On the Hermitian Positive Definite Solutions of Nonlinear Matrix Equation

Nonlinear matrix equation 𝑋 𝑠 + 𝐴 ∗ 𝑋 − 𝑡 1 𝐴 + 𝐵 ∗ 𝑋 − 𝑡 2 𝐵 = 𝑄 has many applications in engineering; control theory; dynamic programming; ladder networks; stochastic filtering; statistics and so forth. In this paper, the Hermitian positive definite solutions of nonlinear matrix equation 𝑋 𝑠 + 𝐴 ∗ 𝑋 − 𝑡 1 𝐴 + 𝐵 ∗ 𝑋 − 𝑡 2 𝐵 = 𝑄 are considered, where 𝑄 is a Hermitian positive definite matrix, 𝐴 , 𝐵 are nonsingular complex matrices, 𝑠 is a positive number, and 0 𝑡 𝑖 ≤ 1 , 𝑖 = 1 , 2 . Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and numerical example is given to show the efficiency of the proposed iterative method.