## 4 Hybrid Systems
### Stability of Autonomous Hybrid Systems  
Firstly, consider the case where the impulsive positive hybrid systems (2) based on FSM are autonomous. **Theorem 5**. : _The constants are \(\mu\geq 1\), \(T,\alpha,\lambda>0\) with \(\lambda\alpha<1\), and suppose \(E\) is the desired set. If there exists a vector-valued function \(v_{q(t)}(t)\succ 0\) such that_  
1. \(\|x_{d}(t_{k})\|_{1}\) _is strictly monotonically decreasing w.r.t._ \(k\)_,_
2. \(\tilde{v}_{q(t)}^{\rm T}(t)+v_{q(t)}^{\rm T}(t)A_{q(t)}(t)\prec 0\)_,_ \(q(t^{-})=q(t)\)__
3. \(v_{q(t)}^{\rm T}(t)H_{q(t)}(t)-\lambda\xi^{\rm T}(t)\preceq 0\)_,_ \(q(t^{-})=q(t)\)__
4. \(v_{q(t)}^{\rm T}(t)D_{q(t^{-})}(t)\preceq\mu_{q(t)}v_{q(t^{-})}^{\rm T}(t)\)_,_ \(q(t^{-})\
eq q(t)\) _and_
5. \(\eta^{\rm T}(t)C_{q(t)}(t)\preceq\alpha v_{q(t)}^{\rm T}(t)\)_,_  
_hold, then system (2) is positive and there is IO-FTS w.r.t. \((E,\omega_{1},\eta(\cdot),T)\) under MDADT_  
\[\tau_{q_{i}}\geq\tau_{q_{i}}^{*}=\frac{(T-t_{0})\ln\mu_{q_{i}}}{-\ln\lambda \alpha}, \tag{13}\]  
_where \(A_{q(t)}(t)\) is a Metzler matrix-valued function, \(H_{q(t)}(t)\succeq 0\), \(D_{q(t)}(t)\succeq 0\), \(C_{q(t)}(t)\succeq 0\), \(x(t_{0})=0\), \(v(\cdot)\in\omega_{1}\), \(q(t)\in\Xi\), and \(t\in[t_{0},T]\)._Proof. : Firstly, we shall prove the stability of FSM. From Assumption 1, we know that the shortest path from \(q(t_{0})\) to \(E\) exists. Meanwhile, \(q(t_{0})\in E\), \(x_{d,1}(t_{0})=0\) and \(x_{d,2}(t_{0})<1\). According to condition (1) of Theorem 5, \(x_{d,2}(t_{k})\) is strictly monotonically decreasing w.r.t. \(k\), i.e., \(E_{0}^{k}[q(t_{i})]\) is strictly monotonically increasing w.r.t. \(k\). Therefore, the condition (2) of Definition 3 is satisfied. Meanwhile, \(q(t_{0})\in\Xi/E\), \(x_{d,1}(t_{0})\geq 1\) and \(x_{d,2}(t_{0})=1\). Because \(x_{d,1}(t_{k})\) is an integer, the integer \(\mathcal{K}>0\) must exist, and \(x_{d,1}(t_{\mathcal{K}})=0\). When the \(k<\mathcal{K}\), condition (1) of Theorem 5 implies that \(x_{d,1}(t_{k})\) is strictly monotonically decreasing w.r.t. \(k\), so condition (1) of Definition 3 is satisfied. When \(k\geq\mathcal{K}\), the condition (1) of Theorem 5 implies that \(x_{d,2}(t_{k})\) is strictly monotonically decreasing w.r.t. \(k\), i.e., \(E_{0}^{k}[q(t_{i})]\) is strictly monotonically increasing w.r.t. \(k\), so the condition (2) of Definition 3 is satisfied. Thus, the FSM is stable w.r.t. \((q(t_{0}),E,\mathcal{K})\), and condition (1) of Definition 5 is satisfied. Now, we shall prove the stability of continuous dynamics. Since \(A_{q(t)}(t)\) is a Metzler matrix-valued function, \(H_{q(t)}(t)\succeq 0\), \(D_{q(t)}(t)\succeq 0\), and \(C_{q(t)}(t)\succeq 0\). System (2) is positive by Lemma 1. Choose a co-positive type Lyapunov function \(V_{q(t_{k})}(t,x(t))=v_{q(t_{k})}^{\mathsf{T}}(t)x(t)\). According to conditions (2) and (3) of Theorem 5, for any \(t\in[t_{k},t_{k+1})\), we have \(q(t)=q(t_{k})=q(t_{k+1}^{-})\), and  
\[ \begin{array}{ll}\dot{V}_{q(t_{k})}(t,x(t))&=\left(v_{q(t)}^{\mathsf{T}}(t)A _{q(t)}(t)+v_{q(t)}^{\mathsf{T}}(t)\right)x+\left(v_{q(t)}^{\mathsf{T}}(t)H_{q (t)}(t)-\lambda\xi^{\mathsf{T}}(t)\right)v(t)\\ &+\lambda\xi^{\mathsf{T}}(t)v(t)<\lambda\xi^{\mathsf{T}}(t)v(t)\end{array}  \tag{14}\]  
Noticing that condition (4) of Theorem 5 and integrating (14) between \(t_{k}\) and \(t\), we obtain  
\[