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

Jódar, Lucas; Navarro, E.

Apl. Mat. 34, No. 6, 429-438 (1989).

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

Keywords:

Cauchy problem; two-point boundary value problem; generalized matrix differential equations; generalized algebraic Riccati equation; error estimates

Software:

RICPAC

References:

[1]	W. F. Arnold A. Laub: Generalized eigenproblem algorithms and software for algebraic Riccati equations. Proc. of the IEEE, Vol. 72, No. 12 (1984), 1746-1754.
[2]	R. Brockett: Finite Dimensional Linear Systems. John Wiley, New York, 1970. · Zbl 0216.27401
[3]	J. Dieudonné: Foundations of Modern Analysis. Academic Press, New York. · Zbl 0122.29702
[4]	N. Dunford J. Schwartz: Linear Operators, Vol. I. Interscience. New York, 1957. · Zbl 0128.34803
[5]	J. Eisenfeld: Operator equations and nonlinear eigenparameter problems. J. Functional Analysis, 12 (1973), 475-490. · Zbl 0255.47018
[6]	G. H. Golub C. F. Van Loan: Matrix Computations. North Oxford Academic, Oxford 1983. · Zbl 0559.65011
[7]	V. Hernández L. Jódar: Sobre la ecuación cuadrática en operadores \(A + BT + TC + TDT=0. Stochastica. Vol. VII, no 2 (1983), 145-154.\)
[8]	V. Hernández L. Jódar: Boundary problems and periodic Riccati equations. IEEE Trans. Autom. Control, Vol. AC-30, No. 11 (1985), 1131-1133. · Zbl 0588.34017
[9]	L. Jódar: Boundary problems for Riccati and Lyapunov equations. Proc. Edinburgh Math. Soc. 29 (1986), 15-21. · Zbl 0569.47040
[10]	L. Jódar: Explicit solutions for Sturm Liouville operator problems. Proc. Edinburg Math. Soc., 30(1987), 301-309. · Zbl 0595.34022
[11]	L. Jódar: Boundary value problems for second order operator differential equations. Linear Algebra and its Appls., 83 (1986), 29-38. · Zbl 0614.34030
[12]	L. Jódar: Boundary value problems and Cauchy problems for the second order Euler operator differential equation. Linear Algebra and its Appls., 91 (1987), 1-14. · Zbl 0624.34014
[13]	H. Kuiper: Generalized Operator Riccati equations. SIAM J. Math. Anal., Vol. 16 (1985), 675-694. · Zbl 0577.34053
[14]	K. Martensson: On the matrix Riccati equation. Inf. Sci. 3 (1971), 17-49. · Zbl 0206.45602
[15]	G. G. L. Meyer H. J. Payne: An iterative method of solution of the algebraic Riccati equation. IEEE Trans. Aut. Control) · Zbl 0261.93003
[16]	J. D. Roberts: Linear Model Reduction and solution of the algebraic Riccati equation by use of the sign function. Int. J. Control, Vol. 32, No. 4 (1980), 677-687. · Zbl 0463.93050