On One Method of Solution of the Matrix Riccati Equation

T. N. Titova

Abstract


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 corresponding

Hamiltonian 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 Hamiltonian

matrix $V$. We prove that the vectors satisfy the symplectic conditions. We obtain sufficient conditions for existence of a solution.


Keywords


Hamiltonian matrix, Jordan normal form, Hamiltonian, eigenvalue, eigenvector

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