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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 431-454, 2020/09/17
We consider a Krylov subspace approximation method for the symmetric differential Riccati equation$\dot{X} = AX + XA^T + Q - XSX$, $X(0)=X_0$. The method we consider is based on projectingthe large-scale equation onto a Krylov subspace spanned by the matrix $A$ and the low-rank factors of $X_0$ and $Q$.We prove that the method is structure preserving in the sense that it preservestwo important properties of the exact flow, