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A solution of the continuous Lyapunov equation by means of power series. (English) Zbl 0631.65073
The author considers the discrete and, especially, the continuous Lyapunov equation \(DX(t)=A^ TX(t)+X(t)A+B.\) The main aim of the paper is to give a stable and efficient computational scheme to solve that equation. Defining the operator \({\mathcal A}: {\mathcal A}B=A^ TB+BA\) (B symmetric) a well-known representation of X is \(X(t)=F(t)X(0)+G(t)\) with \(F(t)=\exp ({\mathcal A}t)\) and \(G(t)=\int^{t}_{0}\exp ({\mathcal A}t)B d\tau.\) The method is by Taylor expansion of F(\(\tau)\), G(\(\tau)\) (for some small \(\tau\) determined by considering \(\| {\mathcal A}\|)\) and applying the formulae \(F(2t)=F^ 2(t)\) and \(G(2t)=G(t)+F(t)G(t).\)
Reviewer: R.Hettich

MSC:
65K10 Numerical optimization and variational techniques
65F30 Other matrix algorithms (MSC2010)
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
15A24 Matrix equations and identities
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References:
[1] F. R. Gantmacher: The Theory of Matrices. vol. 1. Chelsea, New York 1966. · Zbl 0927.15002
[2] P. Lancaster: Theory of Matrices. Academic Press, New York 1969. · Zbl 0186.05301
[3] W. Givens: Elementary divisors and some properties of the Lyapunov mapping \(X \rightarrow AX + XA^*\). Argonne National Laboratory, Argonne, Illinois 1961.
[4] P. Hagander: Numerical solution of \(A^T S + SA + Q = 0\). Lund Institute of Technology, Division of Automatic Control, Lund, Sweden 1969.
[5] V. Kučera: The matrix equation AX + XB = C. SIAM J. Appl. Math. 26 (1974), 1, 15-25.
[6] M. C. Pease: Methods of Matrix Algebra. Academic Press, New York 1965. · Zbl 0145.03701
[7] P. Lancaster: Explicit solution of the matrix equations. SIAM Rev. 12 (1970), 544-566. · Zbl 0209.06502 · doi:10.1137/1012104
[8] J. Štěcha A. Kozáčiková, J. Kozáčik: Algorithm for solution of equations \(PA + A^T P = -Q\) and \(M^T PM - P= -Q\) resulting in Lyapunov stability analysis of linear systems. Kybernetika 9 (1973), 1, 62-71.
[9] S. Barnett: Remarks on solution of AX + XB = C. Electron. Lett. 7 (1971), p. 385.
[10] C. S. Lu: Solution of the matrix equation AX + XB = C. Electron. Lett. 7 (1971), 185-186.
[11] C. S. Berger: A numerical solution of the matrix equation \(P= \Phi P \Phi^T + S\). IEEE Trans. Automat. Control AC-16 (1971), 4, 381-382.
[12] A. Jameson: Solution of the equation AX + XB = C by inversion of an M x M or N X N matrix. SIAM J. Appl. Math. 16 (1968), 1020-1023. · Zbl 0169.35202 · doi:10.1137/0116083
[13] M. Záruba: The Stationary Solution of the Riccati Equation. (in Czech). ÚTIA ČSAV Research Report 371, Prague 1973.
[14] E. C. Ma: A finite series solution of the matrix equation AX - XB = C. SIAM J. Appl. Math. 74 (1966), 490-495. · Zbl 0144.27003 · doi:10.1137/0114043
[15] E. J. Davison, F. T. Man: The numerical solution of \(A'Q + QA = - C\). IEEE Trans. Automat. Control AC-13 (1968), 4, 448-449.
[16] A. Trampus: A canonical basis for the matrix transormation \(X \rightarrow AX+ XB\). J. Math. Anal. Appl. 14 (1966), 242-252. · Zbl 0145.25205 · doi:10.1016/0022-247X(66)90024-2
[17] J. Ježek: UTIAPACK - Subroutine Package for Problems of Control Theory. The User’s Manual. ÚTIA ČSAV, Prague 1984.
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