Finite-Time Stabilization of Stochastic Nonholonomic Systems and Its Application to Mobile Robot

Abstract

This paper investigates the problem of finite-time stabilization for a class of stochastic nonholonomic systems in chained form. By using stochastic finite-time stability theorem and the method of adding a power integrator, a recursive controller design procedure in the stochastic setting is developed. Based on switching strategy to overcome the uncontrollability problem associated with 𝑥₀(0)=0, global stochastic finite-time regulation of the closed-loop system states is achieved. The proposed scheme can be applied to the finite-time control of nonholonomic mobile robot subject to stochastic disturbances. The simulation results demonstrate the validity of the presented algorithm.

1. Introduction

The nonholonomic systems, which can model many classes of mechanical systems such as mobile robots and wheeled vehicles, have attracted intensive attention over the past decades. From Brockett’s necessary condition [1], it is well known that the nonholonomic systems cannot be stabilized to the origin by any static continuous state feedback although it is controllable. As a consequence, the classical smooth control theory cannot be applied directly used to such systems. In order to overcome this obstruction, several approaches have been proposed for the problem, such as discontinuous time-invariant stabilization [2, 3], smooth time-varying stabilization [4–6], and hybrid stabilization [7]. Using these valid approaches, many fruitful results have been developed [8–15]. Particularly, considering the unavoidability of stochastic disturbance, the asymptotic stabilization for stochastic nonholonomic systems was achieved in [16–18]. However, it should be mentioned that those aforementioned papers consider the feedback stabilizer that makes the trajectories of the systems converge to the equilibrium as the time goes to infinity.

Compared to the asymptotic stabilization, the finite-time stabilization, which renders the trajectories of the closed-loop systems convergent to the origin in a finite time, has many advantages such as fast response, high tracking precision, and disturbance-rejection properties. In many practical situations, the finite-time stabilization problem is more meaningful than the classical asymptotical stability. For the deterministic case, a sufficient and necessary condition for finite-time stability has been proposed in [19]. Its improvements and extensions have been given in [20, 21], for continuous systems satisfying uniqueness of solutions in forward time and for nonautonomous continuous systems, respectively. Reference [22] defined finite-time input-to-state stability for continuous systems with locally essentially bounded input. Accordingly, the problem of finite-time stabilization for nonlinear systems has been studied and numerous theoretical control design methods were presented and developed for various types of nonlinear systems over the last years [23–27]. Especially with help of time-rescaling and Lyapunov based method [28] proposed a novel switching finite time control strategy to nonholonomic chained systems in the deterministic setting.

However, the finite-time stabilization for stochastic nonholonomic systems cannot be solved by simply extending the methods for deterministic systems because of the presence of stochastic disturbance. As pointed out by Yin et al. [29], the existence of a unique solution and the nonsatisfaction of local Lipschitz condition are the preconditions of discussing the finite-time stability for a stochastic nonlinear system. Therefore, the finite-time controller design for stochastic nonholonomic systems in this paper should solve the following questions. Under what conditions, the stochastic nonholonomic systems exist possibly finite-time stabilizer? Under these conditions, how can one design a finite-time state-feedback stabilizing controller? Inspired by the works [25, 28], we generalize adding a power integrator design method to a stochastic system and based on stochastic finite-time stability theorem, by skillfully constructing 𝐶² Lyapunov functions, a state feedback controller is successfully achieved to guarantee that the closed-loop system states are globally regulated to zero within a given settling time almost surely.

The remainder of this paper is organized as follows. Section 2 presents some necessary notations, definitions and preliminary results. Section 3 describes the systems to be studied and formulates the control problem. Section 4 gives the main contributions of this paper and presents the design scheme to the controller. Section 5 gives a practical example, the model of which falls into our class of uncertain nonlinear system (3.1) via some technical transformations, to demonstrate the effectiveness of the theoretical results. Finally, concluding remarks are proposed in Section 6.

2. Notations and Preliminary Results

The following notations, definitions, and lemmas are to be used throughout the paper. 𝑅⁺ denotes the set of all nonnegative real numbers and 𝑅^𝑛 denotes the real 𝑛-dimensional space. 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^𝑖 denotes the set of all functions with continuous ith partial derivatives. 𝐾 denotes the set of all functions: 𝑅⁺→𝑅⁺, which are continuous, strictly increasing, and vanishing at zero; 𝐾_∞ denotes the set of all functions which are of class 𝐾 and unbounded.

Consider the stochastic nonlinear system 𝑑𝑥=𝑓(𝑡,𝑥)𝑑𝑡+𝑔(𝑡,𝑥)𝑑𝜔, (2.1) where 𝑥∈𝑅^𝑛 is the system state with the initial condition 𝑥(0)=𝑥₀,𝜔 is an 𝑚-dimensional independent standard Wiener process defined on a complete probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) with Ω being a sample space, ℱ being a 𝜎-field, {ℱ_𝑡}_𝑡≥0 being a filtration, and 𝑃 being a probability measure. The functions: 𝑓∶𝑅⁺×𝑅^𝑛→𝑅^𝑛 and 𝑔∶𝑅⁺×𝑅^𝑛→𝑅^𝑛×𝑚 are piecewise continuous and continuous with respect to the first and second arguments, respectively, and satisfy 𝑓(𝑡,0)≡0 and 𝑔(𝑡,0)≡0.

The following Lemma is a corollary of Theorem 170 in [30], which provides a sufficient condition to ensure the existence and uniqueness of solution for the system (2.1).

Lemma 2.1. Assume that 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥) are continuous in 𝑥. Further, for any 0<𝛿<1, each 𝑁=1,2,…, and each 0≤𝑇<∞, if the following conditions hold: 𝑓(𝑡,𝑥)≤𝑐(𝑡)(1+|𝑥|),𝑔(𝑡,𝑥)²≤𝑐(𝑡)1+|𝑥|²,𝑓𝑡,𝑥₁−𝑓𝑡,𝑥₂≤𝑐^𝑁_𝑇(𝑡)𝑥₁−𝑥₂,𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂≤𝑐^𝑁_𝑇(𝑡)𝑥₁−𝑥₂, (2.2) as 0<𝛿≤|𝑥_𝑖|≤𝑁,𝑖=1,2,𝑡∈[0,𝑇], where 𝑐(𝑡) and 𝑐^𝑁_𝑇(𝑡) are nonnegative functions such that ∫^𝑇₀𝑐(𝑡)𝑑𝑡<∞ and ∫^𝑇₀𝑐^𝑁_𝑇(𝑡)𝑑𝑡<∞. Then for any given 𝑥₀∈𝑅^𝑛, system (2.1) has a pathwise unique strong solution.

