Asymptotic stabilization of stochastic high-order upper-triangular nonlinear systems with input time-varying delay

Abstract

Based on the homogeneous domination approach and stochastic nonlinear time-delay system stability criterion, this paper investigates the global state-feedback stabilization problem for a class of stochastic high-order upper-triangular nonlinear systems with input time-varying delay. By skilfully choosing an appropriate Lyapunov–Krasoviskii functional and successfully solving several troublesome obstacles in the design and analysis procedure, a delay-independent state-feedback controller is designed to render the closed-loop system globally asymptotically stable in probability. The simulation example is given to verify the effectiveness of the proposed design scheme.

Keywords

Stochastic high-order upper-triangular nonlinear systems input time-varying delay state feedback homogeneous domination approach

Introduction

In this paper, we consider the following stochastic high-order upper-triangular nonlinear systems with input time-varying delay described by

\begin{array}{l} d x_{i} = x_{i + 1}^{p_{i}} d t + f_{i} (t, {\bar{x}}_{i + 2}, u (t - τ (t))) d t \\ + g_{i}^{T} (t, {\bar{x}}_{i + 1}, u (t - τ (t))) d ω \\ d x_{n - 1} = x_{n}^{p_{n - 1}} d t + f_{n - 1} (t u (t - τ (t))) d t \\ + g_{n - 1}^{T} (t, {\bar{x}}_{n}, u (t - τ (t))) d ω \\ d x_{n} = u^{p_{n}} d t, i = 1, \dots, n - 2 \end{array}

(1)

where $x = (x_{1}, \dots, x_{n})^{T} \in R^{n}$ and $u \in R$ are the system state and control input, respectively. ${\bar{x}}_{i} = (x_{i}, \dots, x_{n})^{T}$ , $i = 2, \dots, n$ , $τ (t) : R_{+} \to [0, τ_{0}]$ is the time-varying delay. $p_{i} \in R^{*} = : {q \in R_{+} : q \geq 1$ is a ratio of odd integers } is the high-order of system. $ω$ is an m-dimensional standard Wiener process defined on the complete probability space $(Ω, F, {F_{t}}_{t \geq 0}, P)$ with $Ω$ being a sample space, $F$ being a $σ$ -field, ${F_{t}}_{t \geq 0}$ being a filtration, and P being a probability measure. The drift term $f_{i} : R_{+} \times R^{n - i - 1} \times R \to R$ and the diffusion term $g_{i} : R_{+} \times R^{n - i} \times R \to R^{m}$ are assumed to be locally Lipschitz with $f_{i} (t, 0, 0) = 0$ and $g_{i} (t, 0, 0) = 0$ , $i = 1, \dots, n - 1$ .

When the diffusion term $g_{i} (\cdot) = 0$ , system (1) reduces to the deterministic upper-triangular (also called feedforward) nonlinear systems with input delay. Due to the intrinsic phenomenon of system itself or caused by measurement process, it is well known that input delay frequently exists in many real plants, such as network data transmission, fluid flow, telecommunications, etc. In addition, the existence of input delay may be a significant reason of instability and poor performance for the closed-loop system. Therefore, research on deterministic upper-triangular nonlinear systems with input delay has gained much attention and yielded lots of interesting results over the past few decades. Du et al. (2013) considered the sampled-data output feedback control for a class of upper-triangular systems with input delay. Based on a reduction method to remove input delay, Koo et al. (2012) proposed the state feedback controller using dynamic gain for input-delayed systems. Mazenc et al. (2004) studied global uniform asymptotic, and locally exponential, stabilization for feedforward systems with input delay. Mazenc and Malisoff (2013) further solved state feedback control for feedforward systems with a point-wise delay in the input. Wu et al. (2013) and Zhang et al. (2010) respectively investigated asymptotic stabilization of high-order upper-triangular nonlinear systems with input delay and different system structures. Ye (2011) dealt with the adaptive stabilization for feedforward nonlinear time-delay systems by nested saturation feedback. Global asymptotic state-feedback control for a class of feedforward nonlinear systems with input delay was addressed in Zhang et al. (2006).

It is worth mentioning that all the above-mentioned results do not consider the effect of stochastic noise; in fact, stochastic noise and uncertain disturbances often exist in practical applications. Hence, it is necessary and interesting to investigate stabilization problems for stochastic upper-triangular nonlinear systems. Motivated by fruitful deterministic results, the study of stochastic upper-triangular nonlinear systems rapidly became a hot issue in recent years. For the case of system high-order $p_{i} = 1$ , Liu and Xie (2013) and Liu et al. (2015) respectively considered the state feedback and output feedback control for system (1) with state delay; Ai et al. (2016), Liu et al. (2016) and Xie and Zhao (2014) further solved the asymptotic stabilization for system (1) with input delay. Subsequently, the state-feedback stabilization of stochastic high-order feedforward nonlinear systems with different system forms was studied by Jiao et al. (2014a,b) and Zhao and Xie (2013).

However, for the asymptotic stabilization of system (1) with general system high-order and input time-varying delay, there have been no relevant results until now. The main purpose of this paper is to deal with the global state-feedback stabilization problem of stochastic high-order upper-triangular nonlinear systems with input time-varying delay (1). To solve this problem, we introduce the homogeneous domination approach and stochastic nonlinear time-delay system stability criterion. By skilfully choosing an appropriate Lyapunov–Krasovskii functional and the low gain scale in the controller, and overcoming several troublesome obstacles in the design and analysis procedure, a delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is globally asymptotically stable in probability. A simulation example is provided to demonstrate the effectiveness of the proposed control scheme.

The paper is organized as follows. The second section provides some mathematical preliminaries. The controller design and stability analysis are given in the third section and the fourth section, respectively. The fifth section gives the simulation example. The sixth section concludes this paper.

Mathematical preliminaries

The following notations, definitions and lemmas are to be used in the paper.

$R_{+}$ denotes the set of all nonnegative real numbers, $R^{n}$ denotes the real n-dimensional space. $R_{odd}^{+} = {\bar{q} \in R_{+} : \bar{q}$ is a ratio of odd integers}. For a given vector or matrix X, $X^{T}$ denotes its transpose, $Tr {X}$ denotes its trace when X is square, and $| X |$ is the Euclidean norm of a vector X. $C ([- d, 0]; R^{n})$ denotes the space of continuous $R^{n}$ -value functions on $[- d, 0]$ endowed with the norm ∥·∥ defined by $∥ f ∥ = sup_{x \in [- d, 0]} | f (x) |$ for $f \in C ([- d, 0]; R^{n})$ ; $C_{F_{0}}^{b} ([- d, 0]; R^{n})$ denotes the family of all $F_{0}$ -measurable bounded $C ([- d, 0]; R^{n})$ -valued random variables $ξ = {ξ (θ) : - d \leq θ \leq 0}$ . $C^{i}$ denotes the set of all functions with continuous ith partial derivatives; $C^{2, 1}$ denotes the family of all functions which are $C^{2}$ in the first argument and $C^{1}$ in the second argument. $K$ denotes the set of all functions: $R_{+} \to R_{+}$ , which are continuous, strictly increasing and vanishing at zero; $K_{\infty}$ denotes the set of all functions which are of class $K$ and unbounded.

Consider the stochastic time-delay system

dx = f (x, x (t - d (t)), t) dt + g (x, x (t - d (t)), t) d ω, \forall t \geq 0

(2)

with initial data ${x (θ) : - d \leq θ \leq 0} = ξ \in C_{F_{0}}^{b} ([- d, 0]; R^{n})$ , where $d (t) : R_{+} \to [0, d]$ is a Borel measurable function, $ω$ is an m-dimensional standard Wiener process defined on the complete probability space $(Ω, F, {F_{t}}_{t \geq 0}, P)$ , $f : R^{n} \times R^{n} \times R_{+} \to R^{n}$ and $g : R^{n} \times R^{n} \times R_{+} \to R^{n \times m}$ are locally Lipschitz in $(x, x (t - d (t)))$ uniformly in t with $f (0, 0, t) \equiv 0$ and $g (0, 0, t) \equiv 0$ .

For any given $V (x, t) \in C^{2, 1}$ , the differential operator $L$ with system (2) is defined as

L V = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f + \frac{1}{2} Tr {g^{T} \frac{\partial^{2} V}{\partial x^{2}} g}

(3)

where $\frac{1}{2} Tr {g^{T} \frac{\partial^{2} V}{\partial x^{2}} g}$ is given the Hessian term $L$ .

Definition 1. (Xie and Liu, 2013) For fixed coordinates $(x_{1}, \dots,$ $x_{n})^{T} \in R^{n}$ and real numbers $r_{i} > 0$ , $i = 1, \dots, n$ :

the dilation $Δ_{ε} (x)$ is defined by $Δ_{ε} (x) = (ε^{r_{1}} x_{1}, \dots,$ $ε^{r_{n}} x_{n})$ for any $ε > 0$ , $r_{1}, \dots, r_{n}$ are called as the weights of the coordinates. For simplicity, we define dilation weight $Δ = (r_{1}, \dots, r_{n})$ ;

a function $V \in C (R^{n}, R)$ is said to be homogeneous of degree $τ$ if there is a real number $τ \in R$ such that $V (Δ_{ε} (x)) = ε^{τ} V (x_{1}, \dots, x_{n})$ for any $x \in R^{n} \ {0}$ , $ε > 0$ ;

a vector field $h \in C (R^{n}, R^{n})$ is said to be homogeneous of degree $τ$ if there is a real number $τ \in R$ such that $h_{i} (Δ_{ε} (x)) = ε^{τ + r_{i}} h_{i} (x)$ for any $x \in R^{n} \ {0}$ , $ε > 0$ , $i = 1, \dots, n$ ;

a homogeneous p-norm is defined as $∥ x ∥_{Δ, p} = (\sum_{i = 1}^{n} | x_{i} |^{\frac{p}{r_{i}}})^{\frac{1}{p}}$ for any $x \in R^{n}$ , where $p \geq 1$ is a constant. For simplicity, we choose $p = 2$ and write $∥ x ∥_{Δ}$ for $∥ x ∥_{Δ, 2}$ in this paper.

Lemma 1. (Liu et al., 2008) For system (2), if there exists a function $V (x, t) \in C^{2, 1} (R^{n} \times [- d, \infty); R_{+})$ , two class $K_{\infty}$ functions $μ_{1}$ , $μ_{2}$ and a class $K$ function $μ_{3}$ such that

μ_{1} (| x |) \leq V (x, t) \leq μ_{2} (sup_{- d \leq s \leq 0} | x (t + s) |), L V (x, t) \leq - μ_{3} (| x |)

then there is a unique solution on $[- d, \infty)$ for (2), the equilibrium $x (t) = 0$ is globally asymptotically stable in probability and $P {lim_{t \to \infty} | x (t) | = 0} = 1$ .

Lemma 2. (Liu and Xie, 2013) Given a dilation weight $Δ = (r_{1}, \dots, r_{n})$ , suppose that $V_{1} (x)$ and $V_{2} (x)$ are homogeneous functions of degree $τ_{1}$ and $τ_{2}$ , respectively. Then $V_{1} \cdot V_{2}$ is also homogeneous with respect to the same dilation weight $Δ$ . Moreover, the homogeneous degree of $V_{1} \cdot V_{2}$ is $τ_{1} + τ_{2}$ .

Lemma 3. (Liu and Xie, 2013) Suppose that $V : R^{n} \to R$ is a homogeneous function of degree $τ$ with respect to the dilation weight $Δ$ , then

$\frac{\partial V}{\partial x_{i}}$ is homogeneous of degree $τ - r_{i}$ with $r_{i}$ being the homogeneous weight of $x_{i}$ ;

There is a constant c such that $V (x) \leq c ∥ x ∥_{Δ}^{τ}$ . Moreover, if $V (x)$ is positive definite, then $V (x) \geq \underline{c} ∥ x ∥_{Δ}^{τ}$ , where $\underline{c}$ is a positive constant.

Lemma 4. (Liu and Xie, 2011) For any $x, y \in R$ , and $p \geq 1$ is a constant, then $| x + y |^{p} \leq 2^{p - 1} | x^{p} + y^{p} |$ , $(| x | + | y |)^{\frac{1}{p}} \leq | x |^{\frac{1}{p}} + | y |^{\frac{1}{p}}$ . Moreover, if $p \in R^{*}$ , then $| x - y |^{p} \leq 2^{p - 1} | x^{p} - y^{p} |$ , $| x^{\frac{1}{p}} - y^{\frac{1}{p}} | \leq 2^{1 - \frac{1}{p}} | x - y |^{\frac{1}{p}}$ .

Lemma 5. (Liu and Xie, 2011) Let c, d be positive constants. For any positive number $\bar{γ}$ , then $| x |^{c} | y |^{d} \leq \frac{c}{c + d} \bar{γ} | x |^{c + d} + \frac{d}{c + d} {\bar{γ}}^{- \frac{c}{d}} | y |^{c + d}$ .

Lemma 6. (Liu and Xie, 2011) Let $p \in R^{*}$ , x and y be real-valued functions. For a constant $c > 0$ , one has $| x^{p} - y^{p} | \leq c | x - y | | (x - y)^{p - 1} + y^{p - 1} |$ .

Design of state-feedback controller

This paper aims to design a delay-independent state-feedback controller for system (1) such that the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability. To achieve this objective, we need the following assumptions for system (1).

