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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

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
Full Text: EuDML
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