Definition 2.2 (see [31]). For system (2.1), define 𝜏(0,𝑥₀)=inf{𝑇≥0∶𝑥(𝑡,𝑥₀)=0, for all 𝑡≥𝑇}, which is called the stochastic settling time function of system (2.1), where 𝑥₀∈𝑅^𝑛.

Definition 2.3 (see [31]). The equilibrium 𝑥≡0 of the system (2.1) is said to be a stochastic finite-time stable equilibrium if(i)it is stable in probability: for every pair of 𝜀∈(0,1) and 𝑟>0, there exists 𝛿>0 such that 𝑃{|𝑥(𝑡,𝑥₀)|<𝑟,forall𝑡≥0}≥1−𝜀, whenever |𝑥|<𝛿.(ii)its stochastic settling-time function 𝜏(𝑡₀,𝑥₀) exists finitely with probability and 𝐸[𝜏(0,𝑥₀)]<∞.

Lemma 2.4 (see [32]). Consider the stochastic nonlinear system described in (2.1). Suppose there exists a 𝐶² function 𝑉(𝑥), class 𝐾_∞ functions 𝜇₁ and 𝜇₂, real numbers 𝑐>0 and 0<𝛼<1, such that 𝜇₁(|𝑥|)≤𝑉(𝑥)≤𝜇₂𝑥,(𝑥)=(||)ℒ𝑉𝜕𝑉1𝜕𝑥𝑓+2Tr𝑔^𝑇𝜕²𝑉𝜕𝑥²𝑔≤−𝑐𝑉^𝛼(𝑥). (2.3) Then it is globally finite-time stable in probability and the stochastic settling time function 𝜏(0,𝑥₀) satisfies 𝐸𝜏0,𝑥₀≤𝑉^1−𝛼𝑥₀𝑐().1−𝛼 (2.4)

Lemma 2.5 (see [25]). For any real numbers 𝑥_𝑖, 𝑖=1,…,𝑛 and 0<𝑏<1, the following inequality holds: 𝑥₁+⋯+𝑥_𝑛^𝑏≤𝑥₁^𝑏+⋯+𝑥_𝑛^𝑏, (2.5) when 𝑏=𝑝/𝑞<1, where 𝑝>0 and 𝑞>0 are odd integers, 𝑥^𝑏−𝑦^𝑏≤2^1−𝑏𝑥−𝑦^𝑏. (2.6)

Lemma 2.6 (see [25]). Let 𝑐,𝑑 be positive real numbers and 𝜋(𝑥,𝑦)>0 be a real-valued function. Then, |𝑥|^𝑐𝑦^𝑑≤𝑐𝜋𝑥,𝑦|𝑥|^𝑐+𝑑+𝑐+𝑑𝑑𝜋^−𝑐/𝑑𝑥,𝑦𝑦^𝑐+𝑑.𝑐+𝑑 (2.7)

3. Problem Formulation

In this paper, we focus our attention on the following class of stochastic nonholonomic systems: 𝑑𝑥₀=𝑑₀(𝑡)𝑢₀𝑑𝑡,𝑑𝑥_𝑖=𝑑_𝑖(𝑡)𝑥_𝑖+1𝑢₀𝑑𝑡+𝑔^𝑇_𝑖𝑥₀,𝑥_[𝑖]𝑑𝑤,𝑖=1,…,𝑛−1,𝑑𝑥_𝑛=𝑑_𝑛(𝑡)𝑢₁𝑑𝑡+𝑔^𝑇_𝑛𝑥₀,𝑥_[𝑛]𝑑𝑤, (3.1) where 𝑥₀∈𝑅 and 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 are system states, 𝑢₀∈𝑅 and 𝑢₁∈𝑅 are control inputs, respectively; 𝑥_[𝑖]=(𝑥₁,…,𝑥_𝑖)^𝑇, 𝑥_[𝑛]=𝑥; 𝑑_𝑖,𝑖=1,…,𝑛 represent the possible modeling error, refered to as disturbed virtual control coefficients; 𝑔_𝑖∶𝑅×𝑅^𝑖→𝑅^𝑚,𝑖=1,…,𝑛, are uncertain continuous functions satisfying 𝑔_𝑖(0,0)=0; and 𝜔 is an m-dimensional independent standard Wiener process defined on a complete probability space (Ω,𝐹,𝑃) with Ω being a sample space, 𝐹 being a filtration, and 𝑃 being a probability measure.

Remark 3.1. It should be mentioned that the system investigated in this paper, which emphasizes the effect of stochastic disturbance on the x-subsystem, is a special one; however it can be found in many real systems, such as the angular velocity of mobile robot subject to stochastic disturbances (see Section 5).
The objective of this paper is to find a robust state feedback controller of the form 𝑢₀=𝑢₀𝑥₀,𝑢₁=𝑢₁𝑥₀,𝑥, (3.2) such that the stochastic finite-time regulation of closed-loop system states is achieved, that is, 𝑃{lim_𝑡→𝑇(|𝑥₀(𝑡)|+|𝑥(𝑡)|)=0}=1 and 𝑃{(𝑥₀(𝑡),𝑥(𝑡))=(0,0)}=1 for any 𝑡≥𝑇, where 𝑇 is a given settling time.
To achieve the above control objective, we need the following assumptions.