Assumptions and a key lemma

Assumption 1. For $i = 1, \dots, n - 1$ , any $p_{i} \in R^{*}$ satisfying $1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}} - \sum_{k = 1}^{i - 1} \frac{2}{p_{k} \dots p_{i - 1}} > 0$ and $σ \in (σ_{M}, + \infty)$ , there exist positive constants $a_{i 1}$ and $a_{i 2}$ such that

\begin{matrix} | f_{i} | \leq a_{i 1} (| x_{i + 2} |^{\frac{r_{i} + σ}{r_{i + 2}}} + \dots + | x_{n} |^{\frac{r_{i} + σ}{r_{n}}} + | u (t - τ (t {)) |}^{\frac{r_{i} + σ}{r_{n + 1}}}) \\ | g_{i} | \leq a_{i 2} (| x_{i + 1} |^{\frac{2 r_{i} + σ}{2 r_{i + 1}}} + \dots + | x_{n} |^{\frac{2 r_{i} + σ}{2 r_{n}}} + | u (t - τ (t {)) |}^{\frac{2 r_{i} + σ}{2 r_{n + 1}}}) \end{matrix}

where $r_{1} = 1$ , $r_{i + 1} = \frac{r_{i} + σ}{p_{i}}$ $(i = 1, \dots, n)$ , $σ_{M} = max_{1 \leq i \leq n} {σ_{i}}$ , $σ_{1} = \frac{2 - \frac{1}{p_{1} \dots p_{n - 1}}}{1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}}}$ , $σ_{i} = \frac{\frac{2}{p_{1} \dots p_{i - 1}} - \frac{1}{p_{1} \dots p_{n - 1}}}{1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}} - \sum_{k = 1}^{i - 1} \frac{2}{p_{k} \dots p_{i - 1}}}$ .

Assumption 2. The time-varying delay $τ (t)$ satisfies $\overset{\cdot}{τ} (t) \leq γ < 1$ for a constant $γ$ .

In this paper, we assume $σ = \frac{n_{1}}{m_{1}}$ with $n_{1}$ being an even integer and $m_{1}$ being an odd integer. Under this assumption, we know that $r_{i}$ is an odd number in Assumption 1.

Remark 1. Clearly, system (1), satisfying Assumption 1, is a class of stochastic high-order upper-triangular nonlinear systems with input time-varying delay. As discussed in the stochastic references Ai et al. (2016), Jiao et al. (2014a, 2014b), Liu et al. (2015, 2016), Liu and Xie (2013), Xie and Zhao (2014), Zhao and Xie (2013), and deterministic references Du et al. (2013), Koo et al. (2012), Zhang et al. (2006), Assumption 1 is a general and frequently-used condition.

Remark 2. We discuss the value range of $p_{i}$ in Assumption 1.

If $p_{i} = p > 2$ , then $1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}} - \sum_{k = 1}^{i - 1} \frac{2}{p_{k} \dots p_{i - 1}} = \frac{1}{p - 1} (p - 2 + \frac{2}{p^{i - 1}} - \frac{1}{p^{n - 1}}) > 0$ .

Otherwise, for any $p_{i} \geq 3$ , then $1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}} - \sum_{k = 1}^{i - 1} \frac{2}{p_{k} \dots p_{i - 1}} > \frac{1}{3^{i - 1}} > 0$ holds.

For $p_{i}$ takes some values in interval $[1, 3)$ , $1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}} - \sum_{k = 1}^{i - 1} \frac{2}{p_{k} \dots p_{i - 1}} > 0$ may also hold. In the simulation, we will give an example to illustrate this point.

Remark 3. There are two points to be emphasized for Assumption 1.

Assumption 1 is more general than some existing results, such as Ai et al. (2016), Liu et al. (2016), Xie and Zhao (2014) and Zhao and Xie (2013). Furthermore, when $p_{i} = 1$ , Assumption 1 reduces to the conditions used in Ai et al. (2016), Liu et al. (2016) and Xie and Zhao (2014). For $p_{i} = p$ and $τ (t) = 0$ , Assumption 1 changes into the assumption used in Zhao and Xie (2013).

In the controller design and stability analysis, the flexible selection of parameter $σ$ plays an important role in guaranteeing that the closed-loop system satisfies the local Lipschitz condition and ensuries the existence and uniqueness of the solution.

Before giving the state-feedback controller design procedure, we propose the following key lemma, whose role is to guarantee Lemma 8 and Step 1 in the proof of Theorem 1.

Lemma 7. For any $p_{i} \in R^{*}$ and $σ \in (σ_{M}, + \infty)$ in Assumption 1, there exist l and $μ \in R_{odd}^{+}$ such that $r_{n} + σ \geq μ \geq max_{1 \leq i \leq n} {2 r_{i}, \frac{r_{i} + σ}{l}}$ .

Proof. (i) We first prove $r_{n} + σ \geq max_{1 \leq i \leq n} {2 r_{i}}$ . By $r_{1} = 1$ , $r_{i + 1} = \frac{r_{i} + σ}{p_{i}}$ , one has

r_{i} = \sum_{k = 1}^{i - 1} \frac{σ}{p_{k} \dots p_{i - 1}} + \frac{1}{p_{1} \dots p_{i - 1}}, i = 2, \dots, n

(4)

For $i = 1$ , selecting $σ_{1} = \frac{2 - \frac{1}{p_{1} \dots p_{n - 1}}}{1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}}}$ and $σ \geq σ_{1}$ , from (4), it follows that

r_{n} + σ \geq (1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}}) σ_{1} + \frac{1}{p_{1} \dots p_{n - 1}} = 2 r_{1}

(5)

When $i = 2, \dots, n$ , taking $σ_{i} = \frac{\frac{2}{p_{1} \dots p_{i - 1}} - \frac{1}{p_{1} \dots p_{n - 1}}}{1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}} - \sum_{k = 1}^{i - 1} \frac{2}{p_{k} \dots p_{i - 1}}}$ and $σ \geq σ_{i}$ , by (4), one obtains

\begin{matrix} r_{n} + σ \geq (1 + \sum_{k = 1}^{n - 1} \frac{1}{p_{k} \dots p_{n - 1}} - \sum_{k = 1}^{i - 1} \frac{2}{p_{k} \dots p_{i - 1}}) σ_{i} \\ + \frac{1}{p_{1} \dots p_{n - 1}} + \sum_{k = 1}^{i - 1} \frac{2 σ}{p_{k} \dots p_{i - 1}} = 2 r_{i} \end{matrix}

(6)

By (5) and (6), $r_{n} + σ \geq 2 r_{i}$ holds for $i = 1, \dots, n$ .

(ii) For any $p_{i} \in R^{*}$ and $σ \in (σ_{M}, + \infty)$ in Assumption 1, one can choose l to satisfy $r_{n} + σ \geq max_{1 \leq i \leq n} {\frac{r_{i} + σ}{l}}$ .

From (i) and (ii), we have $r_{n} + σ \geq max_{1 \leq i \leq n} {2 r_{i}, \frac{r_{i} + σ}{l}}$ . According to the denseness of real number, there exists $μ \in R_{odd}^{+}$ such that $r_{n} + σ \geq μ \geq max_{1 \leq i \leq n} {2 r_{i}, \frac{r_{i} + σ}{l}}$ .□

State-feedback controller

We first introduce the following coordinate transformation

η_{1} = x_{1}, η_{i} = N^{- κ_{i}} x_{i} = N^{- κ_{n + 1}} u,

(7)

where $κ_{1} = 0$ , $κ_{i + 1} = \frac{κ_{i} + 1}{p_{i}}$ , $i = 1, \dots, n,$ $0 < N < 1$ is a designed scalar. Using (7), system (1) changes into

\begin{matrix} d η_{i} = N η_{i + 1}^{p_{i}} dt + {\bar{f}}_{i} (t, {\bar{η}}_{i + 2}, v (t - τ (t))) dt \\ + {\bar{g}}_{i}^{T} (t, {\bar{η}}_{i + 1}, v (t - τ (t))) d ω \\ d η_{n - 1} = N η_{n}^{p_{n - 1}} dt + {\bar{f}}_{n - 1} (t, v (t - τ (t))) dt \\ + {\bar{g}}_{n - 1}^{T} (t, {\bar{η}}_{n}, v (t - τ (t))) d ω \\ d η_{n} = N v^{p_{n}} dt, i = 1, \dots, n - 2 \end{matrix}

(8)

where ${\bar{f}}_{i} = N^{- κ_{i}} f_{i}$ , ${\bar{g}}_{i} = N^{- κ_{i}} g_{i}$ , $i = 1, \dots, n - 1$ . In the following, we give the state-feedback controller design procedure.

