## 3 Global Stability Analysis
### Stability Analysis for the Second and Third Tanks
Equation (44) leads to:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]0  
Combining with Equation (106) and accounting for the property (102), yields:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]1  
which can be rewritten as:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]2  
where \(V\) is the overall Lyapunov function, defined as:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]3\(\square\)  
**Task 7. ** Integration of the \(dV\,/dt\) expression. Integrating Equation (108), yields:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]4  
Therefore: (BPi) \(V_{2}\in\mathcal{L}_{\infty}\), \(V_{3}\in\mathcal{L}_{\infty}\), \(V_{1}\in\mathcal{L}_{\infty}\), and from the definitions of \(V_{1}\) (35), \(V_{2}\) (81), \(V_{3}\) (98) it follows that \(f_{v1}\in\mathcal{L}_{\infty}\), \(f_{v2}\in\mathcal{L}_{\infty}\), \(f_{v3}\in\mathcal{L}_{\infty}\), and consequently \(f_{v2}^{2}\in\mathcal{L}_{\infty}\); (BPii) \(x_{1}\in\mathcal{L}_{\infty}\), \(x_{2}\in\mathcal{L}_{\infty}\), \(x_{3}\in\mathcal{L}_{\infty}\), what follows from the definitions of \(f_{v1}\)(29), \(f_{v2}\) (75), \(f_{v3}\) (96); (BPiii) \(f_{v2}^{2}\in\mathcal{L}_{1}\). Considering the properties \(f_{v2}^{2}\in\mathcal{L}_{1}\) and \(f_{v2}^{2}\in\mathcal{L}_{\infty}\) and applying the [PERSON]'s lemma (cf. [36]), yields:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]5  
Furthermore, accounting for the definition of \(f_{v2}\) (75), it follows that \(x_{2}\) converges asymptotically to \(\Omega_{x2}=\left\{x_{2}:x_{2}^{l}\leq x_{2}\leq x_{2}^{u}\right\}\). From this convergence result and the definition of \(x_{2}\) (9), it follows that \(S_{2}\) converges asymptotically to \(\Omega_{52}\); where:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]6  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]7  
\(S_{2}^{eq}\) is provided by Equations (5)-(7); the bounds \(x_{2}^{l}\), \(x_{2}^{u}\) are defined in Equations (70) and (71). This completes the proof of the first part of the theorem (Ti). From Equation (111), it follows that \((x_{2}-x_{3})(f_{v2}-f_{v3})\in L_{1}\) and from properties Bpi, BPii it follows that \((x_{2}-x_{3})\big{(}f_{v2}-f_{v3})\in L_{\infty}\). Applying the [PERSON]'s theorem (cf. [22]) yields \(\lim_{t\to\infty}(x_{2}-x_{3})(f_{v2}-f_{v3})=0\). Hence, either \(x_{2}-x_{3}\) or \(f_{v2}-f_{v3}\) converges to zero. This result and the convergence of \(x_{2}\) to \(\Omega_{x2}\), the convergence of \(f_{v2}\) to zero, and the definition of \(f_{v3}\) (96) and \(f_{v2}\) (75), imply:  
(i) if \(x_{2}-x_{3}\) converges to zero, then \(x_{3}\) converges to \(\Omega_{x3}\), \(\Omega_{x3}=\left\{x_{3}:x_{3}^{l}\leq x_{3}\leq x_{3}^{u}\right\}\); (ii) if \(f_{v2}-f_{v3}\) converges to zero, then \(f_{v3}\) converges to zero, which implies the convergence of \(x_{3}\) to \(\Omega_{x3}\). Therefore, as is indicated by both of these possibilities, \(x_{3}\) converges asymptotically to \(\Omega_{x3}\). From this convergence result and the definition of \(x_{3}\) (10) it follows that \(S_{3}\) converges asymptotically to \(\Omega_{53}\), where:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]8  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]9  
\(S_{3}^{eq}\) is provided by Equations (5)-(7), and the bounds \(x_{3}^{l}\), \(x_{3}^{u}\) are defined in Equation (97). This completes the proof of the second part of the theorem (Tii). From Equation (109), it follows that:  
\[V_{3}>0\;for\;\;x_{3}\;\
ot\in\;\;\left[x_{3}^{l},\;\;x_{3}^{u}\right], \tag{99}\]0  
The condition \(V=0\) implies: (i) \(f_{v1}=0\), \(f_{v2}=0\) and \(f_{v3}=0\), as follows from definitions of \(V\) (110), \(V_{1}\) (35), \(V_{2}\) (81), and \(V_{3}\) (98); (ii) \(x_{1}\in\Omega_{x1}\), \(x_{2}\in\Omega_{x2}\), \(x_{3}\in\Omega_{x3}\), as follows from the definition of \(f_{v1}\) (29), \(f_{v2}\) (75) and \(f_{v3}\) (96). Therefore, it follows from (117) that \(\Omega_{x}=\Omega_{x1}\bigcup\ \Omega_{x2}\bigcup\ \Omega_{x3}\) is invariant, and consequently \(\Omega_{\varepsilon}=\Omega_{x1}\bigcup\ \Omega_{x2}\bigcup\ \Omega_{x3}\) is invariant. This completes the proof of Tiii. \(\square\)  
**Remark 1.