Adaptive fixed-time stabilization for a class of nonlinear uncertain systems

1.   Introduction

The fixed-time stabilization studied in this article for the family of nonlinear systems is described by

with the system state $\xi = ({\xi}_1, \cdots, {\xi}_n)^{\rm{T}}$ and the control input $u\in{R}$. The uncertain functions $g_j:\, R^{+}\, {\times}\, R^{n}\, {\times}\, R^{n_{\theta}}\rightarrow {R}, \, j = 1, \cdots, n$, are continuously differentiable with $g_j(t, 0, \theta) = 0$, $\forall\, t\geq0$, and $\theta\in{R^{n_{\theta}}}$ represents the unknown constant vector. The continuous functions $\lambda_j(t, \xi)\neq0$ denote the control coefficients, and specifically, $\lambda_n(t, \xi)$ is also named as the control direction.

The objective of this article is to answer the following questions:

(ⅰ) In the presence of nonlinear parametric uncertainties, is there a globally stabilizing controller that renders the equilibrium $x = 0$ of (1.1) fixed-time globally stable (i.e., global Lyapunov stability with additional properties such as fixed-time convergence)?

(ⅱ) How to design a fixed-time controller which could render the closed-loop system fixed-time stable without residual sets allowing the nonlinear parametric uncertainties?

The stabilizing control of nonlinear uncertain systems is one of the most significant tasks in system control. It is noted that the tracking control, output regulation, or disturbance rejection, could be transformed into a stabilization issue by constructing some sort of "error" variable ^[1]. Asymptotic stabilization together with its enhanced version-exponential stabilization plays a key role in stabilization control for linear or nonlinear systems ^[2]. The settling time in asymptotic or exponential stabilization is infinite in theory. That is, the system states or tracking error approach to its equilibrium point when the time evolves to infinity. In practice, the finite settling time is obviously more appealing, considering the operation is done in a finite time. It is on this background that the finite-time stability (FTS) is proposed in ^[7] with the time-optimal control ^[3]. FTS characterizes that a control system is firstly Lyapunov stable and then exhibits finite-time convergent property of its equilibrium. As noted in ^[4,5,6], finite-time stability has better performances, such as rapid response, high precision, and robust properties. Currently, the finite-time stabilizing control is still one of the most active topics in the community of nonlinear control systems, see ^{[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]} and the references therein.

In last several years, a novel variant of FTS, i.e., fixed-time stability (FxTS), is proposed in ^[23], which is an improved version of finite-time stability. In finite-time stability, the settling time is depending on the system initial values, and it will become large when the initial values are far away from the origin. Fixed-time stability overcomes this drawback, and its settling time is bounded by one fixed constant independent of the initial values ^[24]. Fixed-time control has some advantages in comparison with finite-time control because the controller could be designed in a manner that its control performance is obtained in a predefined time and regardless to its initial conditions. As a result, fixed-time control displays a better behaviour in the sense that it does not require that the design parameters are re-tuned in order to to maintain the settling time. In recent years, fixed-time control is intensively studied in nonlinear control community, and many interesting results have been reported in this area. For example, the works in ^[25] and ^[26] develop the implicit Lyapunov function methodology to investigate the robust FTS and FxTS of nonlinear systems. The work ^[27] develops a fixed-time terminal sliding mode (TSM) controller for second-order nonlinear perturbed systems. The work in ^[28] proposes a finite/fixed-time stabilizing control scheme for nonlinear strict-feedback systems using the AOPI technique. The fixed-time control has found its applications in the multi-agent systems (MAS) ^[29], path-following of automatic vehicles ^[30,31], nonlinear parametrization ^[32], nonholonomic systems ^[33], and robot systems ^[34], etc.