Step 1. Introducing $z_{1} = η_{1}^{\frac{μ}{r_{1}}}$ and $V_{1} (η_{1}) = \int_{η_{1}^{*}}^{η_{1}} (s^{\frac{μ}{r_{1}}} - η_{1}^{*} \frac{μ}{r_{1}})^{\frac{q_{1}}{μ}} ds$ , where $η_{1}^{*} = 0$ , $q_{1} = 4 l μ - σ - r_{1}$ . By (3) and (8), we have $L V_{1} = N η_{1}^{\frac{q_{1}}{r_{1}}} η_{2}^{p_{1}} + \frac{\partial V_{1}}{\partial η_{1}} {\bar{f}}_{1} + \frac{1}{2} Tr {{\bar{g}}_{1} \frac{\partial^{2} V_{1}}{\partial η_{1}^{2}} {\bar{g}}_{1}^{T}}$ . The first virtual controller

η_{2}^{*} = - α_{1}^{\frac{r_{2}}{μ}} z_{1}^{\frac{r_{2}}{μ}}, α_{1} = c_{11}^{\frac{μ}{r_{2} p_{1}}}, c_{11} > 0

(9)

leads to $L V_{1} \leq - N c_{11} z_{1}^{4 l} + {Nz}_{1}^{\frac{q_{1}}{μ}} (η_{2}^{p_{1}} - η_{2}^{p_{1}}) + \frac{\partial V_{1}}{\partial η_{1}} {\bar{f}}_{1} + \frac{1}{2} Tr {{\bar{g}}_{1} \frac{\partial^{2} V_{1}}{\partial η_{1}^{2}} {\bar{g}}_{1}^{T}}$ .

Step $i (i = 2, \dots, n)$ . With the help of Lemma 7, in this step, one can obtain the similar property for the ith subsystem, which is presented by Lemma 8.

Lemma 8. Suppose that at step $i - 1$ , there exist a $C^{2}$ , positive definite and proper Lyapunov function $V_{i - 1} ({\bar{η}}_{i - 1})$ and a series of virtual controllers $η_{1}^{*}, \dots, η_{i}^{*}$ defined by

\begin{matrix} η_{1}^{*} = 0, z_{1} = η_{1}^{\frac{μ}{r_{1}}}, η_{k}^{*} = - α_{k - 1}^{\frac{r_{k}}{μ}} z_{k - 1}^{\frac{r_{k}}{μ}}, \\ z_{k} = η_{k}^{\frac{μ}{r_{k}}} - η_{k}^{\frac{μ}{r_{k}}} k = 2, \dots, i, \end{matrix}

(10)

such that

\begin{matrix} L V_{i - 1} \leq - N \sum_{j = 1}^{i - 1} c_{i - 1, j} z_{j}^{4 l} + {Nz}_{i - 1}^{\frac{q_{i - 1}}{μ}} (η_{i}^{p_{i - 1}} - η_{i}^{p_{i - 1}}) + \sum_{j = 1}^{i - 1} \frac{\partial V_{i - 1}}{\partial η_{j}} {\bar{f}}_{j} \\ + \frac{1}{2} \sum_{p, q = 1}^{i - 1} Tr {{\bar{g}}_{p} \frac{\partial^{2} V_{i - 1}}{\partial η_{p} \partial η_{q}} {\bar{g}}_{q}^{T}}, q_{i - 1} = 4 l μ - σ - r_{i - 1} \end{matrix}

(11)

where $α_{j}$ , $c_{i - 1, j}$ , $j = 1, \dots, i - 1$ , are positive real numbers. Then the ith Lyapunov function

\begin{matrix} V_{i} ({\bar{η}}_{i}) = V_{i - 1} ({\bar{η}}_{i - 1}) + U_{i} ({\bar{η}}_{i}),_{i} ({\bar{η}}_{i}) \\ = \int_{η_{i}^{*}}^{η_{i}} (s^{\frac{μ}{r_{i}}} - η_{i}^{\frac{μ}{r_{i}}})^{\frac{q_{i}}{μ}} ds,_{i} = 4 l μ - σ - r_{i} \end{matrix}

(12)

is $C^{2}$ , positive definite and proper, and there is a virtual control law $η_{i + 1}^{*} = - α_{i}^{\frac{r_{i + 1}}{μ}} z_{i}^{\frac{r_{i + 1}}{μ}}$ such that

\begin{matrix} L V_{i} \leq - N \sum_{j = 1}^{i} c_{ij} z_{j}^{4 l} + {Nz}_{i}^{\frac{q_{i}}{μ}} (η_{i + 1}^{p_{i}} - η_{i + 1}^{* p_{i}}) \\ + \sum_{j = 1}^{i} \frac{\partial V_{i}}{\partial η_{j}} {\bar{f}}_{j} + \frac{1}{2} \sum_{p, q = 1}^{i} Tr {{\bar{g}}_{p} \frac{\partial^{2} V_{i}}{\partial η_{p} \partial η_{q}} {\bar{g}}_{q}^{T}} \end{matrix}

(13)

Proof. We first prove that $V_{i} ({\bar{η}}_{i})$ is $C^{2}$ . According to (10) and (12), it is easy to obtain

\begin{matrix} \frac{\partial U_{i}}{\partial η_{i}} = z_{i}^{\frac{q_{i}}{μ}} \frac{\partial^{2} U_{i}}{\partial η_{i}^{2}} = \frac{q_{i}}{r_{i}} η_{i}^{\frac{μ - r_{i}}{r_{i}}} z_{i}^{\frac{q_{i} - μ}{μ}} \frac{\partial^{2} U_{i}}{\partial η_{i} \partial η_{j}} = \frac{\partial^{2} U_{i}}{\partial η_{j} \partial η_{i}} = - \frac{q_{i}}{μ} \frac{\partial {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}} z_{i}^{\frac{q_{i} - μ}{μ}} \\ \frac{\partial U_{i}}{\partial η_{j}} = - \frac{q_{i}}{μ} \frac{\partial {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}} \int_{η_{i}^{*}}^{η_{i}} (s^{\frac{μ}{r_{i}}} - η_{i}^{*^{\frac{μ}{r_{i}}}})^{\frac{q_{i} - μ}{μ}} ds, \frac{\partial^{2} U_{i}}{\partial η_{j}^{2}} \\ = - \frac{q_{i}}{μ} \frac{\partial^{2} {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}^{2}} \int_{η_{i}^{*}}^{η_{i}} (s^{\frac{μ}{r_{i}}} - η_{i}^{*^{\frac{μ}{r_{i}}}})^{\frac{q_{i} - μ}{μ}} ds \\ + \frac{q_{i} (q_{i} - μ)}{μ^{2}} {(\frac{\partial {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}})}^{2} \int_{η_{i}^{*}}^{η_{i}} (s^{\frac{μ}{r_{i}}} - η_{i}^{*^{\frac{μ}{r_{i}}}})^{\frac{q_{i} - 2 μ}{μ}} ds \\ \frac{\partial^{2} U_{i}}{\partial η_{j} \partial η_{k}} = \frac{\partial^{2} U_{i}}{\partial η_{k} \partial η_{j}} = \frac{q_{i} (q_{i} - μ)}{μ^{2}} \frac{\partial {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}} \frac{\partial {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{k}} \int_{η_{i}^{*}}^{η_{i}} (s^{\frac{μ}{r_{i}}} - η_{i}^{*^{\frac{μ}{r_{i}}}})^{\frac{q_{i} - 2 μ}{μ}} ds \end{matrix}

(14)

where $j, k = 1, \dots, i - 1$ and the last equality is obtained by using $\frac{\partial^{2} {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j} \partial η_{k}} = 0$ , $j \neq k$ . By Lemma 7, $\frac{\partial {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}} = - α_{i - 1} \dots α_{j} \frac{μ}{r_{j}} η_{j}^{\frac{μ - r_{j}}{r_{j}}}$ and $\frac{\partial^{2} {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}^{2}} = - α_{i - 1} \dots α_{j} \frac{μ (μ - r_{j})}{r_{j}^{2}} η_{j}^{\frac{μ - 2 r_{j}}{r_{j}}}$ , one has $\frac{q_{i} - 2 μ}{μ} \geq 0$ , $\frac{μ - r_{j}}{r_{j}} \geq 1$ and $\frac{μ - 2 r_{j}}{r_{j}} \geq 0$ , from which and (14), we can conclude that $V_{i} ({\bar{η}}_{i})$ is $C^{2}$ .

Next, we prove that $V_{i} ({\bar{η}}_{i})$ is positive definite and proper from two aspects.

Case I. For $η_{i}^{*} \leq η_{i}$ , by Lemma 2, we have

U_{i} ({\bar{η}}_{i}) \geq 2^{\frac{q_{i}}{μ} - \frac{q_{i}}{r_{i}}} \int_{η_{i}^{*}}^{η_{i}} (s - η_{i}^{*})^{\frac{q_{i}}{r_{i}}} ds \geq \frac{2^{\frac{q_{i}}{μ} - \frac{q_{i}}{r_{i}}} r_{i}}{4 l μ - σ} (η_{i} - η_{i}^{*})^{\frac{4 l μ - σ}{r_{i}}}

(15)

Case II. When $η_{i}^{*} \geq η_{i}$ , (15) can also be obtained. Hence, $V_{i} ({\bar{η}}_{i}) \geq V_{i - 1} ({\bar{η}}_{i - 1}) + Γ_{i} (η_{i} - η_{i}^{*})^{\frac{4 l μ - σ}{r_{i}}}$ , which implies that $V_{i} ({\bar{η}}_{i})$ is positive definite and proper, where $Γ_{i}$ is a positive constant.

Finally, we prove that inequality (13) holds. From (3), (10)-(12) and (14), it follows that

\begin{matrix} L V_{i} \leq - N \sum_{j = 1}^{i - 1} c_{i - 1, j} z_{j}^{4 l} + {Nz}_{i}^{\frac{q_{i}}{μ}} η_{i + 1}^{p_{i}} + \sum_{j = 1}^{i} \frac{\partial V_{i}}{\partial η_{j}} {\bar{f}}_{j} \\ + \frac{1}{2} \sum_{p, q = 1}^{i} Tr {{\bar{g}}_{p} \frac{\partial^{2} V_{i}}{\partial η_{p} \partial η_{q}} {\bar{g}}_{q}^{T}} \\ + {Nz}_{i - 1}^{\frac{q_{i - 1}}{μ}} (η_{i}^{p_{i - 1}} - η_{i}^{p_{i - 1}}) - \frac{q_{i}}{μ} N \sum_{j = 1}^{i - 1} \frac{\partial {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}} η_{j + 1}^{p_{j}} \\ \int_{η_{i}^{*}}^{η_{i}} {(s^{\frac{μ}{r_{i}}} - {η_{i}^{*}}^{\frac{μ}{r_{i}}})}^{\frac{q_{i} - μ}{μ}} ds \end{matrix}

(16)

We focus on the last two terms on the right-hand side of (16).

For $\frac{r_{i} p_{i - 1}}{μ} \leq 1$ , using (10) and Lemma 2, one leads to

| η_{i}^{p_{i - 1}} - η_{i}^{* p_{i - 1}} | = | {(η_{i}^{\frac{μ}{r_{i}}})}^{\frac{r_{i} p_{i - 1}}{μ}} - {({η_{i}^{*}}^{\frac{μ}{r_{i}}})}^{\frac{r_{i} p_{i - 1}}{μ}} | \leq 2^{1 - \frac{r_{i} p_{i - 1}}{μ}} | z_{i} |^{\frac{r_{i} p_{i - 1}}{μ}}

(17)

When $\frac{r_{i} p_{i - 1}}{μ} \geq 1$ , by (10) and Lemma 4, there are positive constants $\bar{c}$ , ${\bar{b}}_{i 1}$ and ${\tilde{b}}_{i 1}$ such that

\begin{matrix} | η_{i}^{p_{i - 1}} - η_{i}^{* p_{i - 1}} | \leq \bar{c} | z_{i} | | z_{i}^{\frac{r_{i} p_{i - 1}}{μ} - 1} + {(α_{i - 1} z_{i - 1})}^{\frac{r_{i} p_{i - 1}}{μ} - 1} | \\ \leq {\bar{b}}_{i 1} | z_{i - 1} |^{\frac{r_{i} p_{i - 1}}{μ}} + {\tilde{b}}_{i 1} | z_{i} |^{\frac{r_{i} p_{i - 1}}{μ}} \end{matrix}

(18)

Combining (17) and (18), by Lemma 3, we have

\begin{matrix} z_{i - 1}^{\frac{q_{i - 1}}{μ}} (η_{i}^{p_{i - 1}} - η_{i}^{* p_{i - 1}}) \leq | z_{i - 1} |^{\frac{q_{i - 1}}{μ}} ({\bar{b}}_{i 1} | z_{i - 1} |^{\frac{r_{i} p_{i - 1}}{μ}} + b_{i 1} | z_{i} |^{\frac{r_{i} p_{i - 1}}{μ}}) \\ \leq l_{i, i - 1, 1} z_{i - 1}^{4 l} + γ_{i 1} z_{i}^{4 l} \end{matrix}

(19)

where $b_{i 1} = max {2^{1 - \frac{r_{i} p_{i - 1}}{μ}}, {\tilde{b}}_{i 1}}$ , $l_{i, i - 1, 1}$ and $γ_{i 1}$ are positive constants.