Assumption 3.2. For 𝑖=0,1,…,𝑛, there are known positive constants 𝑐_𝑖1 and 𝑐_𝑖2 such that 𝑐_𝑖1≤𝑑_𝑖(𝑡)≤𝑐_𝑖2. (3.3)

Assumption 3.3. For 𝑖=1,…,𝑛, there are constants 𝑏 and 𝜏∈(−2/(4𝑛+1),0) such that 𝑔_𝑖𝑥₀,𝑥_[𝑖]−𝑔_𝑖𝑥₀,𝑥_[𝑖]≤𝑏𝑥₁−𝑥₁^{(2𝑚𝑖+𝜏)/2𝑚1}+⋯+𝑥_𝑖−𝑥_𝑖^{(2𝑚𝑖+𝜏)/2𝑚𝑖}, (3.4) where 𝑚_𝑖=1+(𝑖−1)𝜏.
For simplicity, in this paper we assume 𝜏=−𝑝/𝑞 with 𝑝 being any even integer and 𝑞 being any odd integer, under which and the definition of 𝑚_𝑖 in Assumption 3.3, we know that 𝑚_𝑖 is an odd number.

Remark 3.4. Noting that 𝑔_𝑖(0,0)=0 is assumed, Assumption 3.3 implies that 𝑔_𝑖(𝑡,𝑥,𝑢)≤𝑏𝑥₁^{(2𝑚𝑖+𝜏)/2𝑚1}+⋯+𝑥_𝑖^{(2𝑚𝑖+𝜏)/2𝑚𝑖}. (3.5) In fact, Assumption 3.3 is a generalization of the homogeneous growth condition introduced in [33] where 𝑥_[𝑖]=0 and 𝜏≥0. The assumption is necessary, which plays an essential role in ensuring the existence of finite-time stabilizer for stochastic nonholonomic system (3.1). Furthermore, it is worthwhile to point out that there exist some nonlinearities such as sin𝑥 that can be bounded by a function |𝑥|^𝑚 for any constant 𝑚∈(0,1) and satisfies Assumption 3.3.

4. Finite-Time Stabilization

In this section, we give a constructive procedure for the finite-time stabilizing control of system (3.1) within any given settling time 𝑇. For clarity, the case that 𝑥₀(0)≠0 is considered first. Then, the case where the initial 𝑥₀(0)=0 is dealt later. The inherently triangular structure of system (3.1) suggests that we should design the control inputs 𝑢₀ and 𝑢₁ in two separate stages.

4.1. Control for 𝑥₀(0)≠0

For 𝑥₀-subsystem, we take the following control law: 𝑢₀=−𝑘₀𝑥^𝛼0₀,0<𝛼₀=𝑝𝑞<1, (4.1) where 𝑘₀ is a positive design parameter, and 𝑝, 𝑞 are positive odd numbers.

Taking the Lyapunov function 𝑉₀=𝑥²₀/2, a simple computation gives −𝑐₀₂𝑘₀𝑥^1+𝛼0₀≤𝑉̇₀≤−𝑐₀₁𝑘₀𝑥^1+𝛼0₀≤0, (4.2) which implies |𝑥₀(𝑡)|≤|𝑥₀(0)|.

Furthermore, we have −𝑐₀₂𝑘₀2𝑉₀^(1+𝛼0)/2≤𝑉̇₀≤−𝑐₀₁𝑘₀2𝑉₀^(1+𝛼0)/2. (4.3) Thus, 𝑥₀ tends to 0 within a settling time denoted by 𝑇₀. Moreover, 𝑥₀(0)^1−𝛼0𝑐₀₂𝑘₀1−𝛼₀≤𝑇₀≤𝑥₀(0)^1−𝛼0𝑐₀₁𝑘₀1−𝛼₀. (4.4) To secure finite-time convergence within 𝑇 for any 𝑥₀(0)≠0, we need to keep 𝑇₀≤|𝑥₀(0)|^1−𝛼0/𝑐₀₁𝑘₀(1−𝛼₀)≤𝑇 by taking 𝑘₀>|𝑥₀(0)|^1−𝛼0/𝑐₀₁𝑇(1−𝛼₀). If we take 𝑇_∗=(𝑐₀₁𝑇₀)/(2𝑐₀₂), then we obtain 𝑥₀(𝑡)∈𝑅 does not change its sign when 𝑡<𝑇_∗, 𝑥₀(0)≠0 and moreover 𝑥₀(0)≥𝑥₀(𝑡≥𝑥)⁰(0)2,∀𝑡∈0,𝑇_∗. (4.5)

Therefore, 𝑢₀ is bounded and does not change sign during [0,𝑇_∗]. Furthermore, from this and Assumption 3.2, the following result can be obtained.

Lemma 4.1. For 𝑖=1,…,𝑛, there are positive constants 𝜆_𝑖 and 𝜇_𝑖 such that 𝜆_𝑖≤𝑑_𝑖(𝑡)𝛼_𝑖𝑢₀≤𝜇_𝑖,𝜆_𝑛≤𝑑_𝑛(𝑡)≤𝜇_𝑛, (4.6) where 𝛼_𝑖=−𝑠𝑔𝑛(𝑥₀(0)). Besides, for the simplicity of expression in later use, let 𝛼_𝑛=1.

Since we have already proven that 𝑥₀ can be globally finite-time regulated to zero as 𝑡→𝑇₀. Next, we only need to stabilize the time-varying 𝑥-subsystem 𝑑𝑥_𝑖=𝑑_𝑖(𝑡)𝑥_𝑖+1𝑢₀𝑑𝑡+𝑔^𝑇_𝑖𝑥₀,𝑥_[𝑖]𝑑𝑤,𝑖=1,…,𝑛−1,𝑑𝑥_𝑛=𝑑_𝑛(𝑡)𝑢₁𝑑𝑡+𝑔^𝑇_𝑛𝑥₀,𝑥_[𝑛]𝑑𝑤, (4.7) within the given settling time 𝑇_∗. The control law 𝑢₁ can be recursively constructed by applying the method of adding a power integrator.

Step 1. Let 𝜉₁=𝑥^𝜎₁, where 𝜎>2 is a odd number and choose 𝑉₁=𝑥₁^4𝜎−𝜏/(4𝜎−𝜏) to be the candidate Lyapunov function for this step. Then, along the trajectories of system (4.7), we have ℒ𝑉₁≤𝑑₁𝑢₀𝑥₁^{4𝜎−𝜏−1}𝑥₂−𝑥^∗₂+𝑑₁𝑢₀𝑥₁^{4𝜎−𝜏−1}𝑥^∗₂+12(4𝜎−𝜏−1)𝑏²𝑥₁^4𝜎. (4.8)
Obviously, the first virtual controller 𝑥^∗₂1=−𝜆₁𝛼₁𝑥₁^1+𝜏1𝑀+𝑛−1+2(4𝜎−𝜏−1)𝑏²≜−𝛼₁𝛽₁𝜉^𝑚2₁^/𝜎, (4.9) with design constant 𝑀>0, results in ℒ𝑉₁≤−(𝑀+𝑛−1)𝜉⁴₁+𝑑₁𝑢₀𝑥₁^{4𝜎−𝜏−1}𝑥₂−𝑥^∗₂. (4.10)

Inductive Step (2 ≤ k ≤ n)
Suppose at step 𝑘−1, there is a 𝐶² Lyapunov function 𝑉_𝑘−1, which is positive definite and proper, satisfying 𝑉_𝑘−1≤2𝜉₁^{(4𝜎−𝜏)/𝜎}+⋯+𝜉^{(4𝜎−𝜏)/𝜎}_𝑘−1, (4.11) and a set of virtual controllers 𝑥^∗₁,…,𝑥^∗_𝑘−1 defined by 𝑥^∗₁=0,𝜉₁=𝑥^𝜎/𝑚1₁−𝑥^{∗𝜎/𝑚1}₁,𝑥^∗₂=−𝛼₁𝛽₁𝜉^𝑚2₁^/𝜎,𝜉₂=𝑥^𝜎/𝑚2₂−𝑥^{∗𝜎/𝑚2}₂,𝑥⋮⋮_∗𝑘=−𝛼_𝑘−1𝛽_𝑘−1𝜉^{𝑚𝑘/𝜎}_𝑘−1,𝜉_𝑘=𝑥^{𝜎/𝑚𝑘}_𝑘−𝑥^{∗𝑘𝜎/𝑚}_𝑘, (4.12) with constants 𝛽₁>0,…,𝛽_𝑘−1>0, such that ℒ𝑉_𝑘−1≤−(𝑀+𝑛−𝑘+1)𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑑_𝑘−1𝑢₀𝜉^{(4𝜎−𝜏−𝑚𝑘−1)/𝜎}_𝑘−1𝑥_𝑘−𝑥^∗_𝑘. (4.13)
We claim that (4.11) and (4.13) also hold at step 𝑘. To prove this claim, consider 𝑉_𝑘𝑥_[𝑘]=𝑉_𝑘−1𝑥_[𝑘−1]+𝑊_𝑘𝑥_[𝑘], (4.14) where 𝑊_𝑘𝑥_[𝑘]=^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{(4𝜎−𝜏−𝑚𝑘)/𝜎}𝑑𝑠. (4.15)
Noting that 𝑥^{∗𝜎/𝑚𝑘}_𝑘=−𝛼^{𝜎/𝑚𝑘}_𝑘−1𝛽^{𝜎/𝑚𝑘}_𝑘−1𝜉_𝑘−1,𝜉_𝑘=𝑐_𝑘𝑥^{𝜎/𝑚𝑘}_𝑘+𝑐_𝑘−1𝑥^{𝜎/𝑚𝑘−1}_𝑘−1+⋯+𝑐₁𝑥^𝜎/𝑚1₁, (4.16) where 𝑐_𝑖=𝛼^{𝜎/𝑚𝑘}_𝑘−1𝛽^{𝜎/𝑚𝑘}_𝑘−1⋯𝛼^{𝜎/𝑚𝑖+1}_𝑖𝛽^{𝜎/𝑚𝑖+1}_𝑖,𝑖=1,…,𝑘−1,1,𝑖=𝑘. (4.17)
Similar to the corresponding proof in [25], it is easy to verify that the Lyapunov function 𝑉_𝑘(𝑥_[𝑘]) thus defined has several useful properties collected in the following propositions.

Proposition 4.2. 𝑊_𝑘(𝑥_[𝑘]) is 𝐶². Moreover 𝜕𝑊_𝑘𝜕𝑥_𝑖=−𝑐_𝑖4𝜎−𝜏−𝑚_𝑘𝜎𝑥^{(𝜎−𝑚𝑖)/𝑚𝑖}_𝑖×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝑑𝑠,𝜕𝑊_𝑘𝜕𝑥_𝑘=𝜉^{(4𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎,𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗=𝑐_𝑖𝑐_𝑗4𝜎−𝜏−𝑚_𝑘𝜎⋅3𝜎−𝜏−𝑚_𝑘𝜎𝑥^{𝑖𝑖(𝜎−𝑚)/𝑚}_𝑖𝑥^{(𝜎−𝑚𝑗)/𝑚𝑗}_𝑗×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{(2𝜎−𝜏−𝑚𝑘)/𝜎}𝜕𝑑𝑠,₂𝑊_𝑘𝜕𝑥²_𝑖=𝑐²_𝑖4𝜎−𝜏−𝑚_𝑘𝜎⋅3𝜎−𝜏−𝑚_𝑘𝜎𝑥^{2(𝜎−𝑚𝑖)/𝑚𝑖}_𝑖×^𝑥𝑘_𝑥∗𝑘𝑠^1/𝑚𝑘−𝑥^{∗1/𝑚𝑘}_𝑘^{(2𝜎−𝜏−𝑚𝑘)/𝜎}𝑑𝑠−𝑐²_𝑖4𝜎−𝜏−𝑚_𝑘𝜎𝑥^{(𝜎−2𝑚𝑖)/𝑚𝑖}_𝑖×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝜕𝑑𝑠,₂𝑊_𝑘𝜕𝑥²_𝑘=𝑐_𝑘4𝜎−𝜏−𝑚_𝑘𝜎𝜉^{(3𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥^{(𝜎−𝑚𝑘)/𝑚𝑘}_𝑘,𝜕²𝑊_𝑘𝜕𝑥_𝑘𝜕𝑥_𝑖=𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑘=𝑐_𝑖4𝜎−𝜏−𝑚_𝑘𝜎𝜉^{(3𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥^{(𝜎−𝑚𝑖)/𝑚𝑖}_𝑖, (4.18) where 𝑖,𝑗=1,…,𝑘−1, 𝑖≠𝑗.

