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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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