Positive definiteness of a class of the block symmetric matrix with parameters and its applications (Q2725272)

The author gives a criterion for the positive definiteness of a symmetric block matrix \(Q=(Q_{ij})\), \(Q_{ij}\in {\mathbb R}^{n_i\times n_j}\), \(\sum_{i=1}^N n_i=n\), with \(Q_{ii}>0\). He defines \(\gamma_{ii}=\inf_{\|z_i\|=1, z_i\in {\mathbb R}^{n_i}} z_i^T Q_{ii}z_i/2\), \(\gamma_{ij}=-\sup_{\|z_i\|=\|z_j\|=1, z_i\in {\mathbb R}^{n_i}, z_j\in {\mathbb R}^{n_j}}|z_i^T Q_{ij}z_j|\) \((i\not=j)\), \(\overline{Q}_{ij}=Q_{ij}\) \((i\not=j)\), \(\overline{Q}_i=\alpha_{i} Q_{ii}\), \(\gamma^*_{ij}=\gamma_{ij}\) \((i\not=j)\), \(\gamma^*_{ii}=\alpha_i Q_{ii}\), \(\overline{Q}=(\overline{Q}_{ij})\), and \(\Gamma^*=(\gamma^*_{ij})\), where \(\alpha_i\), \(i=1,\ldots,N\) are adjustable parameters, and then proves that \(\overline{Q}>0\) if \(\Gamma^*>0\). Using the above result, he studies the decentralized stabilization of a time-variant large scale system and obtains that the system can be stabilized by a decentralized state feedback control. A simple algorithm is presented. A numerical example for a decentralized control system is given to illustrate his algorithm.