Using (10) and Lemmas 2–3, one concludes that

\begin{matrix} - \frac{q_{i}}{μ} \sum_{j = 1}^{i - 1} \frac{\partial {η_{i}^{*}}^{\frac{μ}{r_{i}}}}{\partial η_{j}} η_{j + 1}^{p_{j}} \\ \int_{η_{i}^{*}}^{η_{i}} {(s^{\frac{μ}{r_{i}}} - {η_{i}^{*}}^{\frac{μ}{r_{i}}})}^{\frac{q_{i} - μ}{μ}} ds \leq {\bar{ϱ}}_{1} \sum_{j = 1}^{i - 1} | η_{j} |^{\frac{μ - r_{j}}{r_{j}}} | η_{j + 1} |^{p_{j}} | z_{i} |^{\frac{4 l μ - σ - μ}{μ}} \\ \leq {\bar{ϱ}}_{2} \sum_{j = 1}^{i - 1} (| z_{j} |^{\frac{μ - r_{j}}{μ}} + | z_{j - 1} |^{\frac{μ - r_{j}}{μ}}) | z_{i} |^{\frac{4 l μ - σ - μ}{μ}} \\ \cdot (| z_{j + 1} |^{\frac{σ + r_{j}}{μ}} + | z_{j} |^{\frac{σ + r_{j}}{μ}}) \\ \leq \sum_{j = 1}^{i - 1} l_{ij 2} z_{j}^{4 l} + γ_{i 2} z_{i}^{4 l} \end{matrix}

(20)

where ${\bar{ϱ}}_{1}$ , ${\bar{ϱ}}_{2}$ , $l_{ij 2} (j = 1, \dots, i - 1)$ and $γ_{i 2}$ are positive constants. Taking $η_{i + 1}^{*} = - α_{i}^{\frac{r_{i + 1}}{μ}} z_{i}^{\frac{r_{i + 1}}{μ}}$ , $α_{i} = (c_{ii} + γ_{i 1} + γ_{i 2})^{\frac{μ}{r_{i + 1} p_{i}}}$ , $c_{ij} = {\begin{matrix} c_{i - 1, j} - l_{ij 2} > 0, j = 1, \dots, i - 2, \\ c_{i - 1, i - 1} - l_{i, i - 1, 1} - l_{i, i - 1, 2} > 0, j = i - 1 \end{matrix}$ , $c_{ii} > 0$ , and substituting (19)–(20) into (16), the inequality (13) holds.

At step n, choosing $V_{n} ({\bar{η}}_{n}) = \sum_{i = 1}^{n} \int_{η_{i}^{*}}^{η_{i}} (s^{\frac{μ}{r_{i}}} - η_{i}^{\frac{μ}{r_{i}}})^{\frac{q_{i}}{μ}} ds$ and

\begin{matrix} v = η_{n + 1}^{*} = - α_{n}^{\frac{r_{n + 1}}{μ}} z_{n}^{\frac{r_{n + 1}}{μ}} \\ = - ({\bar{α}}_{n} η_{n}^{\frac{μ}{r_{n}}} + {\bar{α}}_{n - 1} η_{n - 1}^{\frac{μ}{r_{n - 1}}} + \dots + {\bar{α}}_{1} η_{1}^{\frac{μ}{r_{1}}})^{\frac{r_{n + 1}}{μ}} \end{matrix}

(21)

from Lemma 8, it follows that

L V_{n} \leq - N \sum_{i = 1}^{n} c_{ni} z_{i}^{4 l} + \sum_{i = 1}^{n} \frac{\partial V_{n}}{\partial η_{i}} {\bar{f}}_{i} + \frac{1}{2} \sum_{i, j = 1}^{n} Tr {{\bar{g}}_{i} \frac{\partial^{2} V_{n}}{\partial η_{i} \partial η_{j}} {\bar{g}}_{j}^{T}}

(22)

where $z_{n} = η_{n}^{\frac{μ}{r_{n}}} - η_{n}^{\frac{μ}{r_{n}}}$ , ${\bar{α}}_{i} = α_{n} \dots α_{i}$ , $c_{ni}$ , $i = 1, \dots, n$ , are positive constants. The system (8) and (21) can be written as the following compact form

d η = NE (η) dt + F (t, η, v (t - τ (t))) dt + G^{T} (t, η, v (t - τ (t))) d ω

(23)

where $η = {\bar{η}}_{n} = (η_{1}, \dots, η_{n})^{T}$ , $E (η) = (η_{2}^{p_{1}}, \dots, v^{p_{n}})^{T}$ , $F (t, η, v (t - τ (t))) = ({\bar{f}}_{1}, \dots, {\bar{f}}_{n - 1}, 0)^{T}$ , $G (t, η, v (t - τ (t))) = ({\bar{g}}_{1}, \dots, {\bar{g}}_{n - 1}, 0)$ . Introducing the dilation weight $Δ = (\underset{for η_{1}, \dots, η_{n}}{\underset{︸}{r_{1}, r_{2}, \dots, r_{n}}})$ , by the definition of $V_{n} (η)$ , we have

\begin{matrix} V_{n} (Δ_{ε} (η)) = \sum_{i = 1}^{n} \int_{ε^{r_{i}} η_{i}^{*}}^{ε^{r_{i}} η_{i}} (s^{\frac{μ}{r_{i}}} - ε^{μ} η_{i}^{*^{\frac{μ}{r_{i}}}})^{\frac{q_{i}}{μ}} ds \\ =^{s = ε^{r_{i}} θ} \sum_{i = 1}^{n} \int_{η_{i}^{*}}^{η_{i}} (ε^{μ} θ^{\frac{μ}{r_{i}}} - ε^{μ} η_{i}^{*^{\frac{μ}{r_{i}}}})^{\frac{q_{i}}{μ}} ε^{r_{i}} d θ \\ = ε^{4 l μ - σ} \sum_{i = 1}^{n} \int_{η_{i}^{*}}^{η_{i}} (θ^{\frac{μ}{r_{i}}} - η_{i}^{*^{\frac{μ}{r_{i}}}})^{\frac{q_{i}}{μ}} d θ \\ = ε^{4 l μ - σ} V_{n} (η) \end{matrix}

from which and Definition 1, one can conclude that $V_{n} (η)$ is homogeneous of degree $4 l μ - σ$ .

Stability analysis

We illustrate the main result of this paper as follows.

Theorem 1. If Assumptions 1–2 hold for the stochastic high-order upper-triangular nonlinear systems with input time-varying delay (1), under the state-feedback controller $u = N^{κ_{n + 1}} v$ and (21), then: (i) the closed-loop system has a unique solution on $[- τ_{0}, \infty)$ ; (ii) the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability.

Proof. The proof procedure of Theorem 1 can be divided into the following three steps.

Step 1. We first prove that $v^{p_{n}} (η)$ in (21) is $C^{1}$ . From (21), we have

\begin{matrix} \frac{\partial v^{p_{n}} (η)}{\partial η_{i}} = - \frac{r_{n + 1} p_{n}}{r_{i}} {\bar{α}}_{i} η_{i}^{\frac{μ - r_{i}}{r_{i}}} \\ ({\bar{α}}_{n} η_{n}^{\frac{μ}{r_{n}}} + {\bar{α}}_{n - 1} η_{n - 1}^{\frac{μ}{r_{n - 1}}} + \dots + {\bar{α}}_{1} η_{1}^{\frac{μ}{r_{1}}})^{\frac{r_{n} + σ - μ}{μ}}, i = 1, \dots, n \end{matrix}

(24)

Using Lemma 7, one obtains $\frac{μ - r_{i}}{r_{i}} \geq 1$ , $\frac{r_{n} + σ - μ}{μ} \geq 0$ , from which and (24), we know that $\frac{\partial v^{p_{n}} (η)}{\partial η_{i}}$ is continuous, then $v^{p_{n}} (η)$ is $C^{1}$ .

Since $f_{i} (\cdot)$ , $g_{i} (\cdot)$ $(i = 1, \dots, n - 1)$ are assumed to be locally Lipschitz, the system consisting of (8) and (21) satisfies the locally Lipschitz condition.

Step 2. Introduce the following Lyapunov function

V (η) = V_{n} (η) + \frac{{\bar{λ}}_{2} + {\bar{λ}}_{3}}{1 - γ} N^{1 + ρ} \int_{t - τ (t)}^{t} ∥ η (θ) ∥_{Δ}^{4 l μ} d θ,

(25)

where ${\bar{λ}}_{2}$ , ${\bar{λ}}_{3}$ and $ρ$ are positive scales to be determined later. Since $V_{n} (η)$ is $C^{2}$ , positive definite and radially unbounded, by Lemma 4.3 in Khalil (2002), there exist two class $K_{\infty}$ functions $β_{1}$ and $β_{21}$ such that

β_{1} (| η |) \leq V_{n} (η) \leq β_{21} (| η |)

(26)

From $τ (t) : R_{+} \to [0, τ_{0}]$ and using the same method in Xie and Liu (2013), there exists a class $K_{\infty}$ function $β_{22}$ such that

\frac{{\bar{λ}}_{2} + {\bar{λ}}_{3}}{1 - γ} N^{1 + ρ} \int_{t - τ (t)}^{t} ∥ η (θ) ∥_{Δ}^{4 l μ} d θ \leq β_{22} (sup_{- τ_{0} \leq s \leq 0} | η (s + t) |)

(27)

Letting $β_{2} = β_{21} + β_{22}$ , by (25)-(27) and $| η | \leq sup_{- τ_{0} \leq s \leq 0} | η (s + t) |$ , one can deduce that

β_{1} (| η |) \leq V (η) \leq β_{2} (sup_{- τ_{0} \leq s \leq 0} | η (s + t) |)

(28)

Step 3. By Lemmas 2–3 and (22), there exists $λ_{1} > 0$ such that

\frac{\partial V_{n}}{\partial η} NE (η) = - N \sum_{i = 1}^{n} c_{ni} z_{i}^{4 l} \leq - λ_{1} N ∥ η ∥_{Δ}^{4 l μ}

(29)

Using Definition 1, Assumption 1 and (7), and ${\frac{k_{j}}{r_{j}}}_{j = i + 2, \dots, n + 1}$ is an increasing sequence, one has

\begin{matrix} | {\bar{f}}_{i} (t, {\bar{η}}_{i + 2}, v (t - τ (t))) | \leq a_{i 1} N^{- κ_{i}} \\ (\sum_{j = i + 2}^{n} | N^{κ_{j}} η_{j} |^{\frac{r_{i} + σ}{r_{j}}} + | N^{κ_{n + 1}} v (t - τ (t {)) |}^{\frac{r_{i} + σ}{r_{n + 1}}}) \\ \leq a_{i 1} N^{κ_{i + 2} \frac{r_{i} + σ}{r_{i + 2}} - κ_{i}} (\sum_{j = i + 2}^{n} | η_{j} |^{\frac{r_{i} + σ}{r_{j}}} + | v (t - τ (t {)) |}^{\frac{r_{i} + σ}{r_{n + 1}}}) \\ \leq {\bar{a}}_{1} N^{1 + ρ_{i 1}} (∥ η ∥_{Δ}^{r_{i} + σ} + ∥ η (t - τ (t)) ∥_{Δ}^{r_{i} + σ}) \end{matrix}

(30)

where $ρ_{i 1} = \frac{p_{i}}{σ + σ \sum_{k = 1}^{i} p_{1} \dots p_{k} + 1}$ and ${\bar{a}}_{1}$ are positive constants. With Lemmas 2,3,5 and (30) yields

\begin{matrix} \sum_{i = 1}^{n} \frac{\partial V_{n}}{\partial η_{i}} {\bar{f}}_{i} (t, {\bar{η}}_{i + 2}, v (t - τ (t))) \\ \leq {\tilde{λ}}_{2} N^{1 + \bar{ρ}} \sum_{i = 1}^{n - 1} ∥ η ∥_{Δ}^{q_{i}} (∥ η ∥_{Δ}^{r_{i} + σ} + ∥ η (t - τ (t)) ∥_{Δ}^{r_{i} + σ}) \\ \leq N^{1 + \bar{ρ}} (λ_{2} ∥ η ∥_{Δ}^{4 l μ} + {\bar{λ}}_{2} ∥ η (t - τ (t)) ∥_{Δ}^{4 l μ}) \end{matrix}

(31)

where $\bar{ρ} = min_{1 \leq i \leq n - 1} {ρ_{i 1}}$ , $λ_{2}$ , ${\bar{λ}}_{2}$ and ${\tilde{λ}}_{2}$ are positive constants. Similar to (30), from Assumption 1, (7) and ${\frac{k_{j}}{r_{j}}}_{j = i + 1, \dots, n + 1}$ is an increasing sequence. It follows that

\begin{matrix} | {\bar{g}}_{i} (t, {\bar{η}}_{i + 1} (t), v (t - τ (t))) | \leq a_{i 2} N^{κ_{i + 1} \frac{2 r_{i} + σ}{2 r_{i + 1}} - κ_{i}} \\ (\sum_{j = i + 1}^{n} | η_{j} |^{\frac{2 r_{i} + σ}{2 r_{j}}} + | v (t - τ (t {)) |}^{\frac{2 r_{i} + σ}{2 r_{n + 1}}}) \\ \leq {\bar{a}}_{2} N^{\frac{1}{2} + ρ_{i 2}} (∥ η ∥_{Δ}^{r_{i} + \frac{σ}{2}} + ∥ η (t - τ (t)) ∥_{Δ}^{r_{i} + \frac{σ}{2}}) \end{matrix}

