# Finite-Time Control of Singular Linear Semi-Markov Jump Systems
## 3 Results
### Stochastically Finite-Time Boundedness
\]According to (14) and (16), by applying the [PERSON] complement lemma, and then multiplying (31a) by \(\underline{m}\) and (31b) by \(\overline{m}\), with \(\overline{m}+\underline{m}=1\) and \(\overline{m}>0,\underline{m}>0\), we can also obtain  
\[\ell V(x(t),r(t))<\alpha\big{(}V(x(t),r(t))+\omega^{T}(t)W_{i}\omega(t)\big{)}. \tag{35}\]  
Multiplying it by \(e^{-\alpha t}\) on the both sides gives  
\[\ell e^{-\alpha t}V(x(t),r(t))<\alpha e^{-\alpha t}\omega^{T}(t)W_{i}\omega(t). \tag{36}\]  
From the Dynkin formula, it is easy to obtain that  
\[e^{-\alpha t}\mathbb{E}\big{\{}V(x(t),r(t))\big{\}}-\mathbb{E}\big{\{}V(x(0),r (0))\big{\}}<\alpha\bigg{(}\int_{0}^{t}e^{-\alpha s}\mathbb{E}\{\omega^{T}(s) W_{i}\omega(s)\}\mathrm{d}s\bigg{)}.\]  
For \(t\in[0,T]\), one can get  
\[\mathbb{E}\big{\{}V(x(t),r(t))\big{\}} <e^{\alpha T}\bigg{(}\mathbb{E}\big{\{}V(x(0),r(0))\big{\}}+ \alpha\bigg{(}\int_{0}^{T}e^{-\alpha s}\mathbb{E}\{\omega^{T}(s)W_{i}\omega(s) \}\mathrm{d}s\bigg{)}\bigg{)}\] \[<e^{\alpha T}\big{(}\mathbb{E}\big{\{}V(x(0),r(0))\big{\}}+ \alpha d\lambda_{\max}(W_{i})\big{)}. \tag{37}\]  
From (31d), it follows \(\lambda_{\max}(W_{i})<\eta\) and therefore (31c) implies  
\[c_{1}e^{\alpha T}+\alpha d\lambda_{\max}(W_{i})\theta e^{\alpha T}<\theta c_{ 2}. \tag{38}\]  
From (31c) and (31d), it follows that \(\theta<\lambda_{\min}\big{(}Q_{i}^{-1}\big{)}<\lambda_{\max}\big{(}Q_{i}^{-1} \big{)}<1\) and then \(\lambda_{\max}\big{(}Q_{i}^{-1}\big{)}c_{1}e^{\alpha T}<c_{1}e^{\alpha T}< \theta\big{(}c_{2}-\alpha d\lambda_{\max}(W_{i})e^{\alpha T}\big{)}<\lambda_{ \min}\big{(}Q_{i}^{-1}\big{)}\big{(}c_{2}-\alpha d\lambda_{\max}(W_{i})e^{ \alpha T}\big{)}\). Bearing in mind that \(\lambda_{\max}(Q_{i})=\frac{1}{\lambda_{\min}\big{(}Q_{i}^{-1}\big{)}}\), \(\lambda_{\min}(Q_{i})=\frac{1}{\lambda_{\max}\big{(}Q_{i}^{-1}\big{)}}\), it can be derived that  
\[\lambda_{\max}(Q_{i})c_{1}e^{\alpha T}<\lambda_{\min}\big{(}Q_{i}\big{)}\big{(} c_{2}-\alpha d\lambda_{\max}(W_{i})e^{\alpha T}\big{)}. \tag{39}\]  
Similarly to Theorem 2, we have  
\[\mathbb{E}\big{\{}V(x(t),r(t))\big{\}} \geq\lambda_{\min}(Q_{i})\mathbb{E}\big{\{}x^{T}(t)E^{T}REx(t) \big{\}}, \tag{40a}\] \[V(x(0),r(0)) \leq\lambda_{\max}(Q_{i})\mathbb{E}\big{\{}x^{T}(0)E^{T}REx(0) \big{\}}. \tag{40b}\]  
If \(\mathbb{E}\big{\{}x^{T}(0)E^{T}REx(0)\big{\}}\leq c_{1}\), from (37), (39) and (40), we get \(\mathbb{E}\big{\{}x^{T}(t)E^{T}REx(t)\big{\}}<c_{2}\). Therefore, the closed-loop system is stochastically finite-time bounded with respect to \((c_{1},c_{2},T,R,d)\), which completes the proof. **Remark 5**. : _The sufficient conditions for mode-dependent controller design of SSMJSs are presented in this theorem, which guarantees the closed-loop SSMJS is stochastically finite-time bounded and of a unique solution. For the existence of scalar \(\alpha\), the matrix inequalities (31) are not linear. If we fix \(\alpha\) to be constant, they are linear ones. With this consideration, we can use the one-dimensional search method to obtain the scalar \(\alpha\in(0,\alpha^{\ast}]\) to resolve (31)._