## 3 Main Results
Then, the following properties hold:_  
_(i) \(A_{dc}\in\mathbf{R}^{n\times n}\) is a closed-loop stability matrix under a linear output-feedback stabilizing controller gain \(K_{d}\in\mathbf{R}^{m\times p}\) if any of the subsequent sufficiency-type conditions holds:_  
_(1) The \(H_{\infty}\)-norm of \((S_{ln}-A_{cc})^{-1}\widetilde{A}_{dc}\) satisfies \(\|(S_{ln}-A_{cc})^{-1}\widetilde{A}_{dc}\|_{\infty}<1\),_  
_(2) \(\|\widetilde{A}_{dc}\|_{2}<1/\sup\limits_{\omega\in\mathbf{R}_{0}}\|(i\omega I_{ n}-A_{cc})^{-1}\|_{2}\)._  
_Other alternative sufficiency-type conditions to Conditions 1 and 2 for the stability of \(A_{dc}\) are:_  
_(3) \(\rho\big{(}A_{cc}^{-1}\widetilde{A}_{dc}\big{)}<1\),_  
_(4) \(\|A_{cc}^{-1}\widetilde{A}_{dc}\|_{2}<1\),_  
_(5) \(\|\widetilde{A}_{dc}\|_{2}<1/\|A_{cc}^{-1}\|_{2}\), that is, \(\lambda_{\max}\big{(}\widetilde{A}_{dc}^{\top}\widetilde{A}_{dc}\big{)}<\lambda _{\min}\big{(}A_{cc}^{\top}A_{cc}\big{)}\),_  
_in the following particular cases:_  
_(a) \(A_{cc}<0\) and \(\widetilde{A}_{dc}>A_{cc}\); and_  
_(b) \(A_{cc}=\big{(}A_{ccij}\big{)}\in M_{E}^{n\times n}\) and \(\widetilde{A}_{dc}=\big{(}\widetilde{A}_{dcij}\big{)}\) fulfills \(\widetilde{A}_{dcij}\leq A_{ccij}\); \(\forall i,j(\
eq i)\in\overline{n}\)._  
_(ii) Assume that Property (i) holds and that the number of inputs and outputs are identical, i.e., \(p=m\), and decompose both the controller gain matrices as sums of their diagonal and off-diagonal parts leading to \(K_{c}=K_{cd}+K_{cd}\) and \(K_{d}=K_{dd}+K_{dd}\), thus \(\overline{K}=K_{c}-K_{d}=(K_{cd}-K_{dd})+(K_{cd}-K_{dd})\). Then, the system is stabilizable under partially decentralized control linear output-feedback control in the sense that Equations (4) and (5), is asymptotically stable under a control law (Equation (6)), if \(K_{d}\in\mathbf{R}^{m\times p}\) is such that, if there_is at least one non-diagonal zero entry in at least one of its rows in the off-diagonal controller error matrix \(\widetilde{K}_{\text{\emph{od}}}=K_{\text{\emph{cod}}}-K_{\text{\emph{dd}}}\). If \(\widetilde{K}_{\text{\emph{od}}}=0\), then the system is stabilizable under decentralized control._  
Proof. : Property (i) is a direct translation of the results of Theorem A1 in Appendix A to the closed-loop system matrices. Property (ii) holds if Property (i) holds with an off-diagonal controller error matrix between the centralized a decentralized controller gain that has at least one non-diagonal zero at some row (or its identically zero) so that a feedback from some crossed output to some of the inputs is not provided to the control law for closed-loop stabilization the stabilization. The following result follows for the time-varying case from Theorem 1 and Theorem A2:  
**Corollary 1**. : _Assume that \(A_{\text{\emph{dc}}}(t)\) and then \(\widetilde{A}_{\text{\emph{dc}}}(t)\) are everywhere piecewise-continuous time-varying. Then, Theorem 1 still holds if Condition 1 is replaced with \(\frac{k_{\text{\emph{dc}}}}{p_{\text{\emph{dc}}}}\sup\limits_{0\leq\tau<t}\| \widetilde{A}_{\text{\emph{dc}}}(\tau)\|<1\); \(\forall t\in\mathbf{R}_{0+}\) with \(k_{\text{\emph{cc}}}\geq 1\) and \(\rho_{\text{\emph{cc}}}>0\) being real constants such that \(e^{-A_{\text{\emph{cd}}}}\leq k_{\text{\emph{cc}}}e^{-\rho_{\text{\emph{dc}}}t}\); \(\forall t\in\mathbf{R}_{0+}\)._  
**Remark 1**. : _Theorem 1 (ii) has been stated for the case \(m=p\). Note that the case \(m>p\) (i.e., there are more inputs than outputs) is irrelevant for the stabilization from the strict algebraic point of view since the \((m-p)\) extra inputs would be redundant.