## 3 Single Linear System
### Stability of Autonomous System  
When \(B(\cdot)\) is a zero matrix, and the system (6) is autonomous, \(\tilde{A}(t)=A(t)\), then the following theorems are obtained. Theorem 3.1: The constants are \(T,\rho,\lambda,\alpha>0\) with \(\rho\alpha<1\), and vector-value function \(\eta(t)\succ 0\). If there exist vector-valued functions \(v(t),\vartheta(t),\pi(t):\mathbb{R}^{+}\to\mathbb{R}^{n}_{+}\) and \(\theta(t):\mathbb{R}^{+}\to\mathbb{R}^{n}\) such that  
1. \(\vartheta^{\rm T}(t)+v^{\rm T}(t)A(t)+\pi^{\rm T}(t)+\vartheta^{\rm T}(t)\prec 0\),
2. \(v^{\rm T}(t)H(t)+\vartheta^{\rm T}(t)-\rho\xi^{\rm T}(t)\preceq 0\),
3. \(\lambda\pi^{\rm T}(t)+\vartheta^{\rm T}(t)+\vartheta^{\rm T}(t)A(t)+\lambda \vartheta^{\rm T}(t)\succeq 0\),
4. \(\lambda\vartheta^{\rm T}(t)+\vartheta^{\rm T}(t)H(t)\succeq 0\), and
5. \(\eta^{\rm T}(t)C(t)-\alpha v^{\rm T}(t)\preceq 0\),  
hold, then the system (6) is positive and there is IO-FTS w.r.t. \((\omega_{1},\eta(\cdot),T)\), where \(A(t)\) is a Metzler matrix-valued function, \(H(t)\succeq 0\), \(C(t)\succeq 0\), \(x(t_{0})=0\), \(v(\cdot)\in\omega_{1}\), and \(t\in[t_{0},T]\). Proof: When \(A(t)\) is a Metzler matrix-valued function, \(H(t)\succeq 0\) and \(C(t)\succeq 0\), system (6) is positive by Lemma 1. Choosing co-positive Lyapunov function  
\[V(t,x(t))=v^{\rm T}(t)x(t)+\int_{t_{0}}^{t}e^{-\lambda(t-s)}\pi^{\rm T}(s)x(s) ds+\int_{t_{0}}^{t}e^{-\lambda(t-s)}\vartheta^{\rm T}(s)v(s)ds,\]according to conditions (1) and (2) of Theorem 1, we get  
\[ \begin{array}{ll}\dot{V}(t,x(t))&=&(v^{\rm T}(t)+v^{\rm T}(t)A(t))x(t)+v^{\rm T}( t)H(t)v(t)+\pi^{\rm T}(t)x(t)-\\ &&\lambda\int_{t_{0}}^{t}e^{-\lambda(t-s)}\pi^{\rm T}(s)x(s)ds+\theta^{\rm T}(t )v(t)-\lambda\int_{t_{0}}^{t}e^{-\lambda(t-s)}\theta^{\rm T}(s)v(s)ds\\ &&=&(\tilde{v}^{\rm T}(t)+v^{\rm T}(t)A(t)+\pi^{\rm T}(t)x(t)+(v^{\rm T}(t)H(t )+\tilde{\theta}^{\rm T}(t)-\rho_{\rm v}^{\rm T}(t))v(t)+\rho_{\rm v}^{\rm T}( t)v(t)-\\ &&\lambda\int_{t_{0}}^{t}e^{-\lambda(t-s)}\pi^{\rm T}(s)x(s)ds-\lambda\int_{t _{0}}^{t}e^{-\lambda(t-s)}\theta^{\rm T}(s)v(s)ds.\end{array}  \tag{8}\]  
Due to \(\int_{t_{0}}^{t}e^{-\lambda(t-s)}d(\theta^{\rm T}(s)x(s))=\theta^{\rm T}(t)x( t)-\lambda\int_{t_{0}}^{t}e^{-\lambda(t-s)}\theta^{\rm T}(s)x(s)ds\), \(\dot{x}(t)=A(t)x(t)+H(t)v(t)\),  
\[ \begin{array}{llll}\int_{t_{0}}^{t}e^{-\lambda(t-s)}d(\theta^{\rm T}(s)x(s) )&=&\int_{t_{0}}^{t}e^{-\lambda(t-s)}(\dot{\theta}^{\rm T}(s)x(s)+\theta^{ \rm T}(s)x(s))ds\\ &=&\int_{t_{0}}^{t}e^{-\lambda(t-s)}(\dot{\theta}^{\rm T}(s)x(s)+\theta^{\rm T }(s)A(s)x(s)+\theta^{\rm T}(s)H(s)v(s))ds,\end{array}  \tag{9}\]  
therefore,  
\[\theta^{\rm T}(t)x(t)-\int_{t_{0}}^{t}e^{-\lambda(t-s)}(\dot{\theta}^{\rm T} (s)+\theta^{\rm T}(s)A(s)+\lambda\theta^{\rm T}(s))x(s)ds-\int_{t_{0}}^{t}e^ {-\lambda(t-s)}\theta^{\rm T}(s)H(s)v(s)ds=0. \tag{10}\]  
Taking (10) into (8), we can obtain  
\[ \begin{array}{llll}\dot{V}(t,x(t))&=&(v^{\rm T}(t)+v^{\rm T}(t)A(t)+\pi^{ \rm T}(t)+\theta^{\rm T}(t)x(t)+(v^{\rm T}(t)H(t)+\theta^{\rm T}(t)-\rho_{\rm v }^{\rm T}(t))v(t)+\\ &&\rho_{\rm v}^{\rm T}(t)v(t)-\int_{t_{0}}^{t}e^{-\lambda(t-s)}(\lambda\pi^{ \rm T}(s)+\theta^{\rm T}(s)+\theta^{\rm T}(s)A(s)+\lambda\theta^{\rm T}(s))x( s)ds-\\ &&\int_{t_{0}}^{t}e^{-\lambda(t-s)}(\lambda\dot{\theta}^{\rm T}(s)+\theta^{ \rm T}(s)H(s))v(s)ds,\end{array}  \tag{11}\]  
According to conditions (1)-(4) of Theorem 1, we find  
\[\dot{V}(t,x(t)<\rho_{\rm v}^{\rm T}(t)v(t). \tag{12}\]  
Integrating (12) between \(t_{0}\) and \(t\), and taking into account that \(v(\cdot)\in\omega_{1}\) and \(x(t_{0})=0\), we obtain  
\[V(t,x(t))<\rho\int_{t_{0}}^{t}\xi^{\rm T}(\tau)v(\tau)dt\leq\rho.\]  
From condition (5) of Theorem 1,  
\[\eta^{\rm T}(t)y(t)=\eta^{\rm T}(t)C(t)x(t)\leq\alpha v^{\rm T}x\leq\alpha V( t,x(t))<1.\]  
Then the system (6) is positive and there is IO-FTS w.r.t. \((\omega_{1},\eta(\cdot),T)\) in the sense of Definition 4 and Definition 5. **Remark 4**.