## 3 Main Results
In the case that \(m\leq p\), Theorem 1 (ii) might be directly generalized to a subsystem's decomposition philosophy if a number \(q\leq m\) of subsystems of inputs and outputs \(\left(u_{1}^{T}\), \(u_{2}^{T},\cdots,u_{q}^{T}\right)^{T}\) and \(\left(y_{1}^{T}\), \(y_{2}^{T},\cdots,y_{q}^{T}\right)^{T}\) with \(u_{i}\in\mathbf{R}^{m_{i}}\), \(y_{i}\in\mathbf{R}^{m_{i}}\); \(\forall i\in\overline{n}\) with \(p=\sum_{i=1}^{q}p_{i}\) and \(m=\sum_{i=1}^{q}m_{i}\)._  
**Remark 2**. : _Theorem 1 can be easily generalized to cases when some dynamics transmission links between state, input or output components (or subsystems, in general) can be suppressed by manipulation. In more general cases, it is possible to extend Theorem 1 to combinations of the subsequent situations with the matrix decompositions having the same sense (in the various modified contexts) as that of Theorem 1 (ii):_  
* **Case 1**. Suppression of some transmission links between the coupled open-loop dynamics by examining the decompositions: \(A_{c}=A_{\text{\emph{cd}}}+A_{\text{\emph{cod}}}\), \(A_{d}=A_{\text{\emph{dd}}}+A_{\text{\emph{dd}}}\), and \(\widetilde{A}=A_{c}-A_{d}=\left(A_{\text{\emph{cd}}}-A_{\text{\emph{dd}}} \right)+\left(A_{\text{\emph{cod}}}-A_{\text{\emph{dd}}}\right)\). (_a_) If there is at least one non-diagonal zero entry in at least one of its rows in the off-diagonal controller error matrix \(\widetilde{A}_{\text{\emph{cd}}}=A_{\text{\emph{cod}}}-A_{\text{\emph{dd}}}\) which is not a corresponding zero in \(A_{\text{\emph{cod}}}\); and (_b_) if there is at least one non-diagonal zero entry in at least one of its rows in the off-diagonal controller error matrix \(\widetilde{K}_{\text{\emph{od}}}=K_{\text{\emph{cod}}}-K_{\text{\emph{dd}}}\), then the closed-loop system is stabilizable under a partial decentralized control even if some links of the dynamics between crossed components are cut if Theorem 1 (ii) holds. If only Condition a is addressed, then the system is stabilizable by centralized control when cutting certain transmission links between coupled dynamics in the open-loop system. This idea can be extended to total decentralized control for a purely diagonal open-loop system's dynamics under full zeroing of the off-diagonal corresponding error dynamics. It can be also generalized to the \"ad hoc\" decompositions between subsystems. Other cases with similar interpretations in the new contexts are:  
* **Case 2**. Suppression of some crossed entries in the open-loop control matrix by examining the decompositions: \(B_{c}=B_{\text{\emph{cd}}}+B_{\text{\emph{cod}}}\), \(B_{d}=B_{\text{\emph{dd}}}+B_{\text{\emph{dd}}}\), and \(\widetilde{B}=B_{c}-B_{d}=\left(B_{\text{\emph{cd}}}-B_{\text{\emph{dd}}} \right)+\left(B_{\text{\emph{cod}}}-B_{\text{\emph{dd}}}\right)\). * **Case 3**. Suppression of some crossed entries in the open-loop output matrix by examining the decompositions: \(C_{c}=C_{\text{\emph{cd}}}+C_{\text{\emph{cod}}}\), \(C_{d}=C_{\text{\emph{dd}}}+C_{\text{\emph{dd}}}\), and \(\widetilde{C}=C_{c}-C_{d}=\left(C_{\text{\emph{cd}}}-C_{\text{\emph{dd}}} \right)+\left(C_{\text{\emph{cod}}}-D_{\text{\emph{dd}}}\right)\). * **Case 4**. Suppression of some crossed entries in the open-loop interconnection matrix by examining the decompositions: \(D_{c}=D_{\text{\emph{cd}}}+D_{\text{\emph{cad}}}\), \(D_{d}=D_{\text{\emph{dd}}}+D_{\text{\emph{dd}}}\), and \(\widetilde{D}=D_{c}-D_{d}=\left(D_{\text{\emph{cd}}}-D_{\text{\emph{dd}}} \right)+\left(D_{\text{\emph{cod}}}-D_{\text{\emph{dd}}}\right)\). * **Case 5**. Any combinations of Cases 1-4. **Problem 1**. : _Find a stabilizing decentralized family of control gains by assuming that \(A_{d}=A_{c}\), such that \(\widetilde{A}=A_{c}-A_{d}=0\) and Assumption 1 holds with \(K_{c}\) being a stabilizing centralized controller gain._  
The following more general result for the eventual case \(\widetilde{A}\
eq 0\) (that is \(A_{c}\
eq A_{d}\) eventually), follows from Theorem 1, Theorem A1 and Theorem A2 and Lemmas B1, B2 and B3:  
**Theorem 2**.