Due to the measurement limitations or the modelling errors in practice, the uncertainties are widespread in control systems. As is well known, the adaptive control plays a key role in dealing with parametric uncertainty ^[35,36,37]. In literature, there have been many results reported in adaptive fixed-time stabilizing control for nonlinear systems with unknown parameters, such as ^{[38,39,40,41,42,43,44]}, etc. It is pointed out that the existing adaptive control results could not realize the zero error stabilization, that is, the system states does only converge to a small neighborhood of the origin in a given finite time. In fact, a weaker version of fixed-time stability, that is, the practical fixed-time stability is achieved in ^{[40,41,42,43,44]}. In view of the unknown parameters in the controlled plants and the finite convergent time regardless to initial values, it brings some residual terms when the control Lyapunov function method is used in control design. This results in that the fixed-time stabilizing without residual sets become extremely difficult. The related methods which effectively handle the adaptive finite-time stability such as ^[6] and ^[22] are inapplicable here.

In this work, we will address the fixed-time stabilizing control with zero residual set for system (1.1). Our aim is to propose a robust adaptive stabilizing controller keeping all the system states globally convergent to zero in some fixed-time. In comparison with the reported results on adaptive fixed-time stabilizing control for uncertain nonlinear systems, the novelties of this article lie in that: 1) This article provides a systematic design of a robust adaptive fixed-time stabilizer to deal with nonlinear parametric uncertainties. The problem of global adaptive fixed-time stabilizing control without residual sets for nonlinear uncertain systems (1.1) is well addressed. 2) We present a constructive procedure to carry out the analysis, since the widely used Barbalat's lemma may fail in nonsmooth feedback control, which brings severe difficulties to the stability analysis in fixed-time control. The constructive analysis gets around this burden, and realizes the zero-error stabilizing control in the presence of unknown parameters in this paper.

The article is structured as follows. Some preliminaries are provided in Section 2. Section 3 gives the adaptive fixed-time stabilizing control design. Main results are presented in Section 4. Section 5 illustrates the proposed control scheme via a pendulum system. Section 6 concludes the article.

Notations The following notations will be used in the paper: $R$ denotes the set of real numbers, and $R^n$ denotes the $n$-dimensional Euclidean space; $|a|$ denotes the absolute value of scalar $a\in{R}$; $\xi^{\rm{T}}$ stands for the transposition of a vector $\xi\in{R^n}$. For a vector $\xi = ({\xi}_1, \cdots, {\xi}_n)^{\rm{T}}\in{R^n}$, we denote $\bar{\xi}_{i} = ({\xi}_1, \cdots, {\xi}_i)^{\rm{T}}$ when $i = 1, \cdots, n$, and we let $\bar{\xi}_{1} = {\xi}_1$ and $\bar{\xi}_{n} = \xi$. The arguments of functions will be sometimes omitted or simplified, for example, a function $f(\xi(t))$ is denoted by $f(\xi)$ or $f(\cdot)$ whenever no confusion can arise from the context.

2.   Preliminaries

Take into account the following nonlinear system

with $f:[0, \, \infty)\times{S}\rightarrow {R^n}$, $h(t, 0) = 0$, and $S\subset{R^n}$ containing the origin.

Definition 1 ^[4]: The equilibrium $\xi = 0$ of (2.1) is globally finite-time stable with $S\, = \, R^{n}$ if it is globally asymptotically stable and moreover any trajectory $\xi(t, \xi_0)$ of (2.1) arrives at the equilibrium at a finite time instant ${T(\xi_0)}$, i.e., $\xi(t, \xi_0) = 0, \, \forall\, t\geq{T(\xi_0)}$, where $T\big(\xi(0)\big)\geq0$ is the settling-time function.

Definition 2 ^[23]: The equilibrium $\xi = 0$ of (2.1) is globally fixed-time stable if it is globally finite-time stable and the settling-time function $T({\xi}_0)$ is bounded by a fixed number, i.e., there is a positive constant ${T_{\max}}$ independent of $\xi(0)$ such that $T({\xi}_0)\, {\leq}\, T_{\max}, \, \forall\, {\xi}_0\in{R^n}$.

