Hermitian Polynomial Matrix Equations and Applications

In this chapter, we consider Hermitian polynomial matrix equations $$ {X}^s\pm {\sum}_{i=1}^{\ell }{\delta}_i{A}_i^{\ast }{X}^{t_i}{A}_i=Q $$ , which frequently arise from linear-quadratic control problems. The latest results of several special cases are reviewed and further developed to the general situation. Based on the spectral analysis of coefficient matrices, sufficient conditions are presented to guarantee the existence and uniqueness of Hermitian positive definite solutions. A general algorithm is designed to compute the maximal or maximal-like solutions. Their feasibility and efficiency are indicated by numerical experiments.