 such that the solution (x 1 (t), y 1 (t)) of system (12) satisfies the condition (17) is called:

Definition 2. The zero solution of Equation
asymptotically mean square stable if it is mean square stable and for each initial value x(0) the solution Z(t) of Equation (17) satisfies the condition lim t→∞ E|Z(t)| 2 = 0.

Remark 6.
Note that the level of nonlinearity of system (12) is higher than one. It is known [14] that in this case, a sufficient condition for asymptotic mean square stability of the zero solution of the linear approximation (17) at the same time is a sufficient condition for stability in probability of the zero solution of system (12). Via Remark 4 to obtain conditions of stability in probability for each of the two equilibria of system (8), it is enough to obtain conditions for asymptotic mean square stability of the zero solution of linear Equation (17). On the other hand, the instability of the zero solution of linear Equation (17) means the instability of the corresponding equilibrium of system (8).

Delay-Independent Condition
Theorem 2. Let there exist positive definite 2 × 2-matrices P and R such that the linear matrix inequality (LMI) holds, where the matrices A, B and C 1 , C 2 are defined in (18). Then, the equilibrium E 1 (x * , y * ) of system (7) and (8) is stable in probability.
Proof. Via Remarks 4 and 6, it is enough to prove that the zero solution of the linear Equation (17) is asymptotically mean square stable. Following the general method of Lyapunov functionals construction [14], let us construct the Lyapunov functional for Equation (17) in the form , P > 0, and the additional functional V 2 (t) will be chosen below. Let L be the generator (see Appendix A) of Equation (17). Then via (19) for V 1 (t), we have =Z T (t)Ψ 1 (P)Z(t) + 2Z T (t)PBZ(t − τ(z * )).
Using the additional functional as a result for the functional V(t) = V 1 (t) + V 2 (t) we obtain where matrix Φ 1 is defined in (19) and η T (t) = (Z T (t), Z T (t − τ(z * ))). The LMI (19), i.e., Φ 1 < 0, holds then there exist c > 0 such that LV(t) ≤ −c|Z(t)| 2 . From Theorem A1 (see Appendix A), it follows that the zero solution of linear Equation (17) is asymptotically mean square stable. The proof is completed.
Theorem 3. Suppose that condition (22) holds, and for some positive definite 2 × 2-matrices P and R the linear matrix inequality (LMI) holds, where matrices A, B and C 1 , C 2 are defined in (18). Then, the equilibrium E 1 (x * , y * ) of system (7) and (8) is stable in probability.
Proof. Via Remarks 4 and 6 it is enough to prove that the zero solution of the linear Equation (21) is asymptotically mean square stable. Following the general method of Lyapunov functionals construction [14], let us construct the Lyapunov functional for Equation (17) in the form V(t) = V 1 (t) + V 2 (t), where V 1 (t) = (Z(t) + BG(t)) T P(Z(t) + BG(t)), P > 0, and the additional functional V 2 (t) will be chosen below.
Let L be the generator (see Appendix A) of Equation (21). Then, via (23) for V 1 (t) we have (24) Note that via Jensen's inequality (Lemma 1) So, for the additional functional Via (23) from (24) and (25) for the functional .

Remark 10.
Note that for stability investigation of the neutral type Equation (21) it is necessary to ensure the exponential stability of the integral equation z(t) = −BG(t) that follows from condition (22). Similarly to [10,19], it can be shown that instead of condition (22) the condition in the form of LMI can be used: if there exists a positive definite matrix S such that the LMI τ 2 (z * )B SB − S < 0 holds. Then, the integral equation z(t) = −BG(t) is exponentially stable. Generally speaking, condition (22) is rougher than this LMI condition, but of course, it is simpler. Moreover, in the scalar case both these conditions coincide.
Via MATLAB for the LMI approach (see [12,13]), it was shown that by the values of the parameters, given in (26), for each of the matrices Φ 1 and Φ 2 there exist positive definite matrices P and R that the LMIs (19) and (23) hold. Moreover, B τ(z * ) = 0.747 < 1. So, via both Theorems 2 and 3, the equilibrium E 1 (7.32, 1.51) of system (8) and (7) is stable in probability.

Example 2. Consider again system
By that, all basic delay properties are preserved: Solving again system (3) with the values of the parameters given in (26), we obtain x * = 8.98, y * = 1.86, τ(z * ) = 3.29.
Via MATLAB it was shown that by the values of the parameters, given in (26), for each of the matrices Φ 1 and Φ 2 there exist positive definite matrices P and R that the LMIs (19) and (23) hold. Moreover, B τ(z * ) = 0.865 < 1. So, via both Theorems 2 and 3, the equilibrium E 1 (8.98, 1.86) of system (8) and (7) is stable in probability.
In Figure 3, 25 trajectories x(t) (blue) and y(t) (green) of the solution of system (7) and (8)   In Figure 4, 25 trajectories x(t) (blue) and y(t) (green) of the solution of system (7)

Remark 11.
Note that by numerical simulation of system (7) and (8) solutions for numerical simulation of trajectories of the standard Wiener processes the special algorithm described in [14] was used.

Conclusions
It is shown how the Lyapunov functionals construction method and the method of linear matrix inequalities (LMIs) can be used for stability and instability investigation of nonlinear systems with a state-dependent delay under stochastic perturbations. Obtained delay-independent and delay-dependent conditions of stability in probability for equilibria of the considered system are formulated in terms of linear matrix inequalities and are illustrated by numerical simulation of solutions of Ito's stochastic differential equation. The proposed method of stability investigation can be successfully used for similar investigations of other types of nonlinear systems with state-dependent delay under stochastic perturbations.
Funding: This research received no external funding.

Conflicts of Interest:
The author declares no conflict of interest.
where x(t) is a value of the solution of Equation (A1) in the time moment t, x t = x(t + s), s < 0, is the trajectory of the solution of Equation (A1) until the time moment t, H 2 is a space of F 0 -adapted functions ϕ(s), s ≤ 0, with continuous trajectories and norm ϕ 2 = sup s≤0 E|ϕ(s)| 2 .
Consider a functional V(t, ϕ) : [0, ∞) × H 2 → R + that can be presented in the form V(t, ϕ) = V(t, ϕ(0), ϕ(s)), s < 0, and for ϕ = x t put V ϕ (t, x) = V(t, ϕ) = V(t, x t ) = V(t, x, x(t + s)), x = ϕ(0) = x(t), s < 0. (A2) Denote by D the set of the functionals, for which the function V ϕ (t, x) defined in (A2) has a continuous derivative with respect to t and two continuous derivatives with respect to x. Let ∇ and ∇ 2 be respectively the first and the second derivatives of the function V ϕ (t, x) with respect to x. For the functionals from D the generator L of Equation (A1) has the form [14,15] LV(t, x t ) = ∂V ϕ (t, x(t)) ∂t + ∇V T ϕ (t, x(t))a(t, x t ) + Theorem A1 ( [14]). Let G(t, ϕ) ≡ 0 and there exist a functional V(t, ϕ) ∈ D, positive constants c 1 , c 2 , c 3 , such that the following conditions hold: Then the zero solution of Equation (A1) is asymptotically mean square stable.