## 2 Problem Formulation and Preliminaries
### Problem Statement
With the control protocol given by (14), if the controller parameters are chosen such that \(k_{1}>\sqrt{\frac{2+|\dot{\phi}|^{2}}{\omega A_{1}}}\), \(k_{1}^{2}\beta<\frac{\lambda_{min}^{2}(L+B)}{w^{2}\|P(L+B)\|^{2}}\), and \(\frac{\lambda_{min}(L+B)-\sqrt{\pi}}{w^{2}\|P(L+B)\|^{2}}<k_{2}<\frac{\lambda _{min}(L+B)+\sqrt{\pi}}{w^{2}\|P(L+B)\|^{2}}\) with \(0<\omega<\frac{1}{k_{1}\lambda_{max}(P)}\), \(P=(\operatorname{diag}((L+B)^{-1}\mathbf{1}))^{-1}\), \(\|\Phi\|\leq\tilde{\Phi}\), \(\alpha=\lambda_{min}^{2}(L+B)-w^{2}k_{1}^{2}\beta\|P(L+B)\|^{2}\), \(\beta=1+2 wk_{1}\|\tilde{\Phi}P\|+\lambda_{max}^{2}(L+B)k_{1}^{2}\), \(\lambda_{1}=\lambda_{min}(P(L+B)+(L+B)^{T}B)\) and let \(\varsigma_{i}(0)>\max\{|e_{ix}(0)|,\frac{5}{4}\}\), then the control objectives mentioned in Section 2.3 can be achieved._  
Proof. : It follows from Assumption 1 that \(L+B\) is a nonsingular M-matrix [40]. According to Lemma 1, there exists a diagonally positive definite matrix \(P=(\operatorname{diag}(q))^{-1}\), with \(q=(L+B)^{-1}\mathbf{1}\) and \(\mathbf{1}=[1,\cdots,1]^{T}\in\mathbb{R}^{N}\). We propose the following Lyapunov function candidate:  
\[V(\varepsilon,e_{v})=\frac{1}{2}\varepsilon^{T}\varepsilon+\frac{1}{2}e_{v} ^{T}e_{v}+\omega\varepsilon^{T}k_{1}Pe_{v} \tag{16}\]  
with \(0<\omega<\frac{1}{k_{1}\lambda_{max}(P)}\) and \(k_{1}>0\). It can be verified that  
\[V=[\varepsilon^{T},e_{v}^{T}]_{v}^{\left[ \begin{matrix}\frac{1}{2}&\frac{ \pi}{2}k_{1}P\\ \frac{\pi}{2}k_{1}P&\frac{1}{2}\end{matrix} \right]}[ \begin{array}{c} \varepsilon\\ e_{v}\end{array} ]>0,\]  
which means that the Lyapunov function is positive definite. Differentiating (16) with respect to \(t\) obtains  
\[\dot{V} =\varepsilon^{T}[\Phi(e_{v}-\dot{\xi}\xi)]\] \[+\omega\varepsilon^{T}k_{1}P(L+B)(-k_{1}\varepsilon-k_{2}e_{v}+d -\mathbf{1}\dot{v}_{0})\] \[+\omega e_{v}{}^{T}k_{1}P[\Phi(e_{v}-\dot{\xi}\xi)]\] \[+e_{v}{}^{T}(L+B)(-k_{1}\varepsilon-k_{2}e_{v}+d-\mathbf{1}\dot{ v}_{0})\]  
Differentiating \(\xi\) versus time obtains  
\[\dot{\xi}=C(t)(e_{v}-\dot{\xi}\xi) \tag{17}\]  
Define the open set  
\[\Omega_{\xi}=\underbrace{(-1,1)\times\cdots\times(-1,1)}_{N\ times} \tag{18}\]  
Since the performance functions \(\varsigma_{i}(t),i=1,\cdots,N\) are selected such that \(\varsigma_{i}(0)>\max\{|e_{ix}(0)|,\frac{5}{4}\},i=1,\cdots,N\), then \(|\xi_{i}(0)|<1,i=1,\cdots,N\) is satisfied. Suppose there is a time instant \(\tau_{max}\) after which \(\xi(t)\) escapes from \(\Omega_{\xi}\). It is deduced  
\[\xi(t)\in\Omega_{\xi},\forall t\in[0,\tau_{max})\]  
From (24), it can be further obtained that \(\|e_{v}(t)\|\leq\|s(t)\|\leq s^{*}\), and \(\|e(t)\|\leq s^{*}\) for all \(t\in[0,t_{s})\). Through (10), (19), and (20), it is obtained that for \(\forall t\in[0,t_{s})\)  
\[-1<R_{1}<\sigma_{1}\leq\xi_{i}(t)\leq\sigma_{2}<R_{2}<1 \tag{25}\]  
where \(\sigma_{1}=\frac{e^{-s^{*}}-1}{e^{-s^{*}}+1}\) and \(\sigma_{2}=\frac{e^{s^{*}}-1}{e^{s^{*}}+1}\). It is noted that if \(\|s(0)\|>\frac{\|b\|}{\lambda_{min}(W)}\), \(R_{1}<\sigma_{1}\) and \(\sigma_{2}<R_{2}\) can be guaranteed by selecting \(\mu\in(0,2)\). Similarly, if \(\frac{\|b\|}{\lambda_{min}(W)}>\|s(0)\|\), \(\mu\) can be chosen such that \(\mu\leq 2(e^{\|s(0)\|}+1)/(e^{\frac{\|b\|}{\lambda_{min}(W)}}+1)\). Define \(\Omega_{\xi}^{**}=[\sigma_{1},\sigma_{2}]^{N}\). From (25) we can obtain that at all \(t\in[0,t_{s})\), \(\xi(t)\in\Omega_{\xi}^{**}\subset\Omega_{\xi}^{s}\subset\Omega_{\xi}\). Hence, assuming \(t_{s}<\infty\), it follows from Lemma 3 that \(\Omega_{\xi}^{**}\subset\Omega_{\xi}^{*}\) implies the existence of \(t^{\prime}\in[0,t_{s})\), such that \(\xi(t^{\prime})\
otin\Omega_{\xi}^{**}\), which contradicts (25). Therefore, \(t_{s}\) and then \(\tau_{max}\) can be extended to \(+\infty\). Thus we can obtain, for any time \(-\varsigma_{i}(t)<e_{ix}<\varsigma_{i}(t)\). Moreover, owing to Lemma 4, it is easy to verify that \(\lim\limits_{t\geq t_{f}}|e_{ix}(t)|<\varsigma_{i}(t_{f}),i=1,\cdots,N\).