(32)

where $0 < ρ_{i 2} = \frac{1}{2 (σ + σ \sum_{k = 1}^{i - 1} p_{1} \dots p_{k} + 1)} < \frac{1}{2}$ , ${\bar{a}}_{2}$ is a positive real number. By Lemmas 2,3,5 and (32), one can obtain

\begin{matrix} \frac{1}{2} \sum_{i, j = 1}^{n} Tr {{\bar{g}}_{i} \frac{\partial^{2} V_{n}}{\partial η_{i} \partial η_{j}} {\bar{g}}_{j}^{T}} \leq {\tilde{λ}}_{3} N^{1 + \tilde{ρ}} \\ \sum_{i, j = 1}^{n - 1} ∥ η ∥_{Δ}^{q_{i} - r_{i}} (∥ η ∥_{Δ}^{r_{i} + \frac{σ}{2}} + ∥ η (t - τ (t)) ∥_{Δ}^{r_{i} + \frac{σ}{2}}) \\ \cdot (∥ η ∥_{Δ}^{r_{j} + \frac{σ}{2}} + ∥ η (t - τ (t)) ∥_{Δ}^{r_{j} + \frac{σ}{2}}) \\ \leq N^{1 + \tilde{ρ}} (λ_{3} ∥ η ∥_{Δ}^{4 l μ} + {\bar{λ}}_{3} ∥ η (t - τ (t)) ∥_{Δ}^{4 l μ}) \end{matrix}

(33)

where $\tilde{ρ} = min_{1 \leq i, j \leq n - 1} {ρ_{i 2} + ρ_{j 2}}$ , $λ_{3}$ , ${\bar{λ}}_{3}$ and ${\tilde{λ}}_{3}$ are positive constants. Choosing $ρ = min {\bar{ρ}, \tilde{ρ}}$ , by Assumption 2, (23), (29), (31) and (33), one leads to

\begin{matrix} L V \leq - λ_{1} N ∥ η ∥_{Δ}^{4 l μ} + N^{1 + ρ} \\ (λ_{2} ∥ η ∥_{Δ}^{4 l μ} + {\bar{λ}}_{2} ∥ η (t - τ (t)) ∥_{Δ}^{4 l μ}) \\ + N^{1 + ρ} (λ_{3} ∥ η ∥_{Δ}^{4 l μ} + {\bar{λ}}_{3} ∥ η (t - τ (t)) ∥_{Δ}^{4 l μ}) \\ + ({\bar{λ}}_{2} + {\bar{λ}}_{3}) N^{1 + ρ} (\frac{1}{1 - γ} ∥ η ∥_{Δ}^{4 l μ} - ∥ η (t - τ (t)) ∥_{Δ}^{4 l μ}) \\ = - N (λ_{1} - (λ_{2} + λ_{3} + \frac{{\bar{λ}}_{2} + {\bar{λ}}_{3}}{1 - γ}) N^{ρ}) ∥ η ∥_{Δ}^{4 l μ} \end{matrix}

(34)

Selecting $0 < N < min {1, (\frac{λ_{1} (1 - γ)}{(λ_{2} + λ_{3}) (1 - γ) + {\bar{λ}}_{2} + {\bar{λ}}_{3}})^{\frac{1}{ρ}}}$ , there always exists a positive constant $λ_{0}$ such that $L V \leq - N λ_{0} ∥ η ∥_{Δ}^{4 l μ}$ . According to Lemma 1, we know that the system consisting of (8) and (21) has a unique solution on $[- τ_{0}, \infty)$ , $η (t) = 0$ is globally asymptotically stable in probability. Since (7) is an equivalent transformation, the closed-loop system consisting of (1), $u = N^{κ_{n + 1}} v$ and (21) has the same properties as the system (8) and (21). Therefore, Theorem 1 holds.

Remark 4. In this Remark, we highlight three points.

In the stability analysis, by adopting the homogeneous domination approach, the effect of input time-varying delay was skilfully dealt with in the inequalities (30) and (32).

Due to the appearance of input time-varying delay, constructing an appropriate Lyapunov–Krasovskii functional (25) to satisfy (34) is not easy to do.

It should be pointed out that the rigorous proof of Theorem 1 is not a trivial task.

Remark 5. The homogeneous domination approach is generalized from deterministic system to stochastic case and solved the global state-feedback stabilization problem for a class of stochastic high-order upper-triangular nonlinear systems with input time-varying delay (1). In contrast to the existing results on stochastic high-order lower-triangular nonlinear systems (Chen et al., 2012; Xie and Liu, 2012, 2013), which introduced high gain into the controller, in this paper, we adopt a low gain scale in the controller to dominate the effect of the drift and diffusion terms as the structures of these two kinds of system are symmetric.

A simulation example

Consider the following stochastic upper-triangular nonlinear system with input delay

\begin{matrix} d x_{1} = x_{2}^{\frac{7}{3}} dt + \frac{1}{20} u (t - τ (t)) \sin u (t - τ (t)) dt + \frac{1}{10} x_{2}^{\frac{5}{3}} d ω \\ d x_{2} = u^{\frac{5}{3}} dt, τ (t) = \frac{1}{8} (1 + \sin t) \end{matrix}

(35)

By $p_{1} = \frac{7}{3}$ , $p_{2} = \frac{5}{3}$ , $n = 2$ , one can deduce that $σ_{1} = \frac{11}{10}$ and $σ_{2} = \frac{3}{4}$ , then $σ \in (\frac{11}{10}, + \infty)$ . Taking $σ = \frac{4}{3}$ in simulation, from $r_{1} = 1$ , we have $r_{2} = 1$ and $r_{3} = \frac{7}{5}$ . Therefore, Assumptions 1–2 are satisfied with $a_{11} = \frac{1}{20}$ , $a_{12} = \frac{1}{10}$ and $\overset{\cdot}{τ} (t) = \frac{1}{8} \cos t < 1$ . Choosing $l = \frac{5}{3}$ , $μ = \frac{11}{5}$ , then Lemma 7 holds. Introducing the change of coordinates $η_{1} = x_{1}$ , $η_{2} = N^{- \frac{3}{7}} x_{2}$ , $υ = N^{- \frac{6}{7}} u$ and using the design procedure in the third section, we can obtain the following state-feedback controller

u = - N^{\frac{6}{7}} (α_{2} N^{- \frac{33}{35}} x_{2}^{\frac{11}{5}} + α_{2} α_{1} x_{1}^{\frac{11}{5}})^{\frac{7}{11}}

(36)

where $α_{1}$ , $α_{2}$ and N are appropriate constants. In simulation, we select $α_{1} = 2$ , $α_{2} = 205$ , $N = 0.15$ , and the initial values $x_{1} (0) = 1$ , $x_{2} (0) = - 1$ . Figure 1 gives the responses of the closed-loop system (35) and (36), which verifies the effectiveness of the proposed design scheme.

Conclusion

This paper solves global state-feedback stabilization for a class of stochastic high-order upper-triangular nonlinear systems with input time-varying delay by combining the homogeneous domination approach with a stochastic nonlinear time-delay system stability criterion. The underlying idea of the homogeneous domination approach is that the homogeneous controller is first designed without considering the drift and diffusion terms, and then a low scaling gain N is introduced to the state-feedback controller to overcome the effects of drift and diffusion terms.

There are still some problems to be investigated: One is to consider the data-driven problem of system (1) as discussed in Yin et al. (2015). Another is to solve the fuzzy or neural control for system (1) as discussed in Chang and Yang (2014), Li et al. (2015) and Liu and Tong (2015a, 2015b). The third is to find a practical example for system (1).

Footnotes

Conflict of interest statement

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant numbers 61403041, 61573172), the Program for Liaoning Excellent Talents in University (grant number LJQ2015001), and the Project Funded by China Postdoctoral Science Foundation (grant number 2015M580435).

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