## 2 Problem Statement  
Consider the following linear and time-invariant system under linear output-feedback centralized control:  
\[\dot{x}_{c}(t)=A_{c}x_{c}(t)+B_{c}\,u_{c}(t);\,x_{c}(0)=x_{c0} \tag{1}\]  
\[y_{c}(t)=C_{c}x_{c}(t)+D_{c}u_{c}(t) \tag{2}\]  
\[u_{c}(t)=K_{c}y_{c}(t) \tag{3}\]  
where \(x_{c}(t)\in\mathbf{R}^{n}\) is the state vector; \(u_{c}(t)\in\mathbf{R}^{m}\) is the centralized control vector; \(y_{c}(t)\in\mathbf{R}^{p}\) is the output; \(A_{c}\), \(B_{c}\), \(C_{c}\) and \(D_{c}\) are the system, control, output and input-output interconnection matrices, respectively, of orders being compatible with the dimensions of the above vectors; and \(K_{c}\in\mathbf{R}^{m\times p}\) is the control matrix. If the system runs in a decentralized control context, we have:  
\[\dot{x}_{d}(t)=A_{d}x_{d}(t)+B_{d}\,u_{d}(t)\ ;\,x_{d}(0)=x_{d0} \tag{4}\]  
\[y_{d}(t)=C_{d}x_{d}(t)+D_{d}u_{d}(t) \tag{5}\]  
\[u_{d}(t)=K_{d}y_{d}(t) \tag{6}\]  
where \(x_{d}(t)\in\mathbf{R}^{n}\) is the state vector; \(u_{d}(t)\in\mathbf{R}^{m}\) is the centralized control vector; \(y_{d}(t)\in\mathbf{R}^{p}\) is the output; \(A_{d}\), \(B_{d}\), \(C_{d}\) and \(D_{d}\) are the system, control, output and input-output interconnection matrices, respectively, of orders being compatible with the dimensions of the above vectors; and \(K_{d}\in\mathbf{R}^{m\times p}\) is the control matrix. Basically, the differences between centralized and decentralized controls are as follows:  
1. In the centralized control, all control components, or more generally, all subsystems if subsystems are considered in the model, have a complete information on the output available for feedback. This means that all control components or block-control inputs are available for controlling each state component (or each individual substate including several state components in the case of a more generic decomposition structure). Basically, the matrix \(K_{c}\) has a complete non-diagonal or block non-diagonal structure. In the decentralized control, the various input components or block-control inputs are not available for controlling each state component. Thus, \(K_{d}\) does not have a complete free design structure of its non-diagonal part and in some cases (completely decentralized disposal) its diagonal or block diagonal. 2. In a more general context, some control or output links can be cut in the decentralized case for the sake of computational simplicity or a more economic control design. In our case, the decentralized input, output and interconnection matrices \(B_{d}\), \(C_{d}\) and \(D_{d}\) can be distinct from the centralized ones and, roughly speaking, to a have a \"more diagonal\" or \"sparser\" structure than their centralized counterparts \(B_{c}\), \(C_{c}\) and \(D_{c}\). If the parameterization of the system (or dynamics) matrix is available to the designer, then some off-diagonal block matrices of \(A_{c}\) could be zeroed or simply re-disposed in a more sparse structure to construct \(A_{d}\). 3. The only strictly necessary condition for the system to be subject to partially (or, respectively, fully) decentralized control is that some (or, respectively, all) of the off-diagonal entries of \(K_{d}\) are forced to be zero even if the system, control, output and interconnection matrices remain identical in Equations (4) and (5) with respect to Equations (1) and (2). **Assumption 1**. : _The system in Equations (1) and (2) is linear output-feedback stabilizable via some centralized control law (Equation (3))._  
Note that Assumption 1 does not hold if the open-loop system in Equations (1) and (2) has unstable or critically stable fixed modes that cannot be removed via linear feedback. **Proposition 1**.