Proposition 4.3. 𝑉_𝑘(𝑥_[𝑘]) is 𝐶², positive definite and proper, satisfying 𝑉_𝑘≤2𝜉₁^{(4𝜎−𝜏)/𝜎}+⋯+𝜉_𝑘^{(4𝜎−𝜏)/𝜎}. (4.19)

Using Proposition 4.2, it is deduced from (4.13) that ℒ𝑉_𝑘≤−(𝑀+𝑛−𝑘+1)𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑑_𝑘−1𝑢₀𝜉^{(4𝜎−𝜏−𝑚𝑘−1)/𝜎}_𝑘−1𝑥_𝑘−𝑥^∗_𝑘+𝑑_𝑘𝑢₀𝜉^{(4𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥_𝑘+1+^𝑘−1_𝑖=1𝜕𝑊_𝑘𝜕𝑥_𝑖𝑑_𝑖𝑢₀𝑥_𝑖+1+12^𝑘−1_{𝑖,𝑗=1,𝑖≠𝑗}𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗𝑔^𝑇_𝑖𝑔^𝑇_𝑗+12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥²_𝑖𝑔^𝑇_𝑖²+12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥_𝑘𝜕𝑥_𝑖𝑔^𝑇_𝑘𝑔^𝑇_𝑖+12𝜕²𝑊_𝑘𝜕𝑥²_𝑘𝑔^𝑇_𝑘². (4.20) To estimate the second term in (4.20), by Lemma 2.5, we have 𝑥_𝑘−𝑥^∗_𝑘=𝑥^{𝜎/𝑚𝑘}_𝑘^{𝑚𝑘/𝜎}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{𝑚𝑘/𝜎}≤2^{(𝜎−𝑚𝑘)/𝜎}𝜉_𝑘^{𝑚𝑘/𝜎}. (4.21) Noting that 𝑚_𝑘=𝑚_𝑘−1+𝜏, by applying (4.6) and Lemma 2.6, we have 𝑑_𝑘−1𝑢₀𝜉^{(4𝜎−𝜏−𝑚𝑘−1)/𝜎}_𝑘−1𝑥_𝑘−𝑥^∗_𝑘≤16𝜉⁴_𝑘−1+𝑙_𝑘1𝜉⁴_𝑘, (4.22) where 𝑙_𝑘1 is a positive constant.

Based on Proposition 4.2 and Lemma 2.6, the following propositions are given to estimate the other terms on the right hand side of inequality (4.20), whose proofs are included in the appendix.

Proposition 4.4. There exists a positive constant 𝑙_𝑘2 such that ^𝑘−1_𝑖=1𝜕𝑊_𝑘𝜕𝑥_𝑖𝑑_𝑖𝑢₀𝑥_𝑖+1≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘2𝜉⁴_𝑘. (4.23)

Proposition 4.5. There exists a positive constant 𝑙_𝑘3 such that 12^𝑘−1_{𝑖,𝑗=1,𝑖≠𝑗}𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗𝑔^𝑇_𝑖𝑔^𝑇_𝑗≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘3𝜉⁴_𝑘. (4.24)

Proposition 4.6. There exists a positive constant 𝑙_𝑘4 such that 12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥²_𝑖𝑔^𝑇_𝑖²≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘4𝜉⁴_𝑘. (4.25)

Proposition 4.7. There exists a positive constant 𝑙_𝑘5 such that 12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥_𝑘𝜕𝑥_𝑖𝑔^𝑇_𝑘𝑔^𝑇_𝑖≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘5𝜉⁴_𝑘. (4.26)

Proposition 4.8. There exists a positive constant 𝑙_𝑘6 such that 12𝜕²𝑊_𝑘𝜕𝑥²_𝑘𝑔^𝑇_𝑘²≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘6𝜉⁴_𝑘. (4.27)

Substituting (4.22)–(4.27) into (4.20) yields ℒ𝑉_𝑘≤−(𝑀+𝑛−𝑘)𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑑_𝑘𝑢₀𝜉^{(4𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥_𝑘+1+𝜉⁴_𝑘𝑙_𝑘1+⋯+𝑙_𝑘6. (4.28) Clearly, the 𝐶⁰ virtual controller 𝑥^∗_𝑘+11=−𝜆_𝑘𝛼_𝑘𝜉^(𝑚𝑘_𝑘^+𝜏)/𝜎𝑀+𝑛−𝑘+𝑙_𝑘1+⋯+𝑙_𝑘6≜−𝛼_𝑘𝛽_𝑘𝜉^𝑚𝑘+1_𝑘, (4.29) with 𝛽_𝑘>0 being constant, results in ℒ𝑉_𝑘≤−(𝑀+𝑛−𝑘)𝜉⁴₁+⋯+𝜉⁴_𝑘+𝑑_𝑘𝑢₀𝜉^{(4𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥_𝑘+1−𝑥^∗_𝑘+1. (4.30)

This completes the proof of the inductive step.

The inductive argument shows that (4.30) holds for 𝑘=𝑛. Hence, we conclude that at the last step the actual controller 𝑢₁=𝑥^∗_𝑛+1=−𝛽_𝑛𝜉^𝑚𝑛+1_𝑛^/𝜎, (4.31) with 𝛽_𝑛>0 being constant and a 𝐶² positive definite and proper Lyapunov function 𝑉_𝑛(𝑥_[𝑛]) of the form (4.14) and (4.15), such that ℒ𝑉_𝑛≤−𝑀𝜉⁴₁+⋯+𝜉⁴_𝑛. (4.32)

We have thus far completed the controller design procedure for 𝑥₀(0)≠0.

4.2. Control for 𝑥₀(0)=0

In the last subsection, we gave the controller expressions (4.1) and (4.31) for 𝑢₀ and 𝑢₁ of system (3.1) in the case of 𝑥₀(0)≠0. Now, we consider finite-time control laws for the case of 𝑥₀(0)=0. In the absence of the disturbances, most of the commonly used control strategies use constant control 𝑢₀=𝑢^∗₀≠0 in time interval [0,𝑡_𝑠). In this paper, we also use this method when 𝑥₀(0)=0, with 𝑢₀ chosen as follows: 𝑢₀=𝑢^∗₀,𝑢^∗₀>0. (4.33)

Since 𝑥₀(0)=0, from the 𝑥₀-subsystem we know that 𝑥̇(0)=𝑢^∗₀(0)=𝑢^∗₀≠0. (4.34) We have 𝑥₀ does not escape and 𝑥(𝑡_𝑠)≠0, for given any finite 𝑡_𝑠>0.

