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 自动化学报  2017, Vol. 43 Issue (7): 1273-1279 PDF

1. 山东大学控制科学与工程学院 济南 250061;
2. 临沂大学自动化与电气工程学院 临沂 276005;
3. 曲阜师范大学自动化研究所 曲阜 273165

Adaptive Control for High-order Nonlinear Feedforward Systems With Input and State Delays
Yaxin Huang1, Xinghui Zhang2, Mengmeng Jiang3
1. School of Control Science and Engineering, Shandong University, Jinan 250061, China;
2. School of Automation and Electrical Engineering, Linyi University, Linyi 276005, China;
3. Institute of Automation, Qufu Normal University, Qufu 273165, China
Abstract: This paper considers the adaptive stabilization of a class of high-order uncertain nonlinear feedforward systems. The growth condition on nonlinearities is further relaxed by allowing not only input and state delays but also the unknown growth rate. By a combined method of adaptive technique, dynamic gain control approach and adding a power integrator technique, a state feedback controller is designed to guarantee that all signals are bounded, the equilibrium point of the closed-loop system is globally stable and the original state converges to zero.
Key words: Adaptive control     dynamic gain     high-order nonlinear feedforward systems     input and state delays
1 Introduction

Feedforward systems have both practical and theoretical importance. Many physical devices, such as the ball and beam with a friction term [1], the planar vertical takeoff and landing aircraft in [2] and the cart-pendulum system [3], can be described by equations with the feedforward structure.

This paper considers a class of high-order uncertain nonlinear feedforward systems with input and state delays described by

 \begin{align}\label{9.1} \dot x_i(t)= & \ x_{i+1}^{p_i}(t)+ f_i(t, x_{[i+2]}(t), u(t), x_{[i+2]}(t-d_i(t)), \cr & \ u(t-d_i(t)), \theta), \quad i=1, \ldots, n-1\nonumber\\ \dot x_n(t)= & \ u^{p_n}(t) \end{align} (1)

where $x_{[i+2]}(t)=[x_{i+2}(t), \ldots, x_n(t)]^{T}\in \mathbb{R}^{n-i-1}$, and $u(t)$ $\in$ $\mathbb{R}$ are the system state and control input, respectively. $\theta$ is an unknown parameter vector, which may be constant or bounded time-varying. For $i=1, \ldots, n-1$, $d_i(t)$ is a time-varying delay with $0\leq d_i(t)\leq h_i$ and $\dot d_i(t)\leq\tau_i<1$, where $h_i$ and $\tau_i$ are known constants; $p_i\in \mathbb{R}_{\rm odd}^{\geq1}:=\{\frac{ p}{q}\in \mathbb{R}^+$: $p$ and $q$ are odd integers, $p\geq q\}$; $f_i$ is continuous in $t$ and locally Lipschitz in $[x_{[i+2]}(t), u(t), x_{[i+2]}(t-d_{i}(t)), u(t-d_{i}(t))]^{T}$ with $f_{i}(t, 0, 0, 0, 0, \theta)=0$. The initial condition $x(r)$ is a continuous function on $[-h, 0]$ with $h\geq \max\{h_1,$ $\ldots,$ $h_{n-1}\}$. System (1) is called as high-order system if there exists at least one $p_i>1$ ($1\leq i\leq n$).

Among the existing results in handling feedforward systems, much of the attention has been focused on the stabilization of a chain of linear integrators perturbed by an upper-triangular nonlinear vector field, which is a special case of system (1) by setting $p_{i}=1$. In this case, for system (1) with $d_{i}=0$ and $\theta=0$, some papers such as [4]-[8] give the global asymptotic stabilization results. For uncertain system (1) with $d_{i}=0$ and $\theta\neq0$, [9] gives a design scheme of global adaptive stabilizer. When $d_{i}\neq0$ and $\theta=0$, though intractable difficulties arise in control design, such as trade-off time-delay effect, identification of time-delay restriction, some results are obtained for system (1), see [10], [11] and references therein. A further result [12] considers the adaptive stabilization problem for feedforward system (1) with $d_{i}$ $\neq$ $0$ and $\theta\neq0$.

For the high-order case when $p_i\geq1$, due to some intrinsic features, e.g., the Jacobian linearization is neither controllable nor feedback linearizable, only a few results are achieved for feedforward system (1). In [13], system (1) with input delays but no unknown parameter $\theta$ is considered. For system (1) with $d_{i}=0$ and $\theta=0$, [14] and [15] establish the global asymptotic stabilization. In [16], an adaptive approach is employed to design a state feedback controller for system (1) with $d_{i}=0$ and $\theta\neq0$. However, to the best of the authors' knowledge, there is no result reported in the literature on the stabilization of high-order feedforward uncertain nonlinear system (1) with both $d_{i}\neq0$ and $\theta$ $\neq$ $0$, and this motivates us to consider this interesting and challenging problem.

In this paper, by using a combined method of adaptive technique, dynamic gain control approach and adding a power integrator technique, and solving several difficulties in the design and analysis, a state feedback controller is designed to guarantee that not only all signals are bounded but also the asymptotic regulation is achieved. Our work has two distinctive features. First, the higher power $p_{i}\geq1$, the existence of input and state delays, and unknown parameter in nonlinearities render system (1) more general. Second, due to the higher power, time-varying delays in both input and state, and unknown parameter, one of the main difficulties dealt in this paper is to determine the adaptive dynamic gain and estimation, based on which the desired adaptive controller is designed by combining the adding a power integrator technique.

This paper is organized as follows: Section 2 gives some preliminaries. Sections 3 and 4 provide the design and analysis of the controller, following a simulation example in Section 5. Section 6 concludes this paper. Proofs of lemmas and propositions are given in Appendices A-C.

2 Mathematical Preliminaries

Notations:$\mathbb{R}^+$ stands for the set of all the nonnegative real numbers. Let $x=[x_1, \ldots, x_n]^{T}\in \mathbb{R}^{n}$, $\bar x_i=[x_1, \ldots,$ $x_i]^{T}$ $\in$ $\mathbb{R}^{i}$, and $x_{[i]}=[x_{i}, \ldots, x_{n}]^{T}\in \mathbb{R}^{n-i+1}$, $i=1, \ldots,$ $n-1$; $\mathcal{C}_h=\mathcal{C}([-h, 0]; \mathbb{R}^n)$ denotes the set of all the continuous functions mapping $[-h, 0]$ into $\mathbb{R}^n$. $\|x\|=(\sum_{i=1}^nx_i^2)^{\frac{1}{2}}$, and $\| x_{t}\|_{\mathcal{C}_h} =\sup_{-h \leq r\leq 0}\| x(t+r)\|$ with $x_t\in \mathcal{C}_h$ being defined by $x_t(r)=x(t+r)$, $r\in [-h, 0]$.

