## 3 Global Stability Analysis
### Stability Analysis for the Second and Third Tanks  
The stability analysis and majorly the proof for asymptotic convergence of \(x_{2}\) and \(x_{3}\) to their compact sets, is based on determining the non-positive nature of \(dV/dt\ =\ dV_{1}/dt+k_{2}dV_{2}/dt+k_{3}dV_{3}/dt\), where \(V=V_{1}+kV_{2}+kV_{3}\), \(V\) is the overall Lyapunov function, \(V_{2}\) is the dead zone quadratic form for the state \(x_{2}\) and \(V_{3}\) the dead zone quadratic form for the state \(x_{3}\), and \(k_{2}\) and \(k_{3}\) are positive constants. To achieve this goal, the functions \(V_{2}\) and \(V_{3}\) and the gradients \(dV_{2}/dx_{2}\) and \(dV_{3}/dx_{3}\) are defined accordingly, and the non-positive nature of the terms of the \(dV/dt\) expression is determined. The procedure comprises the following tasks: (i) determination of the expression for \(dV_{2}/dt+kdV_{3}/dt\); (ii) arrangement of the expression for \(dV_{2}/dt+kdV_{3}/dt\) in terms of \(f_{v1}\); (iii) definition of gradient \(dV_{2}/dx_{2}\) and dead zone quadratic form \(V_{2}\); (iv) arrangement of the expression for \(dV_{2}/dt+kdV_{3}/dt\) in terms of a non-positive function of \(dV_{2}/dx_{2}\); (v) definition of gradient \(dV_{3}/dx_{3}\) and dead zone quadratic form \(V_{3}\); (vi) definition of the overall Lyapunov function \(V\) and determination of the non-positive nature of the expression for \(dV/dt=dV_{1}/dt+kdV_{2}/dt+kdV_{3}/dt\); (vii) integration of the \(dV/dt\) expression. The definition of the gradients \(dV_{2}/dx_{2}\) and \(dV_{3}/dx_{3}\) and the quadratic forms \(V_{2}\) and \(V_{3}\) requires the determination of the expression for \(dV_{2}/dt+kdV_{3}/dt\) and the properties of the terms involved. **Theorem 2. **_Consider the model (11)-(15), subject to assumptions 1 to 3. (Ti) the state \(x_{2}\) converges asymptotically to \(\Omega_{x2}\), \(\Omega_{x2}=\left\{x_{2}:x_{2}^{l}\leq x_{2}\leq x_{2}^{u}\right\}\), where:_  
\(x_{2}^{u}=\left\{x_{2}:\,g_{s2}-max\{\delta_{2}\}=0\right\}\)_, \(x_{2}^{u}>0x_{2}^{l}=\left\{x_{2}:\,g_{s2}-min\{\delta_{2}\}=0\right\}\), \(x_{2}^{l}<0g_{s2}=x_{2}+\frac{r_{s2}}{Q/v_{2}},\delta_{2}=-d_{x1},d_{x1}=\left\{ \begin{array}{c}-x_{1}^{u}\,\,for\,\,x_{1}>x_{1}^{u}\\ -x_{1}\,\,for\,\,x_{1}\,\in\,\left[x_{1}^{l},\,x_{1}^{u}\right]\\ -x_{1}^{l}\,\,for\,\,x_{1}<x_{1}^{l}\end{array} \right.\)_  
\(S_{2}\) _converges asymptotically to \(\Omega_{S2}\) where \(\Omega_{S2}=\left[S_{2}^{l}\,\,\,S_{2}^{u}\right]\), \(S_{2}^{l}=x_{2}^{l}+S_{2}^{eq}\), \(S_{2}^{u}=x_{2}^{u}+S_{2}^{eq}\) and \(S_{2}^{eq}\) is provided by Equations (5)-(7). (Tii) the state \(x_{3}\) converges asymptotically to \(\Omega_{x3}\), \(\Omega_{x3}=\left\{x_{3}:x_{3}^{l}\leq x_{3}\leq x_{3}^{u}\right\}\), where \(x_{3}^{u}=x_{2}^{u}\); \(x_{3}^{l}=x_{2}^{l}\), and \(S_{3}\) converges asymptotically to \(\Omega_{S3}\) where \(\Omega_{S3}=\left[S_{3}^{l}\,\,\,S_{3}^{u}\right]\), \(S_{3}^{u}=x_{3}^{l}+S_{3}^{eq}\), \(S_{3}^{u}=x_{3}^{u}+S_{3}^{eq}\) and \(S_{3}^{eq}\) is provided by Equations (5)-(7). (Tiii) The sets \(\Omega_{x}=\Omega_{x1}\bigcup\,\,\Omega_{x2}\bigcup\,\,\Omega_{x3}\) and \(\Omega_{S}=\Omega_{S1}\bigcup\,\,\Omega_{S2}\bigcup\,\,\Omega_{S3}\) are invariant._  
**Proof. ****Task 1. ** Determination of the expression for \(dV_{2}/dt+kdV_{3}/dt\).