During the time period [0,𝑡_𝑠), using 𝑢₀ defined in (4.33), new control law 𝑢₁=𝑢^∗₁(𝑥₀,𝑥) can be obtained by the control procedure described above to the original 𝑥-subsystem in (3.1). Then we can conclude that the 𝑥-state of (3.1) cannot blow up during the time period [0,𝑡_𝑠). Since 𝑥₀(𝑡_𝑠)≠0 at 𝑡=𝑡_𝑠, we can switch the control inputs 𝑢₀ and 𝑢₁ to (4.1) and (4.31), respectively.

The following theorem summarizes the main result of this paper.

Theorem 4.9. Under Assumptions 3.2 and 3.3, if the proposed control design procedure together with the above switching control strategy is applied to system (3.1), then, for any initial conditions in the state space (𝑥₀,𝑥)∈𝑅^𝑛+1, the closed-loop system is globally finite-time regulated at origin in probability.

Proof. According to the above analysis, it suffices to prove the statement in the case where 𝑥₀(0)≠0.
Since we have already proven that 𝑥₀ can be globally finite-time regulated to zero in Section 4.1, we just need to show that 𝑥(𝑡) is globally stochastically convergence to zero in a finite time. For the system (4.7)+(4.31), it is not hard to verify that all conditions in Lemma 2.1 are satisfied, which means that the closed-loop system admits a unique solution. In this case, choose the Lyapunov function 𝑉=𝑉_𝑛, from (4.32), its time derivative is given by ℒ𝑉≤−𝑀𝜉⁴₁+⋯+𝜉⁴_𝑛. (4.35) Let 𝛼=4𝜎/(4𝜎−𝜏), by Proposition 4.3 and Lemma 2.5, we have 𝑉^𝛼≤2𝜉⁴₁+⋯+𝜉⁴_𝑛. (4.36) Then, putting (4.36) back to (4.35) gives ℒ𝑉≤−𝑀𝑉^𝛼/2. (4.37)
By Lemma 2.4, system (4.7) under control law (4.31) is finite-time stable in probability with its settling time 𝑇_𝑥 satisfying 𝑇_𝑥≤2𝑉^(1−𝛼)(0)𝑀().1−𝛼 (4.38)
Hence with the choice of 𝑀 satisfying 𝑀>2𝑉^(1−𝛼)(0)/[𝑇_∗(1−𝛼)], 𝑇_𝑥<𝑇_∗ is guaranteed. Thus, the conclusion follows.

Remark 4.10. As seen from (4.31) and (4.1), the control law 𝑢₁ may exhibit extremely large value when 𝑥₀(0)≠0 is sufficiently small. This is unacceptable from a practical point of view. It is therefore recommended to apply (4.33) in order to enlarge the initial value of 𝑥₀ before we appeal to the finite-time converging controllers (4.1) and (4.31).

5. Application to Mobile Robot

In this section, we illustrate systematic controller design method proposed above by means of the example of mobile robot.

Consider the tricycle-type mobile robot with parametric uncertainty [34], which is described by 𝑥̇_𝑐=𝑝^∗₁𝑦̇𝑣cos𝜃,_𝑐=𝑝^∗₁̇𝑣sin𝜃,𝜃=𝑝^∗₂𝜔, (5.1) where 𝑝^∗₁ and 𝑝^∗₂ are unknown parameters taking values in a known interval [𝑝_min,𝑝_max] with 0<𝑝_min<𝑝_max<∞, 𝑣 and 𝜔 are two control inputs to denote the linear velocity and angular velocity, respectively.

When the angular velocity 𝜔 is subject to some stochastic disturbances, that is, 𝜔𝑥_𝑐,𝑦_𝑐,𝜃=𝜔₁𝑥_𝑐,𝑦_𝑐,𝜃+𝜔₂𝑥_𝑐,𝑦_𝑐,𝜃𝐵̇(𝑡), (5.2) where 𝐵(𝑡) is the so-called white noise. Then system (5.1) is transformed into 𝑑𝑥_𝑐=𝑝^∗₁𝑣cos𝜃𝑑𝑡,𝑑𝑦_𝑐=𝑝^∗₁𝑣sin𝜃𝑑𝑡,𝑑𝜃=𝑝^∗₂𝜔₁𝑑𝑡+𝑝^∗₂𝜔₂𝑑𝐵. (5.3)

For system (5.3), by taking the following state and input transformation: 𝑥₀=𝑥_𝑐,𝑥₁=𝑦_𝑐,𝑥₂=tan𝜃,𝑢₀=𝑣cos𝜃,𝑢₁=𝜔₁sec²𝜃, (5.4) we obtain 𝑑𝑥₀=𝑝^∗₁𝑢₀𝑑𝑡,𝑑𝑥₁=𝑝^∗₁𝑥₂𝑢₀𝑑𝑡,𝑑𝑥₂=𝑝^∗₂𝑢₁𝑑𝑡+𝑝^∗₂1+𝑥²₂𝜔₂𝑑𝐵. (5.5)

Clearly, system (5.5) is a special case of system (3.1). As discussed in Remark 3.4, there always exist some nonlinearities satisfy Assumption 3.3. For simplicity, it is assumed the 𝜔₂=(𝑥₂sin𝑥₁)/(1+𝑥²₂)². And as in [8], 𝑝_min=1, 𝑝_max=2, which are known for us in contructing control laws to make (5.5) finite-time stable, and parameters 𝑝^∗₁=𝑝^∗₁=1.5 are unknown. When (𝑥₀(0),𝑥₁(0),𝑥₂(0))=(1,1,1), by choosing 𝜏=−2/11 and 𝜎=3, according to the design procedure proposed in Section 4, the following controllers can be obtained for a given settling time 𝑇=6: 𝑢₀1=−2𝑥₀^1/3,𝑢₁=−𝑥₂^11/3−(2𝑀+2)^11/3𝑥³₁^7/33𝑀+𝑙₂₁+𝑙₂₂+𝑙₂₄, (5.6) where 𝑙₂₁, 𝑙₂₂ and 𝑙₂₄ are known positive constants. Choosing design parameter as 𝑀=1, the simulation results in Figures 1 and 2 show that the effectiveness of the controller.