We give some useful lemmas.

Lemma 1 [17]: For $x, y\in \mathbb{R}$, if $p\geq 1$ is a constant, then $|x$ $+$ $y|^{p}\leq 2^{p-1}(|x|^p+|y|^p),$ $(|x|+|y|)^{\frac{1}{p}}\leq |x|^{\frac{1}{p}}+|y|^{\frac{1}{p}}\leq 2^{1-\frac{1}{p}} (|x|+|y|)^{\frac{1}{p}}$. If $p\in \mathbb{R}_{\rm odd}^{\geq1}$, then $|x-y|^p\leq 2^{p-1}|x^p-y^p|$.

Lemma 2 (Young's inequality): Let real numbers $p\geq1$ and $q\geq1$ satisfy $1/p+1/q=1$. Then $xy\leq\gamma|x|^p+\frac{1}{q}(p\gamma)^{-\frac{q}{p}}|y|^{q}\nonumber$ for any $x, y\in \mathbb{R}$ and any given positive number $\gamma$ $>$ $0$.

Lemma 3: For given $p\in \mathbb{R}_{\rm odd}^{\geq1}$ and $x, y\in \mathbb{R}$, there holds $(x-y)^{\frac{1}{p}}y^{2-\frac{1}{p}}\leq-\frac{y^2}{2} +2^{2p-3}p^{-2p}(2p-1)^{2p-1} x^2$.

Proof: See Appendix A.

3 Design of Adaptive State Feedback Controller 3.1 Problem Statement

Throughout this paper, we need the following assumption.

Assumption 1: For $i=1, \ldots, n-1$, there is an unknown positive constant $\Theta$ dependent on $\theta$ such that

 \begin{align}\label{9.5}|f_i(\cdot)|\leq & \ \Theta\sum\limits_{j=i+2}^{n+1}\left(|x_j(t)|^{p_i\ldots p_{j-1}} +|x_j(t-d_i(t))|^{p_i\ldots p_{j-1}}\right)\cr & \ +\Theta\left(|u(t)|^{p_i\ldots p_n}+|u(t-d_i(t))|^{p_i\ldots p_n}\right)\end{align} (2)

where $x_{n+1}(t)=x_{n+1}(t-d_i(t))=0$.

Remark 1: In the existing results on high-order feedforward systems, [13] considers system (1) with input delays but no unknown parameter $\theta$. For system (1) with $d_{i}=0$ and $\theta=0$, [14] and [15] establish the global asymptotic stabilization. In [16], an adaptive approach is employed to design a state feedback controller for system (1) with $d_{i}=0$ and $\theta\neq0$.

The purpose of this paper is to design a continuous adaptive state feedback stabilizing controller

 $\begin{eqnarray}\label{9.4} u=u(x, \hat{\Theta}), \quad \dot{\hat{\Theta}}=\Omega(x, \hat{\Theta})\end{eqnarray}$ (3)

such that all signals of the closed-loop system (1) and (3) are globally uniformly bounded, and system state $x(t)$ converges to zero for any given initial value, where functions $\Omega(\cdot)$ and $u(\cdot)$ are smooth and continuous, respectively, $\hat{\Theta}$ is the estimate of $\Theta$ and will be specified later.

3.2 Design Procedure

We give the design procedure in four parts.

Part Ⅰ: State transformation

Introduce the following rescaling transformation

 \begin{align}\label{9.6} z_i(t)=L^{-m_i}(t)x_i(t), v(t)=L^{-m_{n+1}}(t)u(t), ~~i=1, \ldots, n\end{align} (4)

where $L(t)\geq1$ is a continuously differentiable and monotone increasing function that will be determined later, $m_i$ is defined as $m_1=1+\sum_{j=1}^{n}p_1\ldots p_j$, $m_{i+1}=({m_{i}-1})/{p_{i}}$, $i$ $=$ $1,$ $\ldots,$ $n$.

With transformation (4), system (1) can be converted into

 \begin{align}\label{9.7} \dot z_i= & \ L^{-1} (z_{i+1}^{p_i}-m_i\dot Lz_i)+\phi_i(t, z_{[i+2]}, v, z_{[i+2]}(t-d_i(t)), \cr & \ v(t-d_i(t)), \theta, L, L(t-d_{i}(t))), ~~~i=1, \ldots, n-1\cr \dot z_n= & \ L^{-1}(v^{p_n}-m_n\dot Lz_n)\end{align} (5)

where $\phi_i(\cdot)=L^{-m_i}f_i(\cdot)$, $i=1, \ldots, n-1$.

Part Ⅱ: State feedback controller design for system (5)

Step 1: Define $z_{1}=\xi_{1}$ and choose $V_1(\xi_{1})={\xi_1^2}/{2}$. From (5), it follows that $\dot V_1(\xi_{1})|_(5)=-nL^{-1}\xi_1^2-m_1\dot LL^{-1}\xi_1^2 +L^{-1}\xi_1(z_2^{p_1}-\alpha_1^{p_1}) +L^{-1}\xi_1(\alpha_1^{p_1}+n\xi_1) +\frac{\partial V_{1}}{\partial z_{1}}\phi_{1}$. Choosing $\alpha_1^{p_1}=-g_1\xi_1$, $g_1=n$, $\lambda_1=m_1$, $\gamma_0=1$ yields $\dot V_1(\xi_{1})|_(5)=-n\gamma_0L^{-1}\xi_1^2 -\lambda_1\dot LL^{-1}\xi_1^2+L^{-1}\xi_1(z_2^{p_1}-\alpha_1^{p_1}) +\frac{\partial V_{1}}{\partial z_{1}}\phi_{1}$.

Step k (k = 2, . . . , n): We start with the following proposition.

Proposition 1: Suppose that there exist a continuously differentiable Lyapunov function $V_{k-1}(\bar\xi_{k-1})$ and a series of virtual controllers $\alpha_1, \ldots, \alpha_{k-1}$ defined by

 \begin{align}\label{9.8} & \alpha_0=0 \cr & \alpha_i^{p_1\ldots p_i}=-g_i\xi_i \cr & \xi_i=z_i^{p_1\ldots p_{i-1}}-\alpha_{i-1}^{p_1\ldots p_{i-1}}, \quad i=1, \ldots, k-1\end{align} (6)

such that

 \begin{align}\label{9.13} & \dot V_{k-1}|_(5) \leq-(n-k+2)\gamma_0\ldots\gamma_{k-2}L^{-1}\sum\limits_{i=1}^{k-1}\xi_i^2-\lambda_{k-1}LL^{-1} \cr & \qquad \times\sum\limits_{i=1}^{k-1}\xi_i^2 +L^{-1}\xi_{k-1}^{2-\frac{1}{p_1\ldots p_{k-2}}} \left(z_k^{p_{k-1}}-\alpha_{k-1}^{p_{k-1}}\right) \cr & \qquad +\sum\limits_{i=1}^{k-1}\frac{\partial V_{k-1}}{\partial z_{i}}\phi_{i}\end{align} (7)

where $\gamma_0, \ldots, \gamma_{k-2}$, $g_1, \ldots, g_{k-1}$, $\lambda_{k-1}$ are positive constants. Then, by defining $\xi_k=z_k^{p_1\ldots p_{k-1}}-\alpha_{k-1}^{p_1\ldots p_{k-1}}$ and choosing the $k$th continuous differential function

 \begin{align}\label{9.14} & V_{k}(\bar\xi_{k})=\gamma_{k-1}V_{k-1}(\bar\xi_{k-1})+W_k(\bar\xi_{k})\notag\\ & W_k(\bar\xi_{k})=\int_{\alpha_{k-1}}^{z_k}\left(s^{p_1\ldots p_{k-1}} -{\alpha_{k-1}}^{p_1\ldots p_{k-1}}\right) ^{2-\frac{1}{p_1\ldots p_{k-1}}}ds \end{align} (8)

one can design a virtual controller $\alpha_k^{p_1\ldots p_k}=-g_k\xi_k$ such that

 \begin{align} & {{{\dot{V}}}_{k}}{{|}_{(5)}}\le -(n-k+1){{\gamma }_{0}}\ldots {{\gamma }_{k-1}}{{L}^{-1}}\sum\limits_{i=1}^{k}{\xi _{i}^{2}} \\ & \qquad \quad -{{\lambda }_{k}}\dot{L}{{L}^{-1}}\cdot \sum\limits_{i=1}^{k}{\xi _{i}^{2}}+{{L}^{-1}}\xi _{k}^{2-\frac{1}{{{p}_{1}}\ldots {{p}_{k-1}}}}\left( z_{k+1}^{{{p}_{k}}}-\alpha _{k}^{{{p}_{k}}} \right) \\ & \qquad \quad +\sum\limits_{i=1}^{k}{\frac{\partial {{V}_{k}}}{\partial {{z}_{i}}}}{{\phi }_{i}}. \\ \end{align} (9)

Proof: See Appendix B.

Hence at step $n$, by choosing

 \begin{align}\label{9.15.1} & V_n(\xi)%=\gamma_{n-1}V_{n-1}(\bar\xi_{n-1})+W_n(\xi) =\sum\limits_{k=1}^{n-1}\gamma_{k}\ldots\gamma_{n-1}W_k(\bar\xi_{k})+W_n(\xi)\notag\\ & \alpha_0=0\notag\\ & \alpha_i^{p_1\ldots p_i}=-g_i\xi_i \notag\\ & \xi_{i+1}=z_{i+1}^{p_1\ldots p_{i}}-\alpha_{i}^{p_1\ldots p_{i}}, \quad i=1, \ldots, n-1 \end{align} (10)

one can design the controller as

 \begin{align}\label{9.16} & v=-(g_n\xi_{n})^{\frac{1}{p_1\ldots p_n}}\cr & \xi_{n}=z_{n}^{p_{1}\ldots p_{n-1}}+\sum\limits_{k=1}^{n-1} \bigg(\prod\limits_{l=k}^{n-1}g_{l}\bigg)z_{k}^{p_{1}\ldots p_{k-1}}\end{align} (11)

such that

 \begin{align}\label{9.17} \dot V_n|_(5)\leq-\gamma L^{-1}\sum\limits_{i=1}^{n}\xi_i^2 -\lambda\dot LL^{-1}\sum\limits_{i=1}^{n}\xi_i^2 +\sum\limits_{i=1}^{n}\frac{\partial V_{n}}{\partial z_{i}}\phi_{i}\end{align} (12)

where $\gamma=\gamma_0\ldots\gamma_{n-1}$, $\lambda=\lambda_n$.

Part Ⅲ: Gain assignment

To determine $L$ and $\hat\Theta$, we give the following proposition.

Proposition 2: There are positive constants $D$, $\nu$, such that

 \begin{align*} \sum\limits_{i=1}^{n}\frac{\partial V_n}{\partial z_i}\phi_i \leq & \ n\Theta D L^{-(1+\nu)}\sum\limits_{i=1}^{n}\xi_i^2\notag\\ & +\Theta D L^{-(1+\nu)}\sum\limits_{k=1}^{n-1}\sum\limits_{i=1}^{n}\xi_i^2(t-d_k(t)). \end{align*}

Proof: See Appendix C.

Substituting Proposition 2 into (12) yields

 \begin{align}\label{9.28} \dot V_n|_(5)\leq & -\gamma L^{-1}\sum\limits_{i=1}^{n}\xi_i^2 -\lambda\dot L L^{-1}\sum\limits_{i=1}^{n}\xi_i^2 +n\Theta D L^{-(1+\nu)}\notag\\ & \ \times\sum\limits_{i=1}^{n}\xi_i^2 +\Theta D L^{-(1+\nu)} \sum\limits_{k=1}^{n-1}\sum\limits_{i=1}^{n}\xi_i^2(t\!-\!d_k(t)).\end{align} (13)

Defining a Lyapunov functional

 \begin{align}\label{9.29} V(t, \xi_{t})= & \ V_n(\xi)+\displaystyle\frac{\tilde{\Theta}^2(t)}{2}\cr & +D\Theta\sum\limits_{k=1}^{n-1}\bigg( \displaystyle\frac{1}{1-\tau_k}\int_{t-d_k(t)}^{t} L^{-(1+\nu)}(s)\cr & \times\sum\limits_{i=1}^{n}\xi_i^2(s)ds\bigg)\end{align} (14)

where $\tilde{\Theta}(t)=\Theta-\hat{\Theta}(t)$ is the parameter estimation error. Noting that $L(t)$ is increasing and using (13) and (14) render

 \begin{align*} \dot V|_(5) \leq & -\gamma L^{-1}\sum\limits_{i=1}^{n}\xi_i^2 -\lambda\dot L L^{-1}\sum\limits_{i=1}^{n}\xi_i^2 -\tilde{\Theta}\dot{\hat{\Theta}} \\ & +\frac{2nD\Theta L^{-(1+\nu)}}{1-\bar\tau}\sum\limits_{i=1}^{n}\xi_i^2\\ = & -L^{-(1+\nu)}\left(\gamma L^{\nu}+\lambda\dot{L}L^{\nu} -\frac{2nD}{1-\bar{\tau}}\hat{\Theta}\right) \sum\limits_{i=1}^{n}\xi_i^2\cr & +\tilde{\Theta}\left(\frac{2nD}{1-\bar{\tau}}\sum\limits_{i=1}^{n}\xi_i^2-\dot{\hat{\Theta}}\right) \end{align*}

