Global stabilization of inherently non-linear systems using continuously differentiable controllers

Abstract

This paper concerns the problem of constructing $C^1$ (continuously differentiable) controllers to stabilize a class of uncertain non-linear systems whose linearization around the origin may contain uncontrollable modes. Based on a new definition of homogeneity with monotone degrees, a polynomial Lyapunov function and a $C^1$ global stabilizer are constructed recursively. Moreover, several special cases are investigated to show the advantages of the proposed approaches using the generalized homogeneity compared to the existing approaches using the traditional homogeneity.

Appendix

This appendix collects the proofs of the propositions used in the proof.

Proof of Proposition 1

The proof of this proposition will be separated into two different cases.

Case 1

If ${\frac{r_{i}p_{i-1}}{r_n+k_n}}\le 1$, applying Lemma 1 gives

$$\begin{aligned}&|x_{i}^{p_{i-1}}-x_{i}^{*p_{i-1}}|\nonumber \\&\quad = \left| \left( {x_{i}}^{\frac{r_n+k_n}{r_{i}}}\right) ^{ \frac{r_{i}p_{i-1}}{r_n+k_n}}-\left( {x_{i}^*}^{\frac{r_n+k_n}{r_{i}}} \right) ^{\frac{r_{i}p_{i-1}}{r_n+k_n}}\right| \nonumber \\&\quad \le 2^{1-\frac{r_{i}p_{i-1}}{r_n+k_n}}\left| {x_{i}}^{ \frac{r_n+k_n}{r_{i}}}-{x_{i}^*}^{\frac{r_n+k_n}{r_{i}}} \right| ^{\frac{r_{i}p_{i-1}}{r_n+k_n}}\nonumber \\&\quad \le 2 \left| \xi _i\right| ^{\frac{r_{i}p_{i-1}}{r_n+k_n}}. \end{aligned}$$

(47)

On the other hand, we have

$$\begin{aligned}&{x_{i-1}}^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_{i-1}}}-{x_{i-1}^*}^{ \frac{2\rho -k_{i-1}-r_{i-1}}{r_{i-1}}}\nonumber \\&\quad =\left( {x_{i-1}}^{\frac{r_n+k_n}{r_{i-1}}}\right) ^{ \frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}}\nonumber \\&\quad \quad -\left( {x_{i-1}^*}^{\frac{r_n+k_n}{r_{i-1}}}\right) ^{ \frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}}. \end{aligned}$$

(48)

By the definitions of $\rho$ and $k_j$’s, $j=1,2,\cdots ,i$, one can tell that the power $\frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}\ge 1$. Thus, we can apply the Mean Value Theorem to (48), then using Lemma 1 and by (9) one can obtain the following

$$\begin{aligned}&\left| \left( {x_{i-1}}^{\frac{r_n+_n}{r_{i-1}}}\right) ^{ \frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}}- \left( {x_{i-1}^*}^{ \frac{r_n+k_n}{r_{i-1}}}\right) ^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}} \right| \nonumber \\&\quad \le c'_i|\xi _{i-1}| \Big (\left| {x_{i-1}}^{\frac{r_n+k_n}{r_{i-1}}} \right| ^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}-1}\nonumber \\&\quad +\left| {x_{i-1}^*}^{\frac{r_n+k_n}{r_{i-1}}}\right| ^{ \frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}-1}\Big )\nonumber \\&\quad \le c'_i|\xi _{i-1}| \Big (\left| \xi _{i-1}+\beta _{i-2}^{ \frac{r_n+k_n}{r_{i-2}+k_{i-2}}}(\cdot )\xi _{i-2}\right| ^{ \frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}-1}\nonumber \\&\quad +\left| \beta _{i-2}^{\frac{r_n+k_n}{r_{i-2}+k_{i-2}}} (\cdot )\xi _{i-2}\right| ^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}-1} \Big )\nonumber \\&\quad \le \tilde{c}_i(\cdot )|\xi _{i-1}|\left( |\xi _{i-1}|^{\frac{2\rho -k_{i-1} -r_{i-1}}{r_n+k_n}-1}\right. \nonumber \\&\quad \left. +|\xi _{i-2}|^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}-1}\right) , \end{aligned}$$

(49)

where $c'_i$ is a constant and $\tilde{c}_i(x_1,\cdots ,x_i)$ is a positive function. With the help of (47) and (49), utilizing Lemma 2 and noting $r_{i}p_{i-1}=r_{i-1}+k_{i-1}$, we have

$$\begin{aligned}&\left| \left( {x_{i-1}}^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_{i-1}}}\! -\!{x_{i-1}^*}^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_{i-1}}}\!\right) (x_{i}^{p_{i-1}}-x_{i}^{*p_{i-1}})\right| \nonumber \\&\quad \le 2\tilde{c}_i(\cdot )|\xi _{i-1}|\left( |\xi _{i-1}|^{\frac{2\rho -k_{i-1} -r_{i-1}}{r_n+k_n}-1}\right. \nonumber \\&~~~\left. +|\xi _{i-2}|^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}-1}\right) | \xi _{i}|^{\frac{r_{i-1}+k_{i-1}}{r_n+k_n}}&\nonumber \\&\quad \le \frac{1}{3}\left( \xi _{i-2}^{\frac{2\rho }{r_n+k_n}} +\xi _{i-1}^{\frac{2\rho }{r_n+k_n}}\right) +\underline{c}_{i}(x_1,\cdots ,x_{i-1})\xi _{i}^{\frac{2\rho }{r_n+k_n}}, \end{aligned}$$

(50)

where $\underline{c}_{i}>0$ is a function.

Case 2

If ${\frac{r_{i}p_{i-1}}{r_n+k_n}}\ge 1$, applying the Mean Value Theorem yields and with a similar analysis as (49), we have

$$\begin{aligned}&|x_{i}^{p_{i-1}}-x_{i}^{*p_{i-1}}|\nonumber \\&=\left| \left( {x_{i}}^{\frac{r_n+k_n}{r_{i}}}\right) ^{ \frac{r_{i}p_{i-1}}{r_n+k_n}}-\left( {x_{i}^*}^{\frac{r_n+k_n}{r_{i}}} \right) ^{\frac{r_{i}p_{i-1}}{r_n+k_n}}\right| \nonumber \\&\le \hat{c}_i\left| {x_{i}}^{\frac{r_n+k_n}{r_{i}}}-{x_{i}^*}^{ \frac{r_n+k_n}{r_{i}}}\right| \Big |\left( {x_{i}}^{\frac{r_n+k_n}{r_{i}}} \right) ^{\frac{r_{i}p_{i-1}}{r_n+k_n}-1}\nonumber \\&~~~+\left( {x_{i}^*}^{\frac{r_n+k_n}{r_{i}}}\right) ^{\frac{r_{i} p_{i-1}}{r_n+k_n}-1}\Big |\nonumber \\&\le \check{c}_i(x_1,\cdots ,x_{i-1})|\xi _{i}|\left( |\xi _{i-1}|^{ \frac{r_{i}p_{i-1}}{r_n+k_n}-1}+|\xi _{i}|^{\frac{r_{i}p_{i-1}}{r_n+k_n} -1}\right) , \end{aligned}$$

(51)

where $\hat{c}_i$ is a positive constant and $\check{c}_i(\cdot )$ is a positive function. With the relation $r_{i}p_{i-1}=r_{i-1}+k_{i-1}$ in mind, putting (49) with (51) together and applying Lemma 2 yields

$$\begin{aligned}&\left| \left( {x_{i-1}}^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_{i-1}}}\! -\!{x_{i-1}^*}^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_{i-1}}}\!\right) (x_{i}^{p_{i-1}}-x_{i}^{*p_{i-1}})\right| \nonumber \\&\le \check{c}_i(\cdot )\tilde{c}_i(\cdot )|\xi _{i-1}| \left( |\xi _{i-1}|^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}-1}\right. \nonumber \\&~~~\left. +|\xi _{i-2}|^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_n+k_n}-1}\right) |\xi _{i}|\Big (|\xi _{i-1}|^{\frac{r_{i}p_{i-1}}{r_n+k_n}-1}\nonumber \\&~~~+|\xi _{i}|^{\frac{r_{i}p_{i-1}}{r_n+k_n}-1}\Big )\nonumber \\&\le \frac{1}{3}\left( \xi _{i-2}^{\frac{2\rho }{r_n+k_n}} +\xi _{i-1}^{\frac{2\rho }{r_n+k_n}}\right) +\bar{c}_{i}(x_1,\cdots ,x_{i-1})\xi _{i}^{\frac{2\rho }{r_n+k_n}}, \end{aligned}$$

(52)

where $\bar{c}_i(\cdot )>0$ is a function.

In conclusion, combining (50) and (52), we finish the proof of Proposition 1 as

$$\begin{aligned}&\left| \left( {x_{i\!-1}}^{\frac{2\rho -k_{i-1}\!-r_{i-1}}{r_{i-1}}}\!-\!{x_{i-1}^*}^{\frac{2\rho -k_{i-1}-r_{i-1}}{r_{i-1}}}\!\right) \left( x_{i}^{p_{i-1}}\!-\!x_{i}^{*p_{i-1}}\right) \right| \\&\quad \le \frac{1}{3}\left( \xi _{i-2}^{\frac{2\rho }{r_n+k_n}} +\xi _{i-1}^{\frac{2\rho }{r_n+k_n}}\right) +c_{i}(x_1,\cdots ,x_{i-1})\xi _{i}^{\frac{2\rho }{r_n+k_n}} \end{aligned}$$

by choosing $c_i(\cdot )=\max \{\underline{c}_i(\cdot ),\bar{c}_i(\cdot )\}$.

Proof of Proposition 2

Denote $\tilde{\beta }_j(x_1,\cdots ,x_j)$ $= \beta _j^{\frac{r_n+k_n}{r_j+k_j}}$ $(x_1,\ldots ,x_j),~j=1,\ldots ,i-1$. By applying Lemma 1, Assumption 1 can be rewritten as

$$\begin{aligned} |\phi _i(x_1,\cdots ,x_i)|&\le b_i(\cdot )\Big (|\xi _1|^{\frac{r_i+k_i}{r_n+k_n}}+ |\xi _2-\tilde{\beta }_1(\cdot )\xi _1|^{\frac{r_i+k_i}{r_n+k_n}}\nonumber \\&~~~+\cdots +|\xi _i-\tilde{\beta }_{i-1}(\cdot )\xi _{i-1}|^{ \frac{r_i+k_i}{r_n+k_n}}\Big )\nonumber \\&\le \tilde{b}_i(\cdot )\left( |\xi _1|^{\frac{r_i+k_i}{r_n+k_n}}+ \cdots +|\xi _i|^{\frac{r_i+k_i}{r_n+k_n}}\right) \end{aligned}$$

(53)

with a smooth function $\tilde{b}_i(x_1,\cdots ,x_i)>0$.

With the help of (49) and Lemma , we know

$$\begin{aligned}&\left| \left( ({x_i}^{\frac{r_n+k_n}{r_i}})^{\frac{2\rho -k_i -r_i}{r_n+k_n}}-({x_i^*}^{\frac{r_n+k_n}{r_i}})^{\frac{2\rho -k_i-r_i}{r_n+k_n}}\right) \phi _i(t,x,u)\right| \\&\le \tilde{b}_i(\cdot )\tilde{c}_{i+1}\Big (|\xi _1|^{ \frac{r_i+k_i}{r_n+k_n}}+\cdots \\&~~~+|\xi _i|^{\frac{r_i+k_i}{r_n+k_n}}\Big ) (|\xi _i|^{\frac{2\rho -k_i-r_i}{r_n+k_n}-1} +|\xi _{i-1}|^{\frac{2\rho -k_i-r_i}{r_n+k_n}-1})|\xi _i|\\&\le \check{b}_i(\cdot )\left( |\xi _1|^{\frac{2\rho }{r_n+k_n}-1} +\cdots +|\xi _i|^{\frac{2\rho }{r_n+k_n}-1}\right) |\xi _i|\\&\le \frac{1}{3}\left( |\xi _1|^{\frac{2\rho }{r_n+k_n}}+ \cdots +|\xi _{i-1}|^{\frac{2\rho }{r_n+k_n}}\right) +\hat{b}_i(\cdot )|\xi _{i}|^{\frac{2\rho }{r_n+k_n}}, \end{aligned}$$

where $\check{b}_i(x_1,\cdots ,x_i)>0,~\hat{b}_i(x_1,\cdots ,x_i)>0$ are smooth functions.

Proof of Proposition 3

First, we will show that there exists a smooth function $a_{i,l}(x_1,\cdots ,x_{i-1})>0$, such that the following holds

$$\begin{aligned}&\left| \frac{\partial \xi _{i-1}}{\partial x_l}\dot{x}_l\right| \le a_{i,l}(\cdot )\Big (|\xi _{1}|^{ \frac{r_n+k_n+k_l}{r_n+k_n}}+\cdots \\&\quad +\,|\xi _{i-1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\Big ),~l=1,\cdots ,i-1. \end{aligned}$$

The proof is done by utilizing an inductive argument.

Initial step First we consider the case $i=2$ and $l=1$. With $r_n+k_n\ge r_l$ in mind, by (9), (53), and applying Lemma 2, it yields

$$\begin{aligned}&\left| \frac{\partial \xi _2}{\partial x_1}\dot{x}_1\right| \\&=\left| \frac{\partial {\tilde{\beta }_{1}(x_1)}}{\partial x_1}\xi _{1}+\tilde{\beta }_1(x_1)\frac{\partial \xi _{1}}{\partial x_1}\right| \big |x_2^{p_1}+\phi _1(x_1)\big |\\&\quad \le \left( \left| \frac{\partial {\tilde{\beta }_{1}(x_1)}}{\partial x_1}\right| |\xi _{1}|+\tilde{\beta }_1(x_1)\frac{r_n+k_n}{r_1}|\xi _1|^{ \frac{r_n+k_n-r_1}{r_n+k_n}}\right) \\&~~~\check{a}_{2,1}(x_1)\left( |\xi _1|^{\frac{r_1+k_1}{r_n+k_n}} +|\xi _2|^{\frac{r_1+k_1}{r_n+k_n}}\right) \\&\quad \le \bar{a}_{2,1}(x_1)|\xi _1|^{\frac{r_n+k_n-r_1}{r_n+k_n}} \left( |\xi _1|^{\frac{r_1+k_1}{r_n+k_n}}+|\xi _2|^{\frac{r_1+k_1}{r_n+k_n}}\right) \\&~~~+\hat{a}_{2,1}(x_1)|\xi _1|^{\frac{r_n+k_n-r_1}{r_n+k_n}} \left( |\xi _1|^{\frac{r_1+k_1}{r_n+k_n}}+|\xi _2|^{\frac{r_1+k_1}{r_n+k_n}}\right) \\&\quad \le a_{2,1}(x_1)\left( |\xi _1|^{\frac{r_n+k_n+k_1}{r_n+k_n}} +|\xi _2|^{\frac{r_n+k_n+k_1}{r_n+k_n}}\right) \end{aligned}$$

with smooth functions $\check{a}_{2,1}(\cdot )>0,~\bar{a}_{2,1}(\cdot )>0$, $\hat{a}_{2,1}(\cdot )>0$, and $a_{2,1}(\cdot )>0$.

Inductive step Assume in step $j-1$ that, for $l=1,\cdots ,j-2$, there exists a smooth function $a_{j-1,l}$ ($x_l$, $\cdots$, $x_{j-2}$) $>0$ such that

$$\begin{aligned}&\left| \frac{\partial \xi _{j-1}}{\partial x_l}\dot{x}_l\right| \nonumber \\&\quad \le a_{j-1,l}(\cdot )\left( |\xi _1|^{\frac{r_n+k_n+k_l}{r_n+k_n}} +\cdots +|\xi _{j-1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\right) . \end{aligned}$$

(54)

In what follows, we show (54) will also hold for step $j$, $l=1,2,\cdots ,j-1$.

First for $l=1,2,\cdots , j-2$, from (9) and (53), we have

$$\begin{aligned}&\left| \frac{\partial {\xi _j}}{\partial {x_l}}\dot{x}_l\right| \\&=\left| \frac{\partial \big (x_j^{\frac{r_n+k_n}{r_j}}+\beta _{j-1}^{ \frac{r_n+k_n}{r_jp_{j-1}}}(x_1,x_2,\cdots ,x_{j-1})\xi _{j-1}\big )}{ \partial x_l}\dot{x}_l\right| \\&=\left| \frac{\partial \big (\tilde{\beta }_{j-1}(x_1,\cdots ,x_{j-1}) \xi _{j-1}\big )}{\partial {x_l}}\dot{x}_l\right| \\&\le \left| \frac{\partial \tilde{\beta }_{j-1}(x_1,\cdots ,x_{j-1})}{ \partial x_l}\xi _{j-1}\dot{x}_l\right| \\&~~~+\left| \tilde{\beta }_{j-1}(x_1,\cdots ,x_{j-1})\frac{\partial \xi _{j-1}}{\partial x_l}\dot{x}_l\right| ,\\&\quad ~l=1,2,\cdots ,j-2, \end{aligned}$$

where $\tilde{\beta }_{j-1}(\cdot )=\beta _{j-1}^{\frac{r_n+k_n}{r_jp_{j-1}}}(\cdot )$. Noting that $r_n+k_n\ge r_l$, using Lemma 2 together with (54), yields

$$\begin{aligned}&\left| \frac{\partial {\xi _j}}{\partial {x_l}}\dot{x}_l\right| \nonumber \\&\le \left| \frac{\partial \tilde{\beta }_{j-1}(\cdot )}{\partial x_l}\xi _{j-1}\right| \Big (|\xi _{1}|^{\frac{r_{l}+k_l}{r_n+k_n}}+\cdots +|\xi _{l+1}|^{\frac{r_{l}+k_l}{r_n+k_n}}\Big )\nonumber \\&~~~+\hat{a}_{j-1,l}(\cdot )\left( |\xi _1|^{\frac{r_n+k_n+k_l}{r_n+k_n}} +\cdots +|\xi _{j-1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\right) \nonumber \\&\le \bar{a}_{j,l}(\cdot )|\xi _{j-1}|^{\frac{r_n+k_n-r_l}{r_n+k_n}} \big (|\xi _{1}|^{\frac{r_{l}+k_l}{r_n+k_n}}+\cdots +|\xi _{l+1}|^{ \frac{r_{l}+k_l}{r_n+k_n}}\big )\nonumber \\&~~~+\hat{a}_{j,l}(\cdot )\left( |\xi _1|^{\frac{r_n+k_n+k_l}{r_n+k_n}} +\cdots +|\xi _{j-1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\right) \nonumber \\&\le a_{j,l}(\cdot )\left( |\xi _1|^{\frac{r_n+k_n+k_l}{r_n+k_n}} +\cdots +|\xi _{j-1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\right) ,\nonumber \\&~~~~l=1,2,\cdots ,j-2, \end{aligned}$$

(55)

where $\bar{a}_{j,l}(x_1,\cdots ,x_{j-1})>0,~\hat{a}_{j,l} (x_1,\cdots ,x_{j-1})>0$, and $a_{j,l}(x_1,\cdots ,x_{j-1})>0$ are smooth functions.

Next we shall prove (55) also holds in the case of $l=j-1$. Similarly, using Lemma 2 together with (9) and (53), one can also obtain the following

$$\begin{aligned}&\left| \frac{\partial {\xi _j}}{\partial {x_{j-1}}}\dot{x}_{j-1} \right| \\&=\left| \frac{\partial \Big (x_j^{\frac{r_n+k_n}{r_j}} +\beta _{j-1}^{\frac{r_n+k_n}{r_jp_{j-1}}}(\cdot )\big (x_{j-1}^{ \frac{r_n+k_n}{r_{j-1}}}-x_{j-1}^{*\frac{r_n+k_n}{r_{j-1}}}\big ) \Big )}{\partial x_{j-1}}\dot{x}_{j-1}\right| \\&\le \left| \frac{\partial \tilde{\beta }_{j-1}(\cdot )}{\partial x_{j-1}}\xi _{j-1}\dot{x}_{j-1}\right| +\left| \tilde{\beta }_{j-1}(\cdot ) \frac{\partial x_{j-1}^{\frac{r_n+k_n}{r_{j-1}}}}{\partial x_{j-1}} \dot{x}_{j-1}\right| \\&\le \bar{a}_{j,j-1}(\cdot )\big (|\xi _{j-1}|^{\frac{r_n+k_n-r_{j-1}}{r_n +k_n}}+|\xi _{j-2}|^{\frac{r_n+k_n-r_{j-1}}{r_n+k_n}}\big )\\&~~~\big (|\xi _{1}|^{\frac{r_{j-1}+k_{j-1}}{r_n+k_n}}+\cdots +| \xi _{j}|^{\frac{r_{j-1}+k_{j-1}}{r_n+k_n}}\big )\\&\le a_{j,j-1}(\cdot )\left( |\xi _1|^{\frac{r_n+k_n+k_{j-1}}{r_n+k_n}} +\cdots +|\xi _{j}|^{\frac{r_n+k_n+k_{j-1}}{r_n+k_n}}\right) , \end{aligned}$$

where $\bar{a}_{j,j-1}$ $(x_1,\cdots ,x_{j-1})$ $>0$, $\hat{a}_{j,j-1}$ $(x_1,\!\cdots \!,x_{j-1})$ $>0$, and $a_{j,j-1}(x_1,\cdots ,x_{j-1})>0$ are smooth functions.

In summary, the following holds for step $j$ and $l=1,2,\cdots ,j-1$

$$\begin{aligned} \left| \frac{\partial \xi _j}{\partial x_l}\dot{x}_l\right|&\le a_{j,l}(x_1,\cdots ,x_{j-1})\Big (|\xi _{1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\nonumber \\&~~~+\cdots +|\xi _{j}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\Big ),~l=1,\cdots ,j-1. \end{aligned}$$

(56)

This completes the inductive proof.

Final step Repeatedly using the inductive arguments, we can conclude that for the final step $j=i$, there exists a smooth function $a_{i,l}(x_1,\cdots ,x_{i-1})>0$, such that

$$\begin{aligned} \left| \frac{\partial \xi _i}{\partial x_l}\dot{x}_l\right|&\le a_{i,l}(x_1,\cdots ,x_{i-1})\Big (|\xi _{1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\nonumber \\&~~~+\cdots +|\xi _{i}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\Big ),~l=1,\cdots ,i-1. \end{aligned}$$

(57)

With the help of (57), now we arrive at

$$\begin{aligned}&|x_i+x_i^*||\xi _{i-1}|^{\frac{2\rho -k_i-r_i}{r_n+k_n}-1}\left| {\displaystyle \frac{\partial \xi _{i-1}}{\partial x_l}}\dot{x_l}\right| \nonumber \\&=|x_i-x_i^*+2x_i^*||\xi _{i-1}|^{\frac{2\rho -k_i-r_i}{r_n+k_n}-1} \left| {\displaystyle \frac{\partial \xi _{i-1}}{\partial x_l}}\dot{ x_l}\right| \nonumber \\&\le a_{i,l}(\cdot )\left( |x_i-x_i^*|+2\beta _{i-1}^{\frac{1}{p_i}} (\cdot )|\xi _{i-1}|^{\frac{r_i}{r_n+k_n}}\right) \nonumber \\&~~~|\xi _{i-1}|^{\frac{2\rho -k_i-r_i}{r_n+k_n}-1}\left( |\xi _{1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\!+\!\cdots \!+\!|\xi _{i-1}|^{ \frac{r_n+k_n+k_l}{r_n+k_n}}\right) \nonumber \\&\le \check{a}_{i,l}(\cdot )|\xi _i|^{\frac{r_i}{r_n+k_n}}| \xi _{i-1}|^{\frac{2\rho -k_i-r_i}{r_n+k_n}-1}\Big (| \xi _{1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\nonumber \\&~~~+\cdots +|\xi _{i-1}|^{\frac{r_n+k_n+k_l}{r_n+k_n}}\Big ) \end{aligned}$$

(58)

with a smooth function $\check{a}_{i,l}(\cdot )>0$.

Under the assumption of $k_l\ge k_i$ $(\forall l=1,2,\cdots ,i-1)$, by utilizing Lemma 2, we have

$$\begin{aligned}&|x_i+x_i^*||\xi _{i-1}|^{\frac{2\rho -k_i-r_i}{r_n+k_n}-1}\left| {\displaystyle \frac{\partial \xi _{i-1}}{\partial x_l}}\dot{x_l}\right| \\&\quad \le \frac{1}{3(i-1)}\left( \xi _1^{\frac{2\rho }{r_n+k_n}}+\cdots +\xi _{i-1}^{\frac{2\rho }{r_n+k_n}}\right) \\&~~~+\tilde{a}_{i,l}(x_1,\cdots ,x_{i})\xi _i^{\frac{2\rho }{r_n+k_n}}, \end{aligned}$$

where $\tilde{a}_{i,l}(\cdot )>0$ is a smooth function. Finally, we can always find a smooth function $\bar{b}_i(\cdot )>0$, such that Proposition 3 holds

$$\begin{aligned}&\bar{\beta }(\cdot ) |x_i+x_i^*||\xi _{i-1}|^{\frac{2\rho -k_i-r_i}{r_n+k_n}-1} \sum _{l=1}^{i-1}\left| {\displaystyle \frac{\partial \xi _{i-1}}{\partial x_l}}\dot{x}_l\right| \\&\quad \le \frac{1}{3}\left( \xi _1^{\frac{2\rho }{r_n+k_n}}+\cdots +\xi _{i-1}^{\frac{2\rho }{r_n+k_n}}\right) +\bar{b}_i(x_1,\cdots ,x_i) \xi _i^{\frac{2\rho }{r_n+k_n}}. \end{aligned}$$