Definition 3: The adaptive fixed-time stabilization problem is to find a continuously differential control law

where $\mu(\cdot)$ and $\nu(\cdot)$ are continuous functions, and ${\chi}(t)\, {\in}\, R$ is an adaptive variable to estimate the uncertainties, such that the solutions $\big(\xi(t)^{\rm{T}}, \chi(t)\big)^{\rm{T}}$ of system (1.1) with controller (2.2) is globally uniformly bounded. Additionally, for any $\big(\xi(0)^{\rm{T}}, \chi(0)\big)^{\rm{T}}\, {\in}\, R^{n+1}$, $\xi(t)$ converges to its equilibrium in fixed time. That is, for any $(\xi(0)^{\rm{T}}, \chi(0))^{\rm{T}}$, there exists a time instant $T\, > \, 0$ such that $\xi(t) = 0$ for any $t\, > \, T$, where $T$ is the settling time and does not dependent on $\big(\xi(0)^{\rm{T}}, \chi(0)\big)^{\rm{T}}$.

We provide some useful lemmas for the following control design and analysis.

Lemma 1 ^[28]: For the system: $\dot{z} = -{c_1}z^{\frac{h}{k}}-{c_2}{z^{\frac{m}{n}}}$ with $z\in R$ and $z(0) = z_0$, where $h$, $k$, $m$, $n$ are odd positive integers with $h < k$ and $m > n$, $c_1 > 0$, and $c_2 > 0$. Then the origin of system is globally fixed-time stable and its settling time $T$ is bounded by $T^*\leq\frac{1}{c_1}\frac{k}{k-h}+\frac{1}{c_2}\frac{m}{m-n}$.

Lemma 2 ^[40]: Suppose that the continuous function $W(\xi):R^{n}\rightarrow {R}$ is positive definite and radially unbounded satisfying

where $c_1 > 0$, $c_2 > 0$, $0 < r_1 < 1$ and $r_2 > 1$ and $0 < \eta < \infty$. Then, the system (2.1) is practically fixed-time stable. Additionally, the trajectory converges into the residual set given by

where $0 < \sigma < 1$, and the settling time has the following upper bound

Lemma 3 ^[8]: Given $0 < \nu = \frac{m}{n}\leq1$ with $m, n > 0$ odd integers, the following holds

Lemma 4 ^[8]: Assume $p > 0, q > 0$, $p\in R$, $q\in R$, and the function $\varepsilon(a, b) > 0$, $a, b \in R$. Then,

Lemma 5 ^[28]: Assume $a_1 > 0, \cdots, a_m > 0, b > 0$, and $a_1\in R, \cdots, a_m\in R, b\in R$, then there holds

Throughout this article, we need the following hypothesis.

Assumption 1: For each $g_i(t, \xi, \theta)$, there exists a smooth function $\varphi_i({\xi}_1, \cdots, {\xi}_i)\geq0$, satisfying

where $\sigma\geq1$ is a constant dependent on $\theta$.

Assumption 2: The uncertain control coefficients $\lambda_i(t, \xi)$ satisfy

where $\lambda_{i1}, \lambda_{i2} > 0$ are known positive real numbers.

Remark 1: It is noted that the Assumptions 1–2 are commonly used in literature. Assumption 1 shows that $x = 0$ is the equilibrium of (1.1). It is actually a requirement if the stabilizing control without residual sets. This assumption can also be seen in existing works ^[45,46]. Assumption 2 shows that the uncertain control coefficients $\lambda_i(t, \xi), \, i = 1, \cdots, n,$ have known lower and upper bounds, which avoids the possible singularity ^[47].