where $\bar{\tau}=\max_{1\leq k\leq n-1}\{\tau_k\}$. Choosing

 \begin{align}\label{9.30} & \dot{L}=\max\left\{\displaystyle\frac{L^{-\nu}}{\lambda}\left(-\gamma L^{\nu} +\frac{2Dn}{1-\bar{\tau}}\hat{\Theta}+\mu\right), 0\right\}, \notag\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad L(r)=L_0\geq1 \cr & \dot{\hat{\Theta}}={\displaystyle\frac{2nD}{1-\bar{\tau}}}\sum\limits_{i=1}^{n}\xi_i^2, \hat{\Theta}(r)=\hat{\Theta}_0>0\quad \forall r\in[-h, 0] \end{align} (15)

one obtains

 \begin{align}\label{9.32} \dot{V}|_{(5)}\leq-L^{-(1+\nu)}\mu\sum\limits_{i=1}^{n}\xi_i^2 \end{align} (16)

where $\mu$ is a positive constant.

Part Ⅳ: State feedback controller of system (1)

By (4) and (11), one gets the state feedback controller of system (1) with the form

 \begin{align}\label{9.33} u(t)= & -L^{m_{n+1}}(t)\Bigg(\sum\limits_{k=1}^{n}\bigg(\prod\limits_{j=k}^{n}g_j\bigg) L^{-m_kp_1\ldots p_{k-1}}(t)\cr & \times x_k^{p_1\ldots p_{k-1}}(t)\Bigg)^ {\frac{1}{p_1\ldots p_n}}\end{align} (17)

where $L(t)$ is dynamically updated by (15).

4 Main Result

We state the main result of this paper.

Theorem 1: If Assumption 1 holds for system (1), then under the adaptive state feedback controller (15) and (17),

1) All the closed-loop states are bounded in $[-h, +\infty)$.

2) The equilibrium point $(x, \hat\Theta)=(0, \Theta)$ of the closed-loop system (1), (4), (10) and (15) is globally stable.

3) $\lim_{t\rightarrow +\infty}x(t)=0$.

Proof: 1) Firstly, we prove that the state $\Gamma(t):= [L(t),$ $\tilde{\Theta}(t),$ $z(t)]^{T}$ of the closed-loop system (5), (11) and (15) is bounded in $[-h, +\infty)$. By the existence and the continuation of solution, $\Gamma(t):= [L(t), \tilde{\Theta}(t), z(t)]^{T}$ is defined on a time interval $[-h, t_{M})$, where $t_M>0$ may be finite or infinite. Next we give the proof in three steps.

a) We now prove that $[\tilde{\Theta}(t), z(t)]^{T}$ is bounded in $[-h,$ $t_{M})$. Obviously, $V_n(t, \xi(t))$ is positive definite and radially unbounded on $\xi(t)$. By Lemma 4.3 in [18], there is a $\mathcal{K}_{\infty}$ function $\rho_1(\cdot)$ such that

 \begin{align}\label{9.34} V(t, \Upsilon_{t}))\geq V_n(t, \xi(t))+\frac{{\tilde{\Theta}}^2(t)}{2} \geq\rho_1(\|\Upsilon(t)\|)\end{align} (18)

where $\Upsilon(t)=[\tilde{\Theta}(t), \xi(t)]^{T}$. Noting that

 \begin{align*} & \sum\limits_{k=1}^{n-1}\frac{D}{1-\tau_k}\int_{t-d_k(t)}^{t} L^{-(1+\nu)}(s)\sum\limits_{i=1}^{n}\xi_i^2(s)ds\\ & \qquad \leq \sum\limits_{k=1}^{n-1}\frac{D h_k}{1-\tau_k} \sum\limits_{i=1}^{n}\sup\limits_{-h_k\leq r\leq0} \xi_i^2(t+r)\\ & \qquad \leq\frac{D h n^2}{1-\bar{\tau}}\left(\sup\limits_{-h\leq r\leq0} \|\Upsilon(t+r)\|\right)^2\end{align*}

and using (8), (10), (14) together with Lemma 1 and the mean value theorem, one gets

 \begin{align*} V(t, \Upsilon_{t})\leq & \ 2\sum\limits_{i=1}^{n}\gamma_{i-1}\ldots\gamma_{n}\xi_i^2(t)+ \|\Upsilon(t)\|^2\notag\\ & +n^2 D\frac{h}{1-\bar{\tau}} (\|\Upsilon_t\|_{\mathcal{C}_{h}})^2\\ \leq & \ 2n\sum\limits_{i=1}^{n}\gamma_{i-1}\ldots\gamma_{n} (\|\Upsilon_t\|_{\mathcal{C}_{h}})^2 +(\|\Upsilon_t\|_{\mathcal{C}_{h}})^2\\ & +n^2 D\frac{h}{1-\bar{\tau}}(\|\Upsilon_t\|_{\mathcal{C}_{h}})^2\\ := & \ \rho_2(\|\Upsilon_t\|_{\mathcal{C}_{h}})\end{align*}

where $\rho_2(\cdot)$ is a $\mathcal{K}_{\infty}$ function. Combining this with (18) yields

 \begin{align}\label{9.36} \rho_1(\|\Upsilon(t)\|)\leq V(t, \Upsilon_{t}) \leq\rho_2(\|\Upsilon_t\|_{\mathcal{C}_{h}}).\end{align} (19)

Since $\rho_1(\cdot)$ and $\rho_2(\cdot)$ are $\mathcal{K}_{\infty}$ functions, there is a $\beta=$ $\beta(\varepsilon)> \varepsilon>0$ such that $\rho_2(\varepsilon)\leq\rho_1(\beta)$ for any $\varepsilon>0$. By (15) and (19), one obtains that if $\|\Upsilon_0\|_{\mathcal{C}_h}<\varepsilon$, then $\rho_1(\|\Upsilon(t)\|)$ $\leq$ $V(t, \Upsilon_{t}) \leq V(0, \Upsilon(0))\leq\rho_2(\|\Upsilon_0\|_{\mathcal{C}_h}) <\rho_2(\varepsilon)\leq\rho_1(\beta)$, $\forall t$ $\in$ $[0, t_{M})$, which implies that $\|\Upsilon(t)\|<\beta$, $\forall t\in[-h, t_M)$. Hence, $\Upsilon(t)$ is bounded in $[-h, t_{M})$, so is $[\tilde{\Theta}(t), z(t)]^{T}$ because of (10).

b) We prove that $L(t)$ is bounded in $[-h, t_M)$. Noticing that $L(r)\equiv L_{0}$, $\forall r\in[-h, 0]$, hence, we only need to prove that $L(t)$ is bounded in $[0, t_M)$. Suppose that there is a finite $T\in[0, t_M)$ such that $\gamma L^{\nu}(T)=\frac{2n D}{1-\bar\tau}\hat\Theta(T)+\mu$, which indicates $\dot L(t)=0$, $\forall t\in[T, t_M)$. Hence, $L(t)$ is bounded in $[-h, t_M)$. Otherwise, there are two cases.

