## 4 Stability Analysis
 \tag{31}\]  
where \(F_{i,1}=\frac{1}{k_{i}}(|f_{i,1}(x_{i,1})|+|g_{i,1}(x_{i,1})|\left|k_{i}e_{i,2 }+\sigma(t)x_{i,2}^{0}\right|+|q_{i,2}|+\left|\dot{\sigma}(t)(x_{i,1}^{0}-q_{ i,1}^{0})\right|+\left|\dot{k}_{i}e_{i,1}\right|+b_{i,1}(x_{i,1}))\) and \(E_{i,1}=\frac{g_{i,1}(x_{i,1})}{k_{i}}\). Note that \(x_{i,1}\), \(e_{i,1}\) and \(e_{i,2}\) are bounded on \(\Omega_{e}\) because (23) and (29), respectively. Utilizing the fact that \(k_{i}(t),\dot{k}_{i}(t),\sigma(t),q_{i,1},q_{i,2}\) are bounded and employing the extreme value theorem, owing to the continuity of \(f_{i,1}(\cdot),g_{i,1}(\cdot)\) and \(b_{i,1}(\cdot)\), we arrive at  
\[E_{i,1}\geq c_{1,1}>0 \tag{32}\]  
\[c_{3,1}\geq F_{i,1}\geq c_{2,1}\geq 0 \tag{33}\]  
where \(c_{1,1}\), \(c_{2,1}\), and \(c_{3,1}\) are some unknown positive constants. Then, substituting (32) and (33) into (31) yields  
\[\dot{V}_{i,1}\leq-\lambda_{i,1}c_{1,1}\xi_{i,1}^{2}+c_{3,1}|\xi_{i,1}| \tag{34}\]  
From (34), it follows that \(\dot{V}_{i,1}\) is negative when \(|\xi_{i,1}|\leq c_{3,1}/\lambda_{i,1}c_{1,1}\) and subsequently that  
\[|\xi_{i,1}|\leq\xi_{i,1}^{*}=\frac{c_{3,1}}{\lambda_{i,1}c_{1,1}} \tag{35}\]which implies  
\[|e_{i,1}(t)|\leq c_{4,1}=1-\frac{1}{\xi_{i,1}^{*2}}<1 \tag{36}\]  
As a result, the control signal \(\alpha_{i,1}\) is bounded. Moreover, invoking (24), we also can conclude the boundedness of \(x_{i,2}\). Therefore, the time derivative of \(\alpha_{i,1}\) is  
\[\dot{\alpha}_{i,1}=-\lambda_{i,1}\dot{\xi}_{i,1} \tag{37}\]  
where  
\[ \begin{array}{c}\left|\dot{\xi}_{i,1}\right|\leq\frac{(1+e_{i,1}^{2})}{k_{i} (1-e_{i,1}^{2})}(|f_{i,1}(x_{i,1})|+\left|g_{i,1}(x_{i,1})(k_{i}e_{i,2}+\rho(t )x_{i,2}^{0}+\alpha_{i,1})\right|\\ \qquad\qquad+|q_{i,2}|+\left|\dot{\sigma}(t)(x_{i,1}^{0}-q_{i,1}^{0})\right|+ \left|\dot{k}_{i}e_{i,1}\right|+b_{i,1}(x_{i,1}))\end{array}  \tag{38}\]  
Noting (36) and using the same analysis as (33), it also easy to conclude the boundedness of \(\dot{\xi}_{i,1}\), and hence \(\dot{\alpha}_{i,1}\). Step \(j\) (\(2\leq j\leq n\)): Consider the following positive definite functions  
\[V_{i,j}=\frac{1}{2}\log\frac{1}{1-e_{i,j}^{2}} \tag{39}\]  
Let \(\xi_{i,j}=\frac{1}{1-e_{i,j}^{2}}\). In a similar fashion to that in the former step, by noting Assumption \(1\), it follows from (21), (24), (26) and (39) that the time derivative of \(V_{i,j}\) is  
\[ \begin{array}{c}\dot{V}_{i,j}=\frac{\xi_{i,j}}{k_{i}}(f_{i,j}(\overline{x}_ {i,j})+g_{i,j}(\overline{x}_{i,j})(\alpha_{i,j}+k_{i}e_{i,j+1}+\sigma(t)x_{i, j+1}^{0})\\ \qquad\qquad-\dot{\alpha}_{i,j-1}-\dot{\sigma}(t)x_{i,j}^{0}-\dot{k}_{i}e_{i,j }+d_{i,j}(t,\overline{x}_{i,j}))\\ \leq\frac{1}{k_{i}}g_{i,j}(\overline{x}_{i,j})\alpha_{i,j}\dot{\xi}_{i,j}+F_{ i,j}|\xi_{i,j}|\\ \leq-\lambda_{ij}E_{i,j}\xi_{i,j}^{2}+F_{i,j}\big{|}\xi_{i,j}\big{|}\end{array}  \tag{40}\]  
where \(F_{i,j}=\frac{1}{k_{i}}(\big{|}f_{i,1}(\overline{x}_{i,j})\big{|}+\big{|}g_{i,1}(\overline{x}_{i,j})\big{|}\Big{|}\Big{|}k_{i}e_{i,j+1}+\sigma(t)x_{i,j+1}^ {0}\Big{|}+\left|\dot{\alpha}_{i,j-1}\right|+\left|\dot{\sigma}(t)x_{i,j}^{0} \right|+\left|\dot{k}_{i}e_{i,j}\right|+b_{i,j}(\overline{x}_{i,j}))\) and \(E_{i,j}=\frac{\overline{g}_{i,j}(x_{i,j})}{2k_{i}}\). Noting that \(x_{i,m}\), \(m=1,2,\ldots,j\) are bounded on \(\Omega_{e}\) because the boundedness of \(\alpha_{i,m-1}\), \(e_{i,j}\) and \(e_{i,j+1}\) are bounded on \(\Omega_{e}\) in view of (29). Utilizing the fact that \(k_{i}(t),\dot{k}_{i}(t)\) are bounded and employing the extreme value theorem owing to the continuity of \(f_{i,j}(\cdot)\), \(g_{i,j}(\cdot)\) and \(b_{i,j}(\cdot)\), we arrive at  
\[E_{i,j}\geq c_{1,j}>0 \tag{41}\]  
\[c_{3,j}\geq F_{i,j}\geq c_{2,j}\geq 0 \tag{42}\]  
with \(c_{1,j}\), \(c_{2,j}\) and \(c_{3,j}\) being some unknown positive constants.