Figure 1 

Time history of system states.

Figure 2 

Time history of control inputs.

6. Conclusion

In this paper, the finite-time state feedback stabilization problem has been investigated for a class of nonholonomic systems with stochastic disturbances. With the help of adding a power integrator technique, a systematic control design procedure is developed in the stochastic setting. To get around the stabilization burden associated with nonholonomic systems, a switching control strategy is proposed. It is shown that the designed control laws can guarantee that the closed-loop system states are globally finite-time regulated to zero in probability. In addition, the proposed approach can be applied to mobile robot with stochastic disturbances.

There are some related problems to investigate, for example, how to design a finite-time state-feedback stabilizing controller for stochastic nonholonomic systems when the 𝑥₀-subsystem contains stochastic disturbances. Furthermore, if only partial state vector being measurable, how to design a finite-time output feedback stabilizing controller for stochastic nonholonomic systems.

Appendix

Proof of Proposition 4.4. Recall that 𝜉_𝑖=𝑥^{𝜎/𝑚𝑖}_𝑖−𝑥^{∗𝜎/𝑚𝑖}_𝑖 and 𝑥^∗_𝑖=−𝛼_𝑖−1𝛽_𝑖−1𝜉^{𝑚𝑖/𝜎}_𝑖−1. By Lemma 2.5, for 𝑖=2,…,𝑘, 𝑥_𝑖≤𝜉_𝑖+𝑥^{∗𝜎/𝑚𝑖}_𝑖^{𝑚𝑖/𝜎}≤ℎ𝜉_𝑖−1+𝜉_𝑖^{𝑚𝑖/𝜎}, (A1) where ℎ=max{1,𝛽^𝜎/𝑚1₁,…,𝛽^{𝜎/𝑚𝑛}_𝑛}.
With (4.17), (4.18), and (A1), we get ^𝑘−1_𝑖=1𝜕𝑊_𝑘𝜕𝑥_𝑖𝑑_𝑖𝑢₀𝑥_𝑖+1=^𝑘−1_𝑖=1−𝑐_𝑖4𝜎−𝜏−𝑚_𝑘𝜎𝑥^{(𝜎−𝑚𝑖)/𝑚𝑖}_𝑖×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝑑𝑠×𝑑_𝑖𝑢₀𝑥_𝑖+1≤𝑎_𝑘^𝑘−1_𝑖=1𝜉_𝑘^{(3𝜎−𝜏)/𝜎}𝑥_𝑖^{(𝜎−𝑚𝑖)/𝑚𝑖}𝑥_𝑖+1≤𝑎_𝑘^𝑘−1_𝑖=1𝜉_𝑘^{(3𝜎−𝜏)/𝜎}𝜉_𝑖−1+𝜉_𝑖^{(𝜎−𝑚𝑖)/𝜎}𝜉_𝑖+𝜉_𝑖+1^{𝑚𝑖+1/𝜎}, (A2) where 𝑎_𝑘 and 𝑎_𝑘 are positive constants.
Applying Lemma 2.6(3𝜎−𝜏+𝜎−𝑚_𝑖+𝑚_𝑖+1=4𝜎) to (A2), we can find a positive constant 𝑙_𝑘2 such that ^𝑘−1_𝑖=1𝜕𝑊_𝑘𝜕𝑥_𝑖𝑑_𝑖(𝑡)𝑢₀𝑥_𝑖+1≤13𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘2𝜉^𝑑_𝑘. (A3)

Proof of Proposition 4.5. According to (4.17), (4.18), (A1), and Remark 3.4, we have 12^𝑘−1_{𝑖,𝑗=1,𝑖≠𝑗}𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗𝑔^𝑇_𝑖𝑔^𝑇_𝑗=12^𝑘−1_{𝑖,𝑗=1,𝑖≠𝑗}𝑐_𝑖𝑐_𝑗4𝜎−𝜏−𝑚_𝑘𝜎⋅3𝜎−𝜏−𝑚_𝑘𝜎𝑥^{(𝜎−𝑚𝑖)/𝑚𝑖}_𝑖𝑥^{(𝜎−𝑚𝑗)/𝑚𝑗}_𝑗×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{(2𝜎−𝜏−𝑚𝑘)/𝜎}𝑔^𝑇_𝑖𝑔^𝑇_𝑗≤𝑏_𝑘^𝑘−1_{𝑖,𝑗=1,𝑖≠𝑗}𝜉_𝑘^{(2𝜎−𝜏)/𝜎}𝑥_𝑖^{(𝜎−𝑚𝑖)/𝑚𝑖}𝑥_𝑗^{(𝜎−𝑚𝑗)/𝑚𝑗}𝑔^𝑇_𝑖𝑔^𝑇_𝑗≤𝑏_𝑘^𝑘−1𝑖_{𝑖,𝑗=1,𝑖≠𝑗}^𝑗_𝑙=1𝑟=1𝜉_𝑘^{(2𝜎−𝜏)/𝜎}𝑥^{(𝜎−𝑚𝑖)/𝑚𝑖}_𝑖𝑥^{(𝜎−𝑚𝑗)/𝑚𝑗}_𝑗×𝑥_𝑙^{(2𝑚𝑖+𝜏)/2𝑚𝑙}𝑥_𝑟^{(2𝑚𝑗+𝜏)/2𝑚𝑟}≤𝑏_𝑘^𝑘−1𝑖_{𝑖,𝑗=1,𝑖≠𝑗}^𝑗_𝑙=1𝑟=1𝜉_𝑘^{(2𝜎−𝜏)/𝜎}𝜉_𝑖−1+𝜉_𝑖^{(𝜎−𝑚𝑖)/𝜎}×𝜉_𝑗−1+𝜉_𝑗^{(𝜎−𝑚𝑗)/𝜎}×𝜉_𝑙−1+𝜉_𝑙^{(2𝑚𝑖+𝜏)/2𝜎}×𝜉_𝑟−1+𝜉_𝑟^{(2𝑚𝑗+𝜏)/2𝜎}, (A4) where 𝑏_𝑘, 𝑏_𝑘 and 𝑏_𝑘 are positive constants.
Applying Lemma 2.6 to the above inequality, we have 12^𝑘−1_{𝑖,𝑗=1,𝑖≠𝑗}𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗𝑔^𝑇_𝑖𝑔^𝑇_𝑗≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘3𝜉⁴_𝑘, (A5) where 𝑙_𝑘3 is a positive constant.