Case 1: $\gamma L^{\nu}(t)>\frac{2n D}{1-\bar\tau}\hat\Theta(t)+\mu$, $\forall t\in[0, t_M)$. It follows from (15) that $L(t)\equiv L_{0}$, $\forall t\in[0, t_M)$.

Case 2: $\gamma L^{\nu}(t)\leq\frac{2n D}{1-\bar\tau}\hat\Theta(t)+\mu$, $\forall t\in[0, t_M)$. Obviously, $L(t)$ is bounded in $[0, t_M)$ since $\hat\Theta(t)$ is bounded in $[0, t_M)$.

c) We show that $t_{M}=+\infty$. From above analysis, we know that $\Gamma(t)$ is bounded in $[-h, t_{M})$, which implies that $t_M$ is not an escape time, i.e., $t_{M}=+\infty$.

Secondly, since $z(t)$ and $L(t)$ are bounded in $[-h, +\infty)$, it follows from (10) that $x(t)$ is bounded in $[-h, +\infty)$.

2) For each $\varepsilon>0$, (19) implies that $\rho_{2}(0)<\rho_{1}(\varepsilon)\leq\rho_{2}(\varepsilon)$. Then, by the intermediate value theorem, there is a $0< \delta_{1}\leq\varepsilon$, such that $\rho_{2}(\delta_{1})=\rho_{1}(\varepsilon)$. Since $\rho_{2}$ is increasing, then by choosing $0<\delta=\delta(\varepsilon)<\delta_{1}\leq\varepsilon$, one gets $\rho_2(\delta)<\rho_{2}(\delta_{1})=\rho_1(\varepsilon)$. If $\|\Upsilon_0\|_{\mathcal{C}_h}<\delta$, it follows from (16) and (19) that $\rho_1(\|\Upsilon(t)\|)\leq V(t, \Upsilon_{t}) \leq$ $V(0, \Upsilon_{0})\leq$ $\rho_2(\|\Upsilon_0\|_{\mathcal{C}_h})< \rho_2(\delta)<\rho_1(\varepsilon),$ $t\geq0.$ Hence for $\|\Upsilon_0\|_{\mathcal{C}_h}< \delta$ $<$ $\varepsilon$, there holds $\|\Upsilon(t)\|<\varepsilon$, $t\geq0$. This proves the stability of the closed-loop $\Upsilon$-system at $\Upsilon=0$. Since (4) and (10) are equivalent transformations, the equilibrium point $(x, \hat\Theta)$ $=$ $(0, \Theta)$ of the closed-loop system (1), (4), (10) and (15) is globally stable.

3) We prove $\lim_{t\rightarrow +\infty}x(t)=0$. On one hand, (16) and (19) imply that $V(t, \Upsilon(t))$ is monotonically nonincreasing and bounded below by zero, and hence $\lim_{t\rightarrow+\infty}V(t, \Upsilon(t))$ exists and is finite. Suppose $L(t)\leq l$, $l>0$. Then

 \begin{align*} \int_{0}^{+\infty} & \xi_i^2(s)ds\\ & \leq-\frac{1}{\mu}\int_{0}^{+\infty}L^{1+\nu}(s) \dot V(s, \Upsilon(s))ds\\ & \leq\int_{0}^{+\infty}\xi_i^2(s)ds\leq -\frac{l^{1+\nu}(s)}{\mu}\int_{0}^{+\infty} \dot V(s, \Upsilon(s))ds\\ & =\frac{l^{1+\nu}}{\mu}(V(0, \Upsilon(0))-\lim\limits_{t\rightarrow +\infty}V(t, \Upsilon(t))) <+\infty.\end{align*}

On the other hand, the boundedness of $\Upsilon(t)$ over $[-h,$ $+\infty)$ implies that $\dot \Upsilon(t)$ is bounded uniformly in $t$. Thus, $\Upsilon(t)$ is uniformly continuous in $t$ over $[-h, +\infty)$, and so is $\xi_i^2(t)$. Using Barbalat's lemma in [19], one obtains $\lim_{t\rightarrow+\infty}\xi_i(t)$ $=0$. With (4), (10), for $i=1, \ldots, n$, one has $|x_i(t)|$ $= L^{m_i}(t)|\xi_i(t)-\sum_{l=1}^{i-1}(\prod_{j=l}^{i-1} g_j)z_l^{p_1\ldots p_{l-1}}(t)|^{\frac{1}{p_1\ldots p_{i-1}}}$ $\leq$ $L^{m_i}(t)\sum_{l=1}^{i-1}((\prod_{j=l}^{i-1}g_j)|L^{-m_l}(t)x_l(t)| )^{\frac{1}{p_1\ldots p_{i-1}}}+L^{m_i}(t)$ $\cdot$ $|\xi_i(t)|^{\frac{1}{p_1\ldots p_{i-1}}}$. When $i=1$, one has $\lim_{t\rightarrow+\infty}x_1(t)=0$ from $\alpha_0=0$. Then, one can recursively prove that $\lim_{t\rightarrow+\infty}x_2(t)$ $=\cdots=\lim_{t\rightarrow+\infty}x_n(t)=0$.

5 Simulation Example

Without loss of generality, we consider a simple system:

 \begin{align}\label{9.38} & \dot x_1(t)=x_2^{\frac{11}{9}}(t)+\theta u^{\frac{11}{9}}(t-d_1(t))\cos x_1(t)\notag\\ & \dot x_2(t)=u(t)\end{align} (20)

where $d_1(t)=1+{\sin t}/{2}$ and $\theta$ is an unknown constant. Obviously, $h_1=1.5$, $\tau_1=0.5$, and Assumption 1 holds with $|f_1|$ $\leq$ $\theta|u(t-d_1(t))|^{\frac{11}{9}}$.

