## 4 Hybrid Systems
### Stability of Autonomous Hybrid Systems
\begin{array}{ll}V_{q(t_{k})}(t,x(t))&<\ \lambda\int_{t_{k}}^{t}\xi^{\mathsf{T}} (\tau)v(\tau)\tau d\tau+v_{q(t_{k})}^{\mathsf{T}}(t_{k})x(t_{k})=\lambda\int_{ t_{k}}^{t}\xi^{\mathsf{T}}(\tau)v(\tau)\tau d\tau+v_{q(t_{k})}^{\mathsf{T}}(t_{k}) D_{q(t_{k})}x(t_{k}^{-})\\ &\leq\ \lambda\int_{t_{k}}^{t}\xi^{\mathsf{T}}(\tau)v(\tau)\tau d\tau+\mu_{q(t_{k })}\lambda\int_{t_{k}}^{t}\xi^{\mathsf{T}}(\tau)v(\tau)\tau d\tau+\mu_{q(t_{k })}V_{q(t_{k-1})}(t_{k}-t_{k-1})\\ &\leq\ \lambda\int_{t_{k}}^{t}\xi^{\mathsf{T}}(\tau)v(\tau)\tau d\tau+\mu_{q(t_{k })}V_{q(t_{k-1})}(t_{k-1},x(t_{k-1}))\\ &\leq\ \cdots\leq\lambda\prod_{t_{k}=1}^{t_{k}}\mu_{q(t_{k})}\int_{t_{k}}^{t}\xi^ {\mathsf{T}}(\tau)v(\tau)\tau d\tau+\prod_{t=1}^{t}\mu_{q(t_{k})}V_{q(t_{k})} (t_{0},x(t_{0}))\\ &=\ \lambda\prod_{t=1}^{t}\mu_{q(t_{k})}\int_{t_{0}}^{t}\xi^{\mathsf{T}}(\tau)v (\tau)\tau d\tau.\end{array}  \tag{15}\]  
According to the MDADT (13), Definition 6, and \(v(\cdot)\in\omega_{1}\),  
\[ \begin{array}{ll}V_{q(t_{k})}(t,x(t))&<\ \lambda\prod_{k=1}^{k}\mu_{q(t_{k})}=\lambda\prod_{i=1}^{m} \mu_{q_{i}}^{N_{i}(t_{0},t)}=\lambda e^{\sum\limits_{i=1}^{\frac{m}{T_{0}}(t_{0 },t)}{q_{i}}}\ln\mu_{q_{i}}\\ &\leq\ \lambda e^{\sum\limits_{i=1}^{\frac{T_{0}}{q(t_{0})}\ln\mu_{q_{i}}}}\ln \mu_{q_{i}}\\ &\leq\ \lambda e^{\sum\limits_{i=1}^{\frac{T_{0}}{q(t_{0})}\ln\mu_{q_{i}}}}\ln \mu_{q_{i}}=\lambda e^{\frac{(t-t_{0})\ln\mu_{q_{i}}}{-(T-t_{0})}}\leq\frac{1} {\alpha}\end{array}  \tag{16}\]  
From condition (5) of Theorem 5, we know  
\[\eta^{\mathsf{T}}(t)y(t)=\eta^{\mathsf{T}}(t)C_{q(t)}(t)x(t)\leq\alpha v_{q(t) }^{\mathsf{T}}(t)x(t)<1,\]  
and then the condition (2) of Definition 5 is satisfied. Thus, the system (2) is positive and there is IO-FTS \((E,\omega_{1},\eta(\cdot),T)\) in the sense of Definitions 4 and 5. **Remark 5**. : _We note that the MDADT (13) depends on the parameter \(\mu_{q_{i}}\). When \(\mu_{q_{i}}>1\), obviously, \(\tau_{q_{i}}>0\). This can prevent the hybrid systems from exhibiting Zeno behavior ([41]). If \(\mu_{q_{i}}=1\), then \(\tau_{q_{i}}^{*}=0\), that is to say, the jump can be arbitrary._  
**Theorem 6**. : _The constants are \(\mu\geq 1\), \(T,\alpha,\lambda>0\) with \(\lambda\alpha(T-t_{0})<1\), 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) \(v_{q(t)}^{\mathsf{T}}(t)+v_{q(t)}^{\mathsf{T}}(t)A_{q(t)}(t)\prec 0,\ \ \ \ \ \ \ \ \ \ q(t^{-})=q(t)\)_(3)_ \(v^{\mathsf{T}}_{q(t)}(t)H_{q(t)}(t)-\lambda\delta^{\mathsf{T}}(t)\preceq 0\)_,_ \(q(t^{-})=q(t)\)__  
_(4)_ \(v^{\mathsf{T}}_{q(t)}(t)D_{q(t^{-})}(t)\preceq\mu_{q(t)}v^{\mathsf{T}}_{q(t^{-})} (t)\)_,_ \(q(t^{-})\
eq q(t)\) _and_ _(5)_ \(\eta^{\mathsf{T}}(t)C_{q(t)}(t)\preceq\alpha v^{\mathsf{T}}_{q(t)}(t)\)_,_  
_hold, then system (2) is positive and there is IO-FTS w.r.t. \((E,\omega_{\infty},\eta(\cdot),T)\) under MDADT_  
\[\tau_{q_{i}}\geq\tau^{*}_{q_{i}}=\frac{(T-t_{0})\ln\mu_{q_{i}}}{-\ln(\lambda \alpha(T-t_{0}))}, \tag{17}\]  
_where \(A_{q(t)}(t)\) is a Metzler matrix-valued function, \(H_{q(t)}(t)\succeq 0\), \(D_{q(t)}(t)\succeq 0\), \(C_{q(t)}(t)\succeq 0\), \(x(t_{0})=0\), \(v(\cdot)\in\omega_{\infty}\), \(q(t)\in\Xi\), and \(t\in[t_{0},T]\)._  
Proof. : The proof of the stability for FSM is the same as that of Theorem 5, thus, condition (1) of the Definition 5 is satisfied. Now, let us discuss the stability of the continuous dynamics. From the proof process of Theorem 5, we know the system (2) is positive, and  
\[V_{q(t_{k})}(t,x(t))<\lambda\prod_{i=1}^{k}\mu_{q(t_{i})}\int_{t_{0}}^{t} \delta^{\mathsf{T}}(\tau)v(\tau)d\tau. \tag{18}\]  
Because of \(v(\cdot)\in\omega_{\infty}\) and MDADT (17), we can get  
\[