# Impulse Controllability for Singular Hybrid Coupled Systems
## 3 Results
\begin{array}{cccccc}0&\bar{E}&0&0&\bar{B}_{\varphi_{1 }}&0\\ \bar{E}&\tilde{A}(h(t))&\bar{B}_{j}&\bar{B}_{\varphi}&0&\bar{B}_{\varphi_{2}} \\ 0&\bar{C}_{\mathcal{N}-\varphi}&0&0&0&0\\ 0&\bar{C}_{\varphi-\varphi_{1}}&0&0&0&0\\ \bar{C}_{(\mathcal{N}-\varphi)-\varphi_{2}}&0&0&0&0&0\end{array} \right] \right)\geq Nn+r. \tag{29}\]  
**Remark 1**. : _The proof of Theorem 2 can obviously be concluded by repeated use of Lemma 4 and the impulse controllability conditions._  
**Corollary 1**. : _Consider the system (4). According to Lemma 2(3), the closed-loop system (10) is impulse observable if and only if there is a feedback controller (9) that satisfies this condition for all \(\varphi\in P(\mathcal{N})\), \(\varphi_{1}\in P(\varphi)\), \(\varphi_{2}\in P(\mathcal{N}-\varphi)\)._  
\[\text{rank}\left(\left[ \begin{array}{cccccc}0&\bar{E}&0&\bar{B}_{\varphi_{1 }}&0\\ \bar{E}&\tilde{A}(h(t))&\bar{B}_{\varphi}&0&\bar{B}_{\varphi_{2}}\\ 0&\bar{C}_{\mathcal{N}-\varphi}&0&0&0\\ 0&\bar{C}_{\varphi-\varphi_{1}}&0&0&0\\ \bar{C}_{(\mathcal{N}-\varphi)-\varphi_{2}}&0&0&0&0\\ 0&\bar{C}_{j}&0&0&0\end{array} \right]\right)\geq Nn+r \tag{30}\]  
**Remark 2**. : _In [4], the authors considered the stability problem of singular complex networks. However, for singular systems, the impulsive behavior always has a great influence on the system. Impulsive behavior may cause the system to have poor stability and can even destroy the system. Therefore, it is very important to study pulse controllability, pulse observability, and the elimination of a pulse._  
**Theorem 3**. : _The system (1) is admissible, if_  
\[\bar{E}^{-}=Q\left[ \begin{array}{cc}I_{\tilde{q}}&0\\ 0&0\end{array} \right]P\in\bar{E}\{1\},\alpha\in\mathcal{V}_{H_{n^{2},n^{2}}}, \tag{31}\]  
_and_  
\[trH(\bar{E}^{-}\tilde{A}(\alpha))<-\sqrt{(\tilde{q}-1)tr\big{[}H(\bar{E}- \tilde{A}(\alpha))\big{]}^{2}}, \tag{32}\]  
_where_  
\[P\bar{E}Q=\left[ \begin{array}{cc}I_{\tilde{q}}&0\\ 0&0\end{array} \right],\tilde{q}=\sum_{i=1}^{N}r_{e_{i}},\]  
\[\tilde{A}(\alpha)=\tilde{A}(h(t))+\Omega,\Omega=\sum_{i=1}^{N}\sum_{k=1}^{n_{ i}}\sum_{j=1}^{n}e_{n}(i)(\sum_{l=1}^{i-1}n_{l}+k)e_{n}^{T}(j).\]  
Proof. : According to the convexity principle, as detailed by [28] the inequality presented in (33) suggests that  
\[trH(\bar{E}^{-}\tilde{A}(h(t)))<-\sqrt{(\tilde{q}-1)tr\big{[}H(\bar{E}-\tilde {A}(h(t)))\big{]}^{2}} \tag{33}\]for all \(h(t)\in\mathcal{V}_{H_{\mu_{2},n^{2}}}\), based on a novel restricted system equivalence between the matrix,  
\[PEQ=\left[ \begin{array}{cc}I_{q}&0\\ 0&0\end{array} \right],P\tilde{A}(h(t))Q=\left[ \begin{array}{cc}A_{1}&0\\ 0&I_{n-\bar{q}}\end{array} \right],\]  
where \(A_{1}\in\mathbb{R}^{q\times q}\), and  
\[\det(s\bar{E}-\tilde{A}(h(t)))=0\Leftrightarrow\det\bigl{(}P(s\bar{E}-\tilde{A }(h(t)))Q\bigr{)}=0\Leftrightarrow\det(sl_{q}-A_{1})=0. \tag{34}\]  
Therefore,  
\[\sigma(\bar{E},\tilde{A}(h(t)))=\sigma(A_{1}).\]  
On the other hand, by Lemma 5, when  
\[\bar{E}^{-}=Q\left[ \begin{array}{cc}I_{q}&0\\ 0&0\end{array} \right]P,\]  
then  
\[\bar{E}^{-}\tilde{A}(h(t))=Q\left[ \begin{array}{cc}I_{q}&0\\ 0&0\end{array} \right]PP^{-1}\left[ \begin{array}{cc}A_{1}&0\\ 0&I_{n-\bar{q}}\end{array} \right]Q^{-1}=Q\left[ \begin{array}{cc}A_{1}&0\\ 0&0\end{array} \right]Q^{-1},\]  
we have  
\[tr(\bar{E}^{-}\tilde{A}(h(t))=trA_{1},tr\bigl{[}H(\bar{E}^{-}\tilde{A}(h(t)) \bigr{]}=tr[H(A_{1})]. \tag{35}\]  
By (35) and (33), we can obtain  
\[tr[H(A_{1})]<-\sqrt{(\bar{q}-1)tr[H(A_{1})]^{2}}, \tag{36}\]  
by Lemma 6, we have  
\[\sigma(\bar{E},\tilde{A}(h(t)))=\sigma(A_{1})\subset C^{-},\]  
the system (1) is admissible. \(\square\)  
**4. Illustrative Examples**  
This section explores some illustrative examples to demonstrate the efficacy of the proposed method. **Example 1.