# Finite-Time Control of Singular Linear Semi-Markov Jump Systems
## 3 Results
### Stochastically Finite-Time Stabilization  
**Theorem 1**. : _For given positive scalars \(c_{1}\), \(c_{2}\) and \(T\) with \(c_{1}<c_{2}\) and symmetric positive-definite matrix \(R\), the SSMJS in (1) with \(\omega(t)=0\) is stochastically finite-time stabilizable with respect to \((c_{1},c_{2},T,R)\), if there exists a positive scalar \(\alpha\); a symmetric positive-definite matrix \(P_{i}\); and matrices \(Q_{i}\), \(S_{i}\) and \(Y_{i}\) with appropriate dimensions such that_  
\[ \begin{bmatrix}\Pi_{11}&\Pi_{12}\\ *&\Pi_{22}\end{bmatrix} <0, \tag{5a}\] \[ \begin{bmatrix}\Pi_{11}&\Pi_{12}\\ *&\Pi_{22}\end{bmatrix} <0,\] (5b) \[E^{T}=E^{T}R^{\frac{1}{2}}Q_{i}R^{\frac{1}{2}}EP_{i}E^{T},\] (5c) \[\lambda_{\max}(Q_{i})c_{1}e^{\alpha T}<\lambda_{\min}(Q_{i})c_{2}, \tag{5d}\]  
_where \(H\in\mathbb{R}^{n\times(n-r)}\) is any matrix with full column rank that satisfies \(EH=0\):_  
\[ \begin{array}{l}\underline{\Pi}_{11}=A_{i}(P_{i}E^{T}+HS_{i}^{T})+(P_{i}E^{ T}+HS_{i}^{T})^{T}A_{i}^{T}+B_{i}Y_{i}+Y_{i}^{T}B_{i}^{T}+(\underline{\mu}_{ii}- \alpha)EP_{i}E^{T},\\ \overline{\Pi}_{11}=A_{i}(P_{i}E^{T}+HS_{i}^{T})+(P_{i}E^{T}+HS_{i}^{T})^{T}A_ {i}^{T}+B_{i}Y_{i}+Y_{i}^{T}B_{i}^{T}+(\overline{\mu}_{ii}-\alpha)EP_{i}E^{T}, \\ \underline{\Pi}_{12}=\begin{bmatrix}\sqrt{\mu_{11}}&\cdots&\sqrt{\mu_{i,i-1}}& \sqrt{\mu_{i,i+1}}&\cdots&\sqrt{\mu_{iN}}\end{bmatrix}EP_{i}E^{T},\\ \overline{\Pi}_{12}=\begin{bmatrix}\sqrt{\mu_{i1}}&\cdots&\sqrt{\mu_{i,i-1}}& \sqrt{\mu_{i,i+1}}&\cdots&\sqrt{\mu_{iN}}\end{bmatrix}EP_{i}E^{T},\\ \Pi_{22}=diag\{\Pi_{22,1},\ldots,\Pi_{22,i-1},\Pi_{22,i+1},\ldots,\Pi_{22,N}\}, \end{array} \]with \(\Pi_{22,l}=P_{l}-(P_{l}E^{T}+HS_{l}^{T})-(P_{l}E^{T}+HS_{l}^{T})^{T},l\in\mathcal{S}\). Furthermore, a suitable state feedback control law is given as_  
\[u(t)=Y_{i}(P_{i}E^{T}+HS_{l}^{T})^{-1}x(t). \tag{6}\]  
Proof of Theorem 1.: Firstly, consider the regularity and absence of impulse. Since \(\text{rank}(E)=r\leq n\), there must exist nonsingular matrices \(L\) and \(T\) such that  
\[\bar{E}=LET= \begin{bmatrix}I_{l}&0\\ 0&0\end{bmatrix} . \tag{7}\]  
Similar to this matrix decomposition, we can also define  
\[ \begin{split}\mathcal{A}_{i}=L\mathcal{A}_{i}T=\begin{bmatrix} \mathcal{A}_{i11}&\mathcal{A}_{i12}\\ \mathcal{A}_{i21}&\mathcal{A}_{i22}\end{bmatrix},S_{i}=LS_{i}=\begin{bmatrix} S_{i1}\\ S_{i2}\end{bmatrix},\\ \bar{P}_{i}= T^{-1}P_{i}T^{-T}=\begin{bmatrix}\bar{P}_{i11}&\bar{P}_{i12}\\ \bar{P}_{i21}&\bar{P}_{i22}\end{bmatrix},\quad\bar{H}=T^{-1}H=\begin{bmatrix} 0\\ \bar{H}_{2}\end{bmatrix},\end{split}  \tag{8}\]  
where \(\bar{H}_{2}\in\mathbb{R}^{(n-r)\times(n-r)}\) is nonsingular. By multiplying (5a) or (5b) by \(L\) and \(L^{T}\) on the left and right, respectively, we get  
\[ \begin{bmatrix}\star&\star\\ \star&\mathcal{A}_{i22}\bar{H}_{2}S_{i2}^{T}+S_{i2}\bar{H}_{2}^{T}\mathcal{A} _{i22}^{T}\end{bmatrix} <0, \tag{9}\]  
where \(\star\) represents the matrix block, which is irrelevant to the following discussion. From this inequality, it can be obtained that \(\mathcal{A}_{i22}\) must be nonsingular. Otherwise, there exists a nonzero vector \(\xi\in\mathbb{R}^{n-r}\) such that \(\mathcal{A}_{i22}^{T}\xi=0\) and then \(\xi^{T}(\mathcal{A}_{i22}\bar{H}_{2}S_{i2}^{T}+\bar{S}_{i2}\bar{H}_{2}^{T} \mathcal{A}_{i22}^{T})\xi=0\), which contradicts to (9). Then, the closed-loop SSMJS is regular and impulse-free [19; 21]. Next, a Lyapunov-like function is defined as  
\[V(x(t),r(t))=x^{T}(t)E^{T}(P(r(t))E^{T}+HS(r(t))^{T})^{-1}x(t). \tag{10}\]  
With the fact of \(EH=0\), it is obtained that \((P_{l}E^{T}+HS_{i}^{T})^{T}E^{T}=E(P_{l}E^{T}+HS_{i}^{T})=EP_{l}E^{T}\geq 0\). Multiplying the last equation by \((P_{l}E^{T}+HS_{i}^{T})^{-T}\) and \((P_{l}E^{T}+HS_{i}^{T})^{-1}\) on the left and right, respectively, gives  
\[E^{T}(P_{l}E^{T}+HS_{i}^{T})^{-1}=(P_{i}E^{T}+HS_{i}^{T})^{-T}E\geq 0, \tag{11}\]  
and then the Lyapunov-like function (10) is symmetric semi-positive definite.