Before designing the controller, we introduce the coordinate transformation $z_1=L^{-\frac{31}{9}}x_1$, $z_2=L^{-2}x_2$, $v=L^{-1}u$. Then, system (20) can be rewritten as

 \begin{align}\label{9.47} & \dot z_1=L^{-1}\left(z_2^{\frac{11}{9}}-\frac{31}{9}\dot L z_1\right)+\phi_1\nonumber\\ & \dot z_2=L^{-1}(v-2\dot L z_2)\end{align} (21)

where $\phi_1=L^{-\frac{31}{9}}f_1$. Define $\xi_1=z_1$ and choose $V_1=\frac12\xi_1^2$. Then $\dot V_1|_{(21)}=-2L^{-1}\xi_1^2+L^{-1}\xi_1(z_2^{\frac{11}{9}}-\alpha_1^{\frac{11}{9}}) -\frac{31}{9}\dot LL^{-1}\xi_1^2$ $+$ $L^{-1}\xi_1(\alpha_1^{\frac{11}{9}}+2\xi_1) +\frac{\partial V_1}{\partial z_1}\phi_1$. By choosing $\alpha_1^{\frac{11}{9}}= -2\xi_1$, $\gamma_0=$ $1$, $\lambda_1=\frac{31}{9}$, one has $\dot V_1|_{(21)}=-2L^{-1}\xi_1^2+L^{-1}\xi_1(z_2^{\frac{11}{9}}-\alpha_1^{\frac{11}{9}}) -\frac{31}{9}\dot LL^{-1}\xi_1^2+\frac{\partial V_1}{\partial z_1}\phi_1$.

Define $\xi_2=z_2^{\frac{11}{9}}-\alpha_1^{\frac{11}{9}}$, $W_2=\int_{\alpha_1}^{z_2}( s^{\frac{11}{9}}-\alpha_1^{\frac{11}{9}})^{2-\frac{9}{11}}ds$ and choose $V_2= \gamma_1 V_1+W_2$. Then

 \begin{align}\label{9.44} \dot V_2|_{(21)}\leq & \ \frac{-2\gamma_1}{L}\xi_1^2 -\frac{31\gamma_1\dot L}{9L}\xi_1^2+\frac{v-\alpha_2}{L}\xi_2^{\frac{11}{13}} +\frac{\alpha_2}{L}\xi_2^{\frac{11}{13}} \notag\\ & +\frac{\gamma_1\xi_1}{L}\left(z_2^{\frac{11}{9}}-\alpha_1^{\frac{11}{9}}\right) -\frac{2z_2\dot L}{L}\xi_2^{\frac{13}{11}} \notag\\ & +\frac{8|\xi_2|}{L} \left(|z_2|^{\frac{11}{9}}+\frac{31}{9}\dot L|z_1|\right)+\frac{\partial V_2}{\partial z_1}\phi_1. \end{align} (22)

Using Lemmas 1-3, one deduces that $\gamma_1L^{-1}\xi_1(z_2^{\frac{11}{9}}$-$\alpha_1^{\frac{11}{9}})$ $\leq$ $\frac{\gamma_1}{3}L^{-1}\xi_1^2 +\frac{3\gamma_1}{4}L^{-1}\xi_2^2$, $-2\dot L L^{-1}\xi_2 z_2\leq -\dot LL^{-1}\xi_2^2+5.7\dot LL^{-1} \xi_1^2$, $8L^{-1}|\xi_2|(|z_2|^{\frac{11}{9}}+\frac{31}{9}\dot L|z_1|)\leq\frac{2}{3}\gamma_1L^{-1}\xi_1^2 +(8+\frac{216}{\gamma_1})L^{-1}\xi_2^2 +0.5\dot LL^{-1}\xi_2^2+370\dot LL^{-1}\xi_1^2$, substituting these into (22) and choosing the controller $v(t)=-(g_2\xi_2)^{\frac{9}{11}}$, $g_2$ $=$ $(\frac{7\gamma_1}{4} +8+\frac{216}{\gamma_1})^{p_1}=673.9$, $\gamma_1=\frac{5.7+370+1}{\lambda_1}=112.2$, $\lambda_2$ $=$ $0.5$, one gets $\dot V_2|_{(21)}\leq-\gamma L^{-1}(\xi_1^2+\xi_2^2) -\lambda\dot LL^{-1}(\xi_1^2$ $+$ $\xi_2^2)$ $+$ $\frac{\partial V_2}{\partial z_1}\phi_1 \leq-\gamma L^{-1}(\xi_1^2+\xi_2^2) -\lambda\dot LL^{-1}(\xi_1^2+\xi_2^2)+ DL^{-2}\theta\sum_{i=1}^{2}(\xi_i^2+\xi_i^2(t-d_1(t)))$, where $\gamma=\gamma_0\gamma_1=112.2$, $\lambda$ $=$ $0.5$, $D=2\gamma_1g_2=151 200$. Then, by choosing $V=V_2$ $+$ $\frac{\tilde\theta^2}{2}+ \frac{D\theta}{1-\tau_1}\int_{t-d_1(t)}^{t}(\xi_1^2(s)+\xi_2^2(s))ds$, $\dot L=\max\{\frac{L^{-1}}{\lambda}(-\gamma L+$ $4D\hat{\theta}+2) , 0\}$, $\dot{\hat{\theta}}=4D(\xi_1^2+\xi_2^2)$, it leads to $\dot V|_{(21)}\leq-L^{-2}(\gamma L+\lambda L\dot L -\frac{2D\hat{\theta}}{1-\tau_1})(\xi_1^2+\xi_2^2) +\tilde{\theta}(\frac{2D}{1-\tau_1}(\xi_1^2+\xi_2^2)-\dot{\hat{\theta}})\leq-2L^{-2}(\xi_1^2+\xi_2^2)$.

In summary, the actual adaptive controller is designed as

 \begin{align}\label{9.50} & u=-L\left(2022L^{-\frac{31}{9}}x_1+674L^{-\frac{22}{9}}x_2^{\frac{11}{9}}\right)^{\frac{9}{11}}\cr & \dot L=2L^{-1}\max\left\{-112.2L+604 800\hat{\theta}+2, 0\right\}\cr & L(r)=L_0\geq1 \quad \forall r\in[-1.5, 0]\cr & \dot{\hat{\theta}}=604 800\times\left(5L^{-\frac{62}{9}}x_1^2 +L^{-\frac{44}{9}}x_2^{\frac{22}{9}}+4L^{-\frac{53}{9}}x_1x_2^{\frac{11}{9}}\right)\cr & \hat{\theta}(r)=\hat\theta_0>0 \quad \forall r\in[-1.5, 0]. \end{align} (23)

In this simulation, we choose $\theta=0.12$ and the initial value $[x_{1}(r), x_{2}(r), L(r), \hat\theta(r)]^{T}=$ $[0.01, 0.02, 1.5, 0.15]^{T}$ for any $r\in[-1.5, 0]$. Fig. 1 demonstrates the effectiveness of controller.

 Figure 1 The responses of the closed-loop system (20) and (23).
6 Conclusions

This paper considers the adaptive stabilization of a class of high-order uncertain nonlinear feedforward systems. A state feedback controller is designed which the equilibrium point at the origin, but also guarantees the uniform boundedness of all the other closed-loop signals.

