## On the generalized Riccati matrix differential equation. Exact, approximate solutions and error estimate.(English)Zbl 0695.65050

This paper treats the generalized matrix differential equations of $(1)\quad dX/dt=A+BX(t)-X(t)C-X(t)DX(t),\quad X(0)=P_ 0$ and $(2)\quad dX/dt=A+BX(t)-X(t)C-X(t)DX(t);EX(b)-X(0)F=G;\quad 0\leq t\leq b$ where A, B, C, D, E, F, G, $$P_ 0$$ are given square matrices in $$C^{n\times n}$$. Unlike the known solutions given in terms of the entries of matrix function $$S(t)=\exp (\left[ \begin{matrix} A\\ C\end{matrix} \begin{matrix} B\\ D\end{matrix} \right]t)=(S_{ij}(t))$$, without explicit knowledge of the entries $$S_{ij}(t)$$, it is shown here that for all t in some neighborhood of $$t=0$$, the solutions of problem (1) and (2) can be explicitly expressed in terms of the data A,B,C,..., and also the solution of the generalized algebraic Riccati equation $(3)\quad A+BX- XC-XDX=0.$ Approximate solutions of (1) and (2) are derived and error estimates of them are shown to be proportional to that of the algebraic problem (3).
Reviewer: M.Kono

### MSC:

 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65K10 Numerical optimization and variational techniques 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities 93C15 Control/observation systems governed by ordinary differential equations

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### References:

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