## 3 Main Results  
The first set of technical results which follow are concerned with centralized and decentralized control stabilizability. **Assertion 1**. : _A necessary and sufficient condition for the system to be linear state-stabilizable via some centralized control law is that \(rank[sI_{n}-A_{c}\,B_{c}]=n\) for all Res \(\geq 0\)._  
**Proof. ** Assume that \(rank[sI_{n}-A_{c}\,B_{c}]=n_{1}<n\) for some Res \(\geq 0\). Then, there is some Laplace transform \(\left[\hat{x}^{T}(s)\,,\,\hat{u}^{T}(s)\right]^{T}=Lap\left[x^{T}(t)\,,\,u^{T} (t)\right]^{T}\
eq 0\) such that \([sI_{n}-A_{c}\,B_{c}]\left[\hat{x}^{T}(s)\,,\,\hat{u}^{T}(s)\right]^{T}=0\) for some Res \(\geq 0\) and \([sI_{n}-A_{c}\,B_{c}]\left[ \begin{array}{c}I_{n}\\ K_{c}\end{array} \right]\hat{x}(s)=0\) for any \(K_{c}\in R^{m\times n}\) and some \(\hat{x}(s)\
eq 0\) with Res \(\geq 0\) since \(rank\left(\left[sI_{n}-A_{c}\,B_{c}\right]\left[ \begin{array}{c}I_{n}\\ K_{c}\end{array} \right]\right)\leq\min\left(rank[sI_{n}-A_{c}\,B_{c}]\,,\,rank \left[ \begin{array}{c}I_{n}\\ K_{c}\end{array} \right]\right)\leq\min\left(n_{1}\,,\,n\right)\leq n_{1}<n\) for some Res \(\geq 0\). Therefore, the closed-loop system has some unstable or critically stable solution for any given (centralized) control gain. This proves the necessary part. Sufficiency follows since, if \(rank[sI_{n}-A_{c}\,B_{c}]=n\), then \(\hat{x}(s)\equiv 0\) for all Res \(\geq 0\) and some \(K_{c}\in R^{m\times n}\) which can be found so that \(\min\left(rank[sI_{n}-A_{c}\,B_{c}]\,,\,rank\left[ \begin{array}{c}I_{n}\\ K_{c}\end{array} \right]\right)=\min(n,\,n)=n\). \(\Box\)  
**Assertion 1. ** is a particular adapted ad-hoc test for stabilizability of the celebrated Popov-Belevitch -Hautus rank controllability test [6]. Note that, if there exist unstable or critically stable fixed modes (i.e., those present in the open-loop system that cannot be removed via feedback control), then neither centralized nor decentralized stabilizing control laws can be synthesized. Note that the stabilizability rank test of Assumption 1 can only be evaluated for the critically and unstable eigenvalues of \(A_{c}\) instead for all the open right-hand complex plane. In all the remaining points of such a plane, the test always gives a full rank of the tested matrix. The parallel controllability test should always be applied in the same matrix to any eigenvalues of \(A_{c}\). **Assertion 2. **_A necessary condition for the system to be linear state-stabilizable via some partially or totally decentralized control is that it be stabilizable via centralized control (i.e., Assertion 1 holds)._  
**Proof. ** It is obvious from Assertion 1 since any gain \(K_{d}\) used for centralized or decentralized is sparser than a centralized gain counterpart so that the proof follows from Assertion 1. \(\Box\)  
**Assertion 3. **_A necessary condition for the system to be linear output-stabilizable via some partially or totally decentralized control is that it be stabilizable via centralized control (i.e., that Assertion 1 holds)._  
**Proof. ** It is obvious from Assertions 1 and 2 that, when replacing \(K_{c}\rightarrow(I_{m}-K_{c}D_{c})^{-1}K_{c}C_{c}\) and \(K_{d}\rightarrow(I_{m}-K_{d}D_{d})^{-1}K_{d}C_{d}\) (see Equations (9) and (12)), the second replacement happens under sparser parameterizations. \(\Box\)  
Now, consider the closed-loop system matrices from Equations (7) and (10) for the case \(A=A_{c}=A_{d}\). \[A_{cc}=A+B_{c}(I_{m}-K_{c}D_{c})^{-1}K_{c}C_{c};A_{dc}=A+B_{d}(I_{m}-K_{d}D_{d} )^{-1}K_{d}C_{d}\]  
with its parametrical error being:  
\[\widetilde{A}_{dc}=A_{cc}-A_{dc}=B_{c}(I_{m}-K_{c}D_{c})^{-1}K_{c}C_{c}-B_{d}( I_{m}-K_{d}D_{d})^{-1}K_{d}C_{d}\]  
A first main technical result follows:  
**Theorem 1. **_If Assumption 1 holds, assume also that \(K_{c}\in\mathbf{R}^{m\times p}\) is a centralized linear output-feedback stabilizing controller gain such that the resulting closed-loop system matrix \(A_{cc}\in\mathbf{R}^{n\times n}\) has a stability abscissa \((-\rho_{cc})<0\).