3.   Design of adaptive fixed-time stabilizer

Before the control design, we choose the candidate Lyapunov functions in the form of

with the following positive real numbers:

It can be shown that $U_i$'s are positive definite functions (see ^[8]). In the following control design, we construct the continuously differentiable function

Choose the following virtual control laws together as well as the errors

and the parameter updating law

with $\Gamma_1 = {z}_1^r{L}_1(\xi_1), \, \Gamma_i = \Gamma_{i-1}+{z}_i^r{L}_i(\bar{\xi}_i, \widehat{\Psi}), \, 2\leq i\leq n$. Particularly, $u = \xi_{n+1}^{\ast}$, $\zeta_i(\cdot)\geq0$ and ${L}_i(\cdot)\geq0$ are some smooth functions determined later.

The Propositions 1–6 are used in control design. We provide their proofs in Appendices A–E.

Proposition 1: Consider

and one have

Proposition 2: One can find a positive number $H_{i1}$ satisfying

Proposition 3: There is a continuously differentiable function ${L}_{i1}(\bar{\xi}_i, \widehat{\Psi})\geq0$ such that

Proposition 4: There are positive continuously differentiable functions $H_{i2}(\bar{\xi}_{i-1}, \widehat{\Psi})\geq0$ and ${L}_{i2}(\bar{\xi}_i, \widehat{\Psi})$ such that

Proposition 5: There exist smooth functions $\omega_i(\bar{\xi}_i, \widehat{\Psi})\geq0$ and $\varpi_i(\bar{\xi}_i, \widehat{\Psi})\geq0$ satisfying

Then, the AOPI method is invoked to show the controller design procedure.

Step $1$: Let

In what follows, we denote $\widehat{\Psi}$ as the estimate of $\Psi$ with the error $\widetilde{\Psi} = \Psi-\widehat{\Psi}$.

We choose the candidate Lyapunov function

Then, from (3.1), the time-derivative of $W_1$ along the $\xi_1$-subsystem in (1.1) is

For notational convenience, let ${L}_1(\xi_1) = {z}_1^{2-d}\phi_1(\xi_1)\geq0$, then, we know from Assumption 1 that

Substitute (3.15) into (3.14), and denote $\zeta_1(\xi_1, \widehat{\Psi}) = n+l_1+\bar{l}_1{z}_1^{\frac{r_0+1}{r_0}}+{L}_1(\xi_1)\sqrt{1+{\widehat{\Psi}}^2}$, with $l_1 > 0$ a design constant, $\bar{r} = r+\frac{r_0+1}{r_0}$, and $r_0\, > \, 0$ an odd positive integer, and then one get

As a result, we choose the following virtual controller

and from (3.16), we get

Step $i(2\leq{i}\leq{n})$: Suppose that in Step $i-1$, the following holds

where the notation of $\Omega_{i-1}, \, i = 2, \cdots, n+1$, is defined by

Then, we choose the function

and in terms of (3.19), its derivative satisfies

The following Proposition 6 is used to derive the virtual controller $\xi_{i+1}^{*}$, whose proof can be seen in Appendix F.

Proposition 6: There exists a nonnegative continuous function $H_{i3}(\bar{\xi}_i, \widehat{\Psi})$ satisfying

Define $\zeta_{i}(\bar{\xi}_{i}, \widehat{\Psi}) = n-i+1+l_i+H_{i}(\bar{\xi}_{i}, \widehat{\Psi}) +{L}_{i}(\bar{\xi}_{i}, \widehat{\Psi})\sqrt{1+{\widehat\Psi}^2}+\bar{l}_i\, {z}_{i}^{\frac{r_0+1}{r_0}}$, with $H_i(\bar{\xi}_{i}, \widehat{\Psi}) = H_{i1}+H_{i2}(\bar{\xi}_{i-1}, \widehat{\Psi})+H_{i3}(\bar{\xi}_{i}, \widehat{\Psi})$ and ${L}_i(\bar{\xi}_{i}, \widehat{\Psi}) = {L}_{i1}(\bar{\xi}_{i}, \widehat{\Psi})+{L}_{i2}(\bar{\xi}_{i}, \widehat{\Psi})$, $l_i, \bar{l}_i > 0$. Then, the virtual control law is taken as follows

Substitute (3.24) into (3.22), and one get

In particular, we design the actual controller when $i = n$ in (3.24) as follows

as well as the parameter updating law

Then, based on the above calculations, we know that the Lyapunov function

satisfy

So far, we complete the adaptive fixed-time stabilizer design.

