On One Method of Solution of the Matrix Riccati Equation

T. N. Titova


In this paper we find a symmetric solution of the matrix Riccati equation: $$XCX+XB^T+BX+A=0,$$where $A$, $B$, $C$, $X$ are real square matrices of order $n$, $A$ and $C$ are symmetric. This equation appears when one finds the normal form of a quadratic Hamiltonian with the help of a canonical transformation. We consider the case when zero eigenvalues of the correspondingHamiltonian matrix $$V=\begin{pmatrix} B^T & C \\ -A & -B \end{pmatrix}$$form blocks of even order in the matrix of Jordan normal form (with the exception, perhaps, of one pair,forming two blocks of the first order). Matrix Riccati equation arises in optimal control problems.Solution method relies on finding appropriate eigenvectors and generalized eigenvectors of the Hamiltonianmatrix $V$. We prove that the vectors satisfy the symplectic conditions. We obtain sufficient conditionsfor existence of a solution.



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