Two interesting problems remain unexplored: a) Under Assumption 1, how to design an output feedback controller to stabilize nonlinear feedforward system (1)? b) For stochastic nonlinear systems, many results have been obtained in recent years, e.g., [19]-[37], but most of these papers consider lower-triangular systems. To our knowledge, only [38] and [39] consider the control design of stochastic nonlinear feedforward systems, and requires known growth rate for the system nonlinearities. Therefore, anther important problem is whether the result in this paper can be extended to stochastic nonlinear feedforward systems with unknown growth rate.

Appendix A Proof of Lemma 3

Proof: From Lemma 1, one has $|(x-y)^{\frac{1}{p}}+y^{\frac{1}{p}}| =|(x-y)^{\frac{1}{p}}-(-y)^{\frac{1}{p}}|\leq 2^{1-\frac{1}{p}}|x|^{\frac{1}{p}}$, which implies that $-2^{1-\frac{1}{p}}|x|^{\frac{1}{p}}-y^{\frac{1}{p}}\leq (x$-$y)^{\frac{1}{p}} \leq2^{1-\frac{1}{p}}|x|^{\frac{1}{p}}-y^{\frac{1}{p}}$. When $y\geq0$, by Lemma 2, it can be deduced that $(x-y)^{\frac{1}{p}}y^{2-\frac{1}{p}}\leq(2^{1-\frac{1}{p}}|x|^{\frac{1}{p}} -y^{\frac{1}{p}})y^{2-\frac{1}{p}} \leq -\frac{y^2}{2}$ $+$ $2^{2p-3}p^{-2p}(2p-1)^{2p-1}x^2$. When $y<0$, following the above analysis, one can obtain the same conclusion.

Appendix B Proof of Proposition 1

Proof: Using (5)-(8), Lemma 1 and the mean value theorem yields

 \begin{align}\label{9.18} \dot V_k| & _{(5)}\leq-(n-k+2)\gamma_0\ldots\gamma_{k-1}L^{-1}\sum\limits_{i=1}^{k-1}\xi_i^2 -\lambda_{k-1}\gamma_{k-1}\frac{\dot L}{L}\sum\limits_{i=1}^{k-1}\xi_i^2\cr & +L^{-1}\xi_{k}^{2-\frac{1}{p_1\ldots p_{k-1}}} \left(z_{k+1}^{p_k}-\alpha_k^{p_k}\right)+L^{-1}\xi_{k}^{2-\frac{1}{p_1\ldots p_{k-1}}}\alpha_k^{p_k}\cr & +\sum\limits_{i=1}^{k}\frac{\partial V_{k}}{\partial z_{i}}\phi_{i} +\gamma_{k-1}L^{-1}\xi_{k-1}^{2-\frac{1}{p_1\ldots p_{k-2}}} \left(z_k^{p_{k-1}}-\alpha_{k-1}^{p_{k-1}}\right)\cr & -m_k\xi_{k}^{2-\frac{1}{p_1\ldots p_{k-1}}}\dot LL^{-1}z_k+\bigg(2-\frac{1}{p_1\ldots p_{k-1}}\bigg) 2^{1-\frac{1}{p_1\ldots p_{k-1}}}\notag \\ & \times L^{-1}|\xi_{k}| \times\sum\limits_{i=1}^{k-1}\bigg|\frac{\partial\alpha_{k-1}^{p_1\ldots p_{k-1}}}{\partial z_i}\bigg| \cdot(|z_{i+1}|^{p_i}+m_i\dot L|z_i|).\end{align} (24)

Using (6) and Lemmas 1 and 2 lead to

 \begin{align}\label{9.26} \frac{\gamma_{k-1}}{L}\xi_{k-1}^{2-\frac{1}{p_1\ldots p_{k-2}}} (z_k^{p_{k-1}}-\alpha_{k-1}^{p_{k-1}}) \leq\frac{\gamma_0\ldots\gamma_{k-1}}{3L}\xi_{k-1}^2 +\frac{\pi_1}{L}\xi_k^2\end{align} (25)

where

 \begin{align*}\pi_{1} = & \ 4^{p_1\ldots p_{k-2}-1} \frac{\gamma_{k-1}}{2p_1\ldots p_{k-2}}\\ & \times (\frac{3(2p_1\ldots p_{k-2}-1)}{2p_1\ldots p_{k-2}\gamma_0\ldots\gamma_{k-2}} )^{2p_1\ldots p_{k-2}-1}.\end{align*}

With Lemma 3, one has

 \begin{align}\label{9.27} -\frac{m_k\dot L}{L}\xi_{k}^{2-\frac{1}{p_1\ldots p_{k-1}}}z_k \leq-\frac{m_k\dot L}{2L}\xi_k^2+\frac{\pi_2\dot L}{L}\xi_{k-1}^2 \end{align} (26)

where

 \begin{align*} \pi_2= & \ 2^{2p_1\ldots p_{k-1}-3}(p_1\ldots p_{k-1})^ {-2p_1\ldots p_{k-1}}\\ & \times(2p_1\ldots p_{k-1}-1)^ {2p_1\ldots p_{k-1}-1}m_kg_{k-1}^2.\end{align*}

From (10) and Lemmas 1 and 2, it follows that

 \begin{align}\label{9.20} \bigg(2- & \frac{1}{p_1\ldots p_{k-1}}\bigg) 2^{1-\frac{1}{p_1\ldots p_{k-1}}}L^{-1}|\xi_{k}| \sum\limits_{i=1}^{k-1}\bigg|\frac{\partial\alpha_{k-1}^{p_1\ldots p_{k-1}}}{\partial z_i}\bigg|\cr & \times(|z_{i+1}|^{p_i}+m_i\dot L|z_i|) \leq3\pi_3 L^{-1}|\xi_k|\sum\limits_{i=1}^{k}|\xi_i|\!+\!2\pi_3 m_1 \dot L L^{-1}\cr & \times|\xi_k|\sum\limits_{i=1}^{k-1}|\xi_i| % & & =3\pi_3 L^{-1}|\xi_k|^2\!+\!3\pi_3 L^{-1}|\xi_k\|\xi_{k-1}|\!+\!3\pi_3 L^{-1}|\xi_k|\sum\limits_{i=1}^{k-2}|\xi_i| %\!+\!2\pi_3 m_1\dot L L^{-1}|\xi_k\|\xi_{k-1}|\!+\!2\pi_3 m_1 \dot LL^{-1}|\xi_k|\sum\limits_{i=1}^{k-2}|\xi_i|\cr \leq AL^{-1}|\xi_k|^2 +\frac{2\gamma_0\ldots\gamma_{k-1}}{3}L^{-1}|\xi_{k-1}|^2\cr & +\gamma_0\ldots\gamma_{k-1}\sum\limits_{i=1}^{k-2}L^{-1}|\xi_i|^2 +\frac{m_k}{4}\dot LL^{-1}|\xi_k|^2\cr & +\frac{8m_1^2\pi_3^2}{m_k}\dot LL^{-1}|\xi_{k-1}|^2 +\frac{8m_1^2\pi_3^2}{m_k}\dot LL^{-1}\sum\limits_{i=1}^{k-2}|\xi_i|^2\end{align} (27)