Proof of Proposition 4.6. From (4.17), (4.18), (A1), and Remark 3.4, with Lemma 2.6, it is deduced that 12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥²_𝑖𝑔^𝑇_𝑖²=12^𝑘−1_𝑖=1𝑐²_𝑖4𝜎−𝜏−𝑚_𝑘𝜎⋅3𝜎−𝜏−𝑚_𝑘𝜎𝑥^{2(𝜎−𝑚𝑖)/𝑚𝑖}_𝑖×^𝑥𝑘_𝑥∗𝑘𝑠^1/𝑚𝑘−𝑥^{∗1/𝑚𝑘}_𝑘^{(2𝜎−𝜏−𝑚𝑘)/𝜎}𝑑𝑠−𝑐²_𝑖4𝜎−𝜏−𝑚_𝑘𝜎𝑥^{(𝜎−2𝑚𝑖)/𝑚𝑖}_𝑖×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝑑𝑠𝑔^𝑇_𝑖²≤𝑑_𝑘^𝑘−1_𝑖=1𝜉_𝑘^{(2𝜎−𝜏)/𝜎}𝑥_𝑖^{2(𝜎−𝑚𝑖)/𝑚𝑖}+𝜉_𝑘^{(3𝜎−𝜏)/𝜎}𝑥^{(𝜎−2𝑚𝑖)/𝑚𝑖}_𝑖𝑔^𝑇_𝑖²≤𝑑_𝑘^𝑘−1𝑖_𝑖=1𝑗=1𝜉_𝑘^{(2𝜎−𝜏)/𝜎}𝑥_𝑖^{2(𝜎−𝑚𝑖)/𝑚𝑖}+𝜉_𝑘^{(3𝜎−𝜏)/𝜎}𝑥^{(𝜎−2𝑚𝑖)/𝑚𝑖}_𝑖𝑥^{(2𝑚𝑖+𝜏)/𝑚𝑗}_𝑗≤𝑑_𝑘^𝑘−1𝑖_𝑖=1𝑗=1𝜉_𝑘^{(2𝜎−𝜏)/𝜎}𝜉_𝑖−1+𝜉_𝑖^{2(𝜎−𝑚𝑖)/𝜎}+𝜉_𝑘^{(3𝜎−𝜏)/𝜎}𝜉_𝑖−1+𝜉_𝑖^{(𝜎−2𝑚𝑖)/𝜎}×𝜉_𝑗−1+𝜉_𝑗^{(2𝑚𝑖+𝜏)/𝜎}≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘4𝜉⁴_𝑘, (A6) where 𝑑_𝑘, 𝑑_𝑘, 𝑑_𝑘 and 𝑙_𝑘4 are positive constants.

Proof of Proposition 4.7. From (4.17), (4.18), (A1), and Remark 3.4, with Lemma 2.6, it follows that 12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥_𝑘𝜕𝑥_𝑖𝑔^𝑇_𝑘𝑔^𝑇_𝑖=12^𝑘−1_𝑖=1𝑐_𝑖4𝜎−𝜏−𝑚_𝑘𝜎𝜉^{(3𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥^{(𝜎−𝑚𝑖)/𝑚𝑖}_𝑖𝑔^𝑇_𝑘𝑔^𝑇_𝑖≤𝑒_𝑘^𝑘−1_𝑖=1𝜉_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝑥_𝑖^{(𝜎−𝑚𝑖)/𝑚𝑖}𝑔^𝑇_𝑘𝑔^𝑇_𝑖≤𝑒_𝑘^𝑘−1𝑘_𝑖=1^𝑖_𝑙=1𝑟=1𝜉_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝑥_𝑖^{(𝜎−𝑚𝑖)/𝑚𝑖}𝑥_𝑙^{(2𝑚𝑘+𝜏)/2𝑚𝑙}𝑥_𝑟^{(2𝑚𝑖+𝜏)/2𝑚𝑟}≤𝑒_𝑘^𝑘−1𝑘_𝑖=1^𝑖_𝑙=1𝑟=1𝜉_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝜉_𝑖−1+𝜉_𝑖^{(𝜎−𝑚𝑖)/𝜎}×𝜉_𝑙−1+𝜉_𝑙^{(2𝑚𝑘+𝜏)/2𝜎}×𝜉_𝑟−1+𝜉_𝑟^{(2𝑚𝑖+𝜏)/2𝜎}≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘5𝜉⁴_𝑘, (A7) where 𝑒_𝑘, 𝑒_𝑘, 𝑒_𝑘 and 𝑙_𝑘5 are positive constants.

Proof of Proposition 4.8. By (4.17), (4.18), (A1), and Remark 3.4, with Lemma 2.6, we can obtain 12𝜕²𝑊_𝑘𝜕𝑥²_𝑘𝑔^𝑇_𝑘²=12𝑐_𝑘4𝜎−𝜏−𝑚_𝑘𝜎𝜉^{(3𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥^{(𝜎−𝑚𝑘)/𝑚𝑘}_𝑘𝑔^𝑇_𝑘²≤𝑓_𝑘^𝑘_𝑖=1𝜉_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝑥_𝑘^{(𝜎−𝑚𝑘)/𝑚𝑘}𝑥_𝑖^{(2𝑚𝑘+𝜏)/2𝑚𝑖}≤𝑓_𝑘^𝑘_𝑖=1𝜉_𝑘^{(3𝜎−𝜏−𝑚𝑘)/𝜎}𝜉_𝑘−1+𝜉_𝑘^{(𝜎−𝑚𝑘)/𝜎}×𝜉_𝑖−1+𝜉_𝑖^{(2𝑚𝑘+𝜏)/𝜎}≤16𝜉⁴₁+⋯+𝜉⁴_𝑘−1+𝑙_𝑘6𝜉⁴_𝑘, (A8) where 𝑓_𝑘, 𝑓_𝑘, and 𝑙_𝑘6 are positive constants.

Acknowledgments

This work has been supported in part by National Nature Science Foundation of China under Grant 61073065 and Nature Science Foundation of Henan Province under Grant 092300410145. The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper.