State-Feedback Stabilization for a Class of Stochastic Feedforward Nonlinear Time-Delay Systems

Definition 1 (see [6].)For any given V(x(t), t) ∈ 𝒞^2,1 associated with system (2), the differential operator ℒ is defined as

Definition 2 (see [6].)The equilibrium x(t) = 0 of system (2) is said to be globally asymptotically stable (GAS) in probability if for any ϵ > 0 there exists a function β(·, ·) ∈ 𝒦ℒ such that P{|x(t)| ≤ β(∥ξ∥, t)} ≥ 1 − ϵ for any t ≥ 0, $$, where ∥ξ∥ = sup _{θ∈[−d,0]}|x(θ)|.

Definition 3 (see [26].)For fixed coordinates (x₁, …,  x_n) ^T ∈ Rⁿ and real numbers r_i > 0, i = 1, …, n, one has the following.

• (i)

  The dilation Δ_ɛ(x) is defined by $$ for any ɛ > 0; r₁, …, r_n are called as the weights of the coordinates. For simplicity, we define dilation weight Δ = (r₁, …, r_n).

• (ii)

  A function V ∈ 𝒞(Rⁿ, R) is said to be homogeneous of degree τ if there is a real number τ ∈ R such that V(Δ_ɛ(x)) = ɛ^τV(x₁, …, x_n) for any x ∈ Rⁿ∖{0}, ɛ > 0.

• (iii)

  A vector field h ∈ 𝒞(Rⁿ, Rⁿ) is said to be homogeneous of degree τ if there is a real number τ ∈ R such that $$ for any x ∈ Rⁿ∖{0}, ɛ > 0, i = 1, …, n.

• (iv)

  A homogeneous p-norm is defined as $$ for any x ∈ Rⁿ, where p ≥ 1 is a constant. For simplicity, in this paper, one chooses p = 2 and writes ∥x∥_Δ for ∥x∥_Δ,2.

Lemma 4 (see [6].)For system (2), if there exist a function V(x(t), t) ∈ 𝒞^2,1(Rⁿ × [−d, ∞); R₊), two class 𝒦_∞ functions α₁, α₂, and a class 𝒦 function α₃ such that

then there exists a unique solution on [−d, ∞) for (2), the equilibrium x(t) = 0 is GAS in probability, and P{lim _t→∞|x(t)| = 0} = 1.

Lemma 5 (see [26].)Given a dilation weight Δ = (r₁, …, r_n), suppose that V₁(x) and V₂(x) are homogeneous functions of degrees τ₁ and τ₂, respectively. Then V₁(x)V₂(x) is also homogeneous with respect to the same dilation weight Δ. Moreover, the homogeneous degree of V₁ · V₂ is τ₁ + τ₂.

Lemma 6 (see [26].)Suppose that V : Rⁿ → R is a homogeneous function of degree τ with respect to the dilation weight Δ; then (i) ∂V/∂x_i is homogeneous of degree τ − r_i with r_i being the homogeneous weight of x_i; (ii) there is a constant c such that $$. Moreover, if V(x) is positive definite, then $$, where $$ is a positive constant.

Lemma 7 (see [5].)Let c and d be positive constants. For any positive number $$, then |x|^c|y|^d≤$$.

Theorem 12. If Assumptions 8 and 9 hold for the stochastic feedforward nonlinear time-delay system (1), under the state-feedback controller u = κⁿv and (13), then

• (i)

  the closed-loop system has a unique solution on [−d, ∞);

• (ii)

  the equilibrium at the origin of the closed-loop system is GAS in probability.

Proof. We prove Theorem 12 by four steps.

Step  1. Since f_i, g_i, i = 1, …, n, are assumed to be locally Lipschitz, so the system consisting of (13) and (15) satisfies the locally Lipschitz condition.

Step  2. We consider the following entire Lyapunov function for system (15):

where $$ and $$ are positive parameters to be determined. It is easy to verify that V(η) is 𝒞² on η. Since V_n(η) is continuous, positive definite, and radially unbounded, by Lemma 4.3 in [31], there exist two class 𝒦_∞ functions β₁ and α₂₁ such that

By Lemma 4.3 in [31] and Lemma 6, there exist positive constants $$ and $$, class 𝒦_∞ functions $$ and $$, and a positive definite function U(η) whose homogeneous degree is 4 such that

From d(t) : R₊ → [0, d] and (19), it follows that

where $$, c are positive constants and α₂₂ is a class 𝒦_∞ function. Since |η| ≤ sup _−d≤s≤0|η(s + t)|, α₂₁(|η|) ≤ α₂₁(sup _−d≤s≤0|η(s + t)|). Defining β₂ = α₂₁ + α₂₂, by (17), (18), and (20), one gets

Step  3. By Lemma 6 and (14), there exists a positive constant c₀₁ such that

By Assumption 8, (6), and 0 < κ < 1, one has

where δ₁ is a positive constant. Using Lemmas 5–7 and (23), one gets where c₀₂, $$, and $$ are positive constants. Similar to (23), there is a positive constant δ₂ such that from which and Lemmas 5–7, one gets where c₀₃, $$, and $$ are positive constants. With the help of (3), (15), (17), (22), (24), (26), and Assumption 9, one has

Since c₀₁ is a constant independent of c₀₂, c₀₃, $$, $$, and γ, by choosing

Equation (27) becomes $$, where c₀ is a positive constant. By (19), one obtains

By Steps 1–3 and Lemma 4, the system consisting of (13) and (15) has a unique solution on [−d, ∞), η = 0 is GAS in probability, and P{lim _t→∞|η| = 0} = 1.

Step  4. Since (6) is an equivalent transformation, so the closed-loop system consisting of (1), u = κⁿv, and (13) has the same properties as the system (13) and (15). Theorem 12 holds.

Remark 13. In this paper, the homogeneous domination idea is generalized to stochastic feedforward nonlinear time-delay systems (1). The underlying philosophy of this approach is that the state-feedback controller is first constructed for system (7) without considering the drift and diffusion terms, and then a low gain κ in (6) (whose the value range is (28)) is introduced to state-feedback controller to dominate the drift and diffusion terms.

Remark 14. Due to the special upper-triangular structure and the appearance of time-varying delay, there is no efficient method to solve the stabilization problem of system (1). By combining the homogeneous domination approach with stochastic nonlinear time-delay system criterion, the state-feedback stabilization of system (1) was perfectly solved in this paper.

Remark 15. One of the main obstacles in the stability analysis is how to deal with the effect of time-varying delay. In this paper, by constructing an appropriate Lyapunov-Krasovskii functional (17), this problem was effectively solved.

Remark 16. It is worth pointing out that the rigorous proof of Theorem 12 is not an easy job.