where

 \begin{align*} \pi_3= & \ 12\max\limits_{1\leq i\leq k-1}( p_1\ldots p_{i-1}\Pi_{j=i}^{k-1}g_j(1+g_{i-1}^{1-\frac{1}{p_1\ldots p_{k-1}}}))\\ & \times \max\limits_{1\leq i\leq k-1}( 2+g_i^{p_1\ldots p_{i-1}}+g_{i-1}^{p_1\ldots p_{i-1}})\end{align*}

and

 \begin{align*} A=3\pi_3+\frac{27 \pi_3^2}{8\gamma_0\ldots \gamma_{k-1}} +\frac{9 \pi_3^2}{4\gamma_0\ldots \gamma_{k-1}}.\end{align*}

Substituting (25)-(27) into (24) yields

 \begin{align*} & \dot V_k|_{(5)}\leq-(n-k+1)\gamma_0\ldots\gamma_{k-1}L^{-1} \sum\limits_{i=1}^{k}\xi_k^2+\frac{\left(z_{k+1}^{p_k}-\alpha_k^{p_k}\right)}{L}\cr & \times\xi_k^{2-\frac{1}{p_1\ldots p_{k-1}}} +\sum\limits_{i=1}^{k}\frac{\partial V_{k}}{\partial z_{i}}\phi_{i} +\frac{\alpha_k^{p_k}}{L}\xi_k^{2-\frac{1}{p_1\ldots p_{k-1}}} +(A+\pi_1\cr & +\!(n-k\!+\!1)\gamma_0\ldots\gamma_{k-1})\frac{\xi_k^2}{L} \!-\!\frac{m_k\dot L}{4L}\xi_k^2\!-\! \bigg(\gamma_{k-1}\lambda_{k-1}\!-\!\pi_2\!-\!\frac{8m_1^2\pi_3^2}{m_k}\bigg)\cr & \times\frac{\dot L}{L}\xi_{k-1}^2 -\bigg(\gamma_{k-1}\lambda_{k-1}-\frac{8{m_1}^2\pi_3^2}{m_k}\bigg)\frac{\dot L}{L} \sum\limits_{i=1}^{k-2}\xi_i^2. \end{align*}

Choosing $\gamma_{k-1}$ such that $\gamma_{k-1}\lambda_{k-1}-\pi_2-\frac{8m_1^2\pi_3^2}{m_k}>0$ and defining $\lambda_k=\min\{\frac{m_k}{4}, \gamma_{k-1}\lambda_{k-1}-\pi_2 -\frac{8m_1^2\pi_3^2}{m_k}\}$, $g_k= (A+\pi_1$ $+$ $(n-k+1)\gamma_0\ldots\gamma_{k-1})^{p_1\ldots p_{k-1}}$, one gets (9).

Appendix C Proof of Proposition 2

Proof: From the definition of $m_i$, it can be deduced that $m_jp_i\ldots p_{j-1}-m_i =-(1+\sum_{k=i}^{j-2}p_i\ldots p_k)\leq-(1+p_i)$ for $i=$ $1,$ $\ldots, j-1$, $j= i+2, \ldots, n+1$, which together with $L\geq 1$ imply that $L^{-m_i+m_jp_i\ldots p_{j-1}}\leq L^{-(1+p_i)}$. Hence, by (2), (4), (10), Lemmas 1 and 2, the mean value theorem and $L(t-d_{i}(t))\leq L(t)$, we have

 \begin{align}\label{9.51} \frac{\partial V_n}{\partial z_i} & \phi_i=\left(\bigg(\prod\limits_{k=i}^{n-1} \gamma_k\bigg)\frac{\partial W_i}{\partial z_i}+\sum\limits_{j=i+1}^{n} \bigg(\prod\limits_{k=j}^{n-1}\gamma_k\bigg)\frac{\partial W_j}{\partial z_i}\right)\phi_i\notag\\ % & \leq & \left(\prod\limits_{k=i}^{n-1}\gamma_k\right)\xi_i^{2-\frac{1}{p_1\ldots p_{i-1}}} %+4\sum\limits_{j=i+1}^{n}\left(\prod\limits_{k=j}^{n-1}\gamma_k\right)\pi_{i-1, j-1}|\xi_j|\cdot\left(|\xi_i|^{1-\frac{1}{p_1\ldots p_{i-1}}}+ %|\xi_{i-1}|^{1-\frac{1}{p_1\ldots p_{i-1}}}\right)\cr \leq & \ \frac{B_{i} b_i\Theta}{L^{1+p_i}}\sum\limits_{j=i-1}^{n}|\xi_j|^{2-\frac{1}{p_1\ldots p_{i-1}}} \sum\limits_{j=i+1}^{n}\bigg(|\xi_j|^{\frac{1}{p_1\ldots p_{i-1}}}+|\xi_j(t \notag\\ & -d_i(t))|^{\frac{1}{p_1\ldots p_{i-1}}}\bigg) \leq\frac{\Theta D_{i}}{L^{1+p_i}} \sum\limits_{j=i-1}^{n}(|\xi_j|^2+|\xi_j(t-d_i(t))|^2)\end{align} (28)

where

 \begin{align*} b_i & =\max\limits_{{i+2}\leq j \leq{n+1}}\{1+g_{j-1}^{\frac{1}{p_1\ldots p_{i-1}}}\}\\ B_{i} & =4C_i\sum\limits_{j=i}^{n}\prod\limits_{k=j}^{n-1}\gamma_k\pi_{i-1, j-1}, \pi_{i-1, j-1}\\ & =p_1\ldots p_{i-1}(\prod\limits_{l=i}^{j-1}g_l)(1+g_{i-1}^{1-\frac{1}{p_1\ldots p_{j-1}}})\\ D_{i} & =B_{i} b_i(1-\frac{1}{2p_1\ldots p_{i-1}})\\ C_i & =1+\frac{2p_1\ldots p_{i-1}}{2p_1\ldots p_{i-1}-1}(\frac{p_1\ldots p_{i-1}-1}{2p_1\ldots p_{i-1}-1})^ {1-\frac{1}{p_1\ldots p_{i-1}}}.\end{align*}

Defining $D=\max_{1\leq i\leq n-1}\{D_{i}\}$ and $\nu=\min_{1\leq i\leq n-1}\{p_i\}$, one gets Proposition 2.

References