4.   Main results

Now we summarize the main results contributed in this article in the following Theorem 1.

Theorem 1: The considered system (1.1) with the adaptive controller (3.26), is globally fixed-time stable in the context of Definition 3.

Proof: According to (3.28) and (3.29), we can conclude that all solutions $\big(\xi(t), \widetilde{\Psi}(t)\big)$ are bounded. In view of $\Psi$ a constant, it can be concluded that the adaptive estimate $\widehat{\Psi}(t)$ is also bounded. Specially, $\widehat{\Psi}(t)\geq0$ if $\widehat{\Psi}(0)\geq0$ according to $\dot{\widehat{\Psi}} = \Gamma_n(\xi, \widehat{\Psi})\geq0$. Then, one can find a positive constant $\Lambda$ such that

In accordance of

we know from (3.29) that

Considering $d < 2$ and $2^{\frac{r}{2}-1} < 1$, we know from Lemma 5 that

Similarly, from $\bar{r} > 2$ and $2^{\frac{\bar{r}}{2}-1} > 1$, we have

For notational convenience, let

then, (4.3) turns into

Define the function

According to Proposition 5 with $i = n$ and (4.1), the following calculations hold

In view of

and the continuous property of $\widetilde{W}(\xi, \widehat{\Psi})$, it is known that there is a real number $\rho > 0$ satisfying for each $\xi\in{N_1}$ with

there holds

As a result, if $(\xi, \widehat{\Psi})\in{N_1}$, that is $W_n^{*}(\xi, \widehat{\Psi})\leq{\rho}$, in view of the definition of $N_1$, which further implies that

This shows that once $(\xi, \widehat{\Psi})\in{N_1}$, it will be always in $N_1$.

In what follows, the fixed-time convergence analysis is separated into the following Case Ⅰ and Ⅱ.

Case Ⅰ: If $(\xi(0), \widehat{\Psi}(0))\in{N_1}$, it can be seen that, if $(\xi, \widehat{\Psi})\in{N_1}$, there holds

Then, $W_n^{*}$ is fixed-time convergent with local property. Since $W_n^{*} = 0$ if and only if $x = 0$, according to Lemma 1, one can conclude that $x$ turns to be $0$ within $T_1$:

Clearly, the real constants $c_1, \, c_2, \, r, \, \bar{r}$ are independent on the initial values $\big(\xi(0), \widehat{\Psi}(0)\big)$.

Case Ⅱ: If $\big(\xi(0), \widehat{\Psi}(0)\big)\not\in{N_1}$, one can calculate its maximum arriving time moment $T_2$ into $N_1$.

Since $\xi$, $\widehat{\Psi}$ and $\widetilde{\Psi}$ are bounded, then there is a real number $C > 0$ satisfying

which leads to

Motivated by the recent work ^[40,42], we define the following set:

with any positive constant $\epsilon\in(0, 1)$.

Choose the constants $l_i$'s and $\bar{l}_i$'s large enough, $i = 1, \cdots, n$, which in terms of (4.6) renders $c_1$ and $c_2$ large, and further implies $\Big(\frac{C}{(1-\epsilon)c_1}\Big)^{\frac{2}{r}}$ and $\Big(\frac{C}{(1-\epsilon)c_2}\Big)^{\frac{2}{\bar{r}}}$ sufficiently small, such that

and then, we can get

As a result, in view of the Lemma 2, after the fixed-time $T_1$:

$(\xi, \widehat{\Psi})$ goes into $N_2$, and hence enters $N_1$. According to the analysis in Case Ⅰ, when $(\xi, \widehat{\Psi})\in{N_1}$, after the fixed-time $T_2$:

$(\xi, \widehat{\Psi})$ arrives at zero. Thus, in this situation, $\xi$ approaches the origin within $T = T_1+T_2$:

Noting that fact of the positive design parameters $c_1, \, c_2, \, r, \, \bar{r}$ irrespective of the initial values, we can see that the closed-loop system states $\xi$ globally converge to zero in fixed-time. Consequently, the problem of global adaptive fixed-time stabilization stated in Definition 3 is well addressed.

5.   Simulation results

In this subsection, a practical example of a pendulum system is used to illustrate the proposed fixed-time control strategy. It is known that the simple pendulum motion can be described by ^[48]

where the torque $u\in{R}$ is viewed as the control variable, the angular displacement $\upsilon\in{R}$ is the state. The constants $M$, $L$, $k$, and $g$ denote the mass, length, friction coefficient of the rod, and the gravity acceleration, respectively. It does not require that the parameters $M$, $L$, and $k$ are known a priori.

The control task is to construct a control input torque using the presented control methodology developed here, so that the angular displacement of pendulum is regulated at the angle $\upsilon = \pi$ in a finite time irrespective of system initial conditions.

Towards this end, we need the following additional Assumption 3 to characterize the unknown parameters $M$, $L$, and $k$.

Assumption 3: The parameters $m$ and $l$ are assumed to satisfy

According to Assumption 3, we have $\frac{1}{\overline{M}\, \bar{L}^2}\, \leq\, \frac{1}{M L^2}\, \leq\, \frac{1}{\underline{M}\, \underline{L}^2}$.

To be first, we perform the following coordinates changes

and we get

It follows that the new system (5.4) has the same form with the considered system (1.1) with $g_1(t, \xi, \theta) = 0$, $g_2(t, \xi, \theta) = \frac{g}{L}\sin({\xi}_1)-\frac{k}{M}{\xi}_2$, and $\theta = \max\{\frac{g}{L}, \frac{k}{M}\}$.

According to the developed controller design algorithm presented in Section 3, we construct the following adaptive fixed-time controller

with $\zeta_1({\xi}_1) = 2+l_1+\bar{l}_1{\xi}_1^{\frac{r_0+1}{r_0}}$, $\widehat{\mu}_1({\xi}_1, \widehat{\Psi}) = \zeta_1^{\frac{1}{r_2}}({\xi}_1)+\frac{1}{r_2}\frac{r_0+1}{r_0}\bar{l}_1\zeta_1^{{\frac{1}{r_2}}-1}({\xi}_1){\xi}_1^{\frac{r_0+1}{r_0}}$, $\zeta_2(\bar{\xi}_2, \widehat{\Psi}) = 1+l_2+\bar{l}_2\xi_2^{\frac{r_0+1}{r_0}} +H_2(\bar{\xi}_2,$ $\widehat{\Psi})+{L}_2(\bar{\xi}_2, \widehat{\Psi})\sqrt{1+{\widehat\Psi}^2}$.

In simulation, we take the design constants: $r_1 = 1, \; r_2 = \frac{3}{5}, \; r_3 = \frac{1}{5}, \; r = \frac{8}{5}, \; r_0 = 5, \; \bar{r} = \frac{14}{5}$, and the gain functions: $H_2(\bar{\xi}_2, \widehat{\Psi}) = \frac{r_2}{r}(\frac{2}{5})^{-\frac{1}{r_2}}2^{\frac{(1-r_2)(1+r_2)}{r_2}} +(2-r_2)2^{2(1-r_2)}\widehat{\mu}_1({\xi}_1,$ $\widehat{\Psi}) +\frac{1}{r}(\frac{2}{3})^{-r_2}\Big((2-r_2)2^{1-r_2}\widehat{\mu}_1({\xi}_1, \widehat{\Psi})\Big)^r$, ${L}_2(\bar{\xi}_2, \widehat{\Psi}) = \Big(\frac{2}{5}\Big)^{-\frac{5}{8}(1+\frac{1}{r_2})}\frac{r_2}{r}\xi_2^{\frac{2-2r_2}{r_2}r}$$+2^{1-r_2}\xi_2^{2-r}+\frac{1}{r}(\frac{2}{3})^{-r_2}\Big(\xi_2^{1-r_2}\zeta_1({\xi}_1, \widehat{\Psi})\Big)^{1+r_2}$.

For simulation use, the parameters $M$, $L$, and $k$ are chosen as $ML^2 = 1$, $L = g$, $k = M$, and $l_1 = 1$, $l_2 = 0.1$, $\bar{l}_1 = 1$, $\bar{l}_2 = 0.1$, and the initial values $\xi_{1}(0) = 0.1$, $\xi_2(0) = 0.5$, $\widehat{\Psi}(0) = 0.5$. The simulation results are shown in Figures 1–2. Particularly, Figure 1 depicts the profiles of the angular displacement $\upsilon$, its desired angular displacement $\pi$, the velocity $\dot{\upsilon}$, parameter estimate $\widehat{\Psi}$, and input torque $u$ in (5.1). Figure 2 verifies fixed-time convergence property of the system states in closed-loop system (5.4)–(5.6). From the simulation results, we can see that the designed fixed-time stabilizer could achieve the fixed-time stabilization with zero error for the pendulum.

6.   Conclusions

The paper presents an adaptive fixed-time stabilization strategy for a kind of nonlinear systems perturbed by nonlinear parametric uncertainty and unknown control coefficients. We combine the adding one power integrator tool and backstepping method to present a systematic fixed-time controller design scheme. The proposed adaptive stabilizer guarantees that the states can converge to its equilibrium in fixed-time, and all closed-loop solutions are bounded. It provides a basic fixed-time stable approach to realize the adaptive stabilizing control for the class of nonlinear uncertain systems with parametric uncertainty and uncertain control coefficients. The obtained result in this article is an improvement of the existing results in the adaptive fixed-time control direction. The simulation results demonstrate the efficacy of the proposed control scheme by means of a pendulum system.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 61803229, the Development Plan of Youth Innovation Team of University in Shandong Province under Grant 2021KJ067, and the National Natural Science Foundation of Shandong Province under Grant ZR2021MF009.

Conflict of interest

The authors declare that there is no conflict of interest.

Appendix

A. Proof of Proposition 1:

This proposition can be referred to Proposition 2 together with its proof in ^[8].

B. Proof of Proposition 2:

Using Lemma 3, one can verify that

with $H_{i1} = \frac{r_i}{r}{\Big(\frac{r}{4(2-r_{i-1})}\Big)}^{-\frac{2-r_{i-1}}{r_i}}2^{\frac{(1-r_2)(2-r_{i-1}+r_i)}{r_i}}\lambda_{i-1, 2}^{\frac{2-r_{i-1}+r_i}{r_i}}$, and then, the proof is completed.

C. Proof of Proposition 3:

In terms of Assumption 1 and ${{\xi}_j^{*}} = -\frac{1}{\lambda_{{j-1}, 1}}\xi_{j-1}^{r_i}\zeta_{j-1}(\bar{\xi}_{j-1}, \widehat{\Psi})$, $j = 2, \cdots, i$, one obtain

Then, there exists a continuous function $\bar{\phi}_i(\bar{\xi}_i, \widehat{\Psi})\geq0$ such that

Using Lemma 4, we have the following calculations

with ${L}_{i1}(\bar{\xi}_i, \widehat{\Psi}) = \sum\limits_{j = 1}^{i}\frac{2-r_i}{d}\Big(\frac{d}{4r_{i+1}}\Big)^{-\frac{r_{i+1}}{2-r_i}}\Big(\bar{\phi}_i(\bar{\xi}_i, \widehat{\Psi})\Big)^{\frac{d}{2-r_{i}}}$, and then we complete the proof of proposition.

D. Proof of Proposition 4:

Firstly, it can be verified that

Secondly, we know from ${{\xi}_i^{*}} = -\frac{1}{\lambda_{{i-1}, 1}}\xi_{i-1}^{r_i}\zeta_{i-1}(\bar{\xi}_{i-1}, \widehat{\Psi})$ that

Thanks to the inductive method, one can find a smooth function $\widehat{\mu}_{ij}(\bar{\xi}_{i-1}, \widehat{\Psi})\geq0$ satisfying

Additionally, according to (A7), one get

where $\widehat{H}_{ij}(\bar{\xi}_{j}\widehat{\Psi})$ and $\widehat{L}_{ij}(\bar{\xi}_{j}, \widehat{\Psi})$ are some nonnegative functions.

From

one can find two nonnegative $C^1$ functions $H_{i2}(\bar{\xi}_{i-1}, \widehat{\Psi})$, ${L}_{i2}(\bar{\xi}_{i}, \widehat{\Psi})$, such that

Then, we complete the proof.

E. Proof of Proposition 5:

Firstly, one can choose

Then, the function $\omega_i(\cdot)$ can be chosen as $\omega_i(\bar{\xi}_{i}, \widehat{\Psi}) = \max\{{L}_1({\xi}_1), {L}_{2}(\bar{\xi}_{2}, \widehat{\Psi}), \cdots\!, {L}_i(\bar{\xi}_{i}, \widehat{\Psi})\}$, implying

Considering ${\xi}_{j}^{\ast} = -\frac{1}{\lambda_{j-1, 2}}\xi_{j-1}^{r_{j}}\zeta_{j-1}(\bar{\xi}_{j-1}, \widehat{\Psi})$, and $\xi_{j} = {\xi}_{j}^{\frac{1}{r_{j}}}-{{\xi}_{j}^{\ast}}^{\frac{1}{r_{j}}}$, $j = 2, \cdots, i$, and then, we have

Then, in view of (A13), there exists a nonnegative function $\varpi_i(\bar{\xi}_{i}, \widehat{\Psi})$ satisfying

Then, we complete the proof of Proposition 5.

F. The proof of Proposition 6:

Firstly, the following holds

Then, considering ${{\xi}_i^{*}}^{\frac{1}{r_i}} = -\Big(\frac{1}{\lambda_{{i-1}, 1}}\Big)^{\frac{1}{r_i}}\xi_{i-1}\zeta_{i-1}^{\frac{1}{r_i}}(\bar{\xi}_{i-1}, \widehat{\Psi})$ and $\frac{1}{r_i} > 1$, one can find a smooth function $\widehat{\nu}_i(\bar{\xi}_{i}, \widehat{\Psi})\geq0$ satisfying

Furthermore, it is known from (A9) that

According to Lemma 4, we have

Define $\bar{H}_{i3}(\bar{\xi}_{i}, \widehat{\Psi}) = \Big(\sum\limits_{j = 1}^{i}\xi_j^r\Big)\Big(\frac{r}{4(r-1)}\Big)^{-(r-1)}\frac{1}{r}\Big((2-r_i)2^{1-r_i}$ ${\cdot}\widehat{\nu}_i(\bar{\xi}_{i}, \widehat{\Psi})\omega_i(\bar{\xi}_{i}, \widehat{\Psi})\Big)^r$, and then, we know from Proposition 5 that

Let $H_{i3}(\bar{\xi}_{i}, \widehat{\Psi}) = \bar{H}_{i3}(\bar{\xi}_{i}, \widehat{\Psi}) +{L}_i(\bar{\xi}_{i}, \widehat{\Psi})\cdot(2-r_{i-1})2^{1-r_{i-1}}\sqrt{1+\xi_{i-1}^2}\, \widehat{\nu}_{i-1}(\bar{\xi}_{i-1}, \widehat{\Psi})$, and one can verify that

Then, the proof is completed.