We consider the Hermitian positive definite solution of the nonlinear matrix equation . Some new sufficient conditions and necessary conditions for the existence of Hermitian positive definite solutions are derived. An iterative method is proposed to compute the Hermitian positive definite solution. In the end, an example is used to illustrate the correctness and application of our results. 1. Introduction In this paper we consider the Hermitian positive definite solution of the nonlinear matrix equation where are complex matrices; and are Hermitian positive definite matrices. Here, denotes the conjugate transpose of the matrix . The nonlinear matrix equation (1) plays an important role in linear optimal and robust control. For instance, the solvability of the discrete-time linear quadratic optimal control problem with , depends on the solvability of (1) in some special cases [1–4]. Due to the important applications in system and control theory, in the past decades, (1) with has been extensively studied, and the research results mainly concentrated on the following: (a)sufficient conditions and necessary conditions for the existence of an Hermitian solution [5–9];(b)numerical methods for computing the Hermitian solution [4, 10–13];(c)properties of the Hermitian solution [14, 15];(d)perturbation analysis for the discrete algebraic Riccati equation [16–18].(e)connection with symplectic matrix pencil [9, 19, 20];(f)connection with stochastic realization and spectral factorization [21–23].Nonetheless, (1) with has not been studied as far as we know. In this paper we study the generalized nonlinear matrix equation (1). Firstly, we transform (1) into an equivalent nonlinear matrix equation. By Sherman-Woodbury-Morrison formula [24, Page 50], we have then Set then That is, Therefore, the nonlinear matrix equation (1) can be equivalently rewritten as (7). So we first investigate the Hermitian positive definite solution of (7) in Section 2 and then derive some new results on the nonlinear matrix equation (1) by using the matrix transformations (5) in Section 3. Finally, we use an example to illustrate the correctness and application of the results of Section 3. Throughout this paper, we write if the matrix is Hermitian positive definite (semidefinite). If is Hermitian positive definite (semidefinite), then we write . If an Hermitian positive definite matrix satisfies , we denote by . We use to denote all eigenvalues (each repeated as many times as its algebraic multiplicity) of an Hermitian matrix . The symbol denotes the spectral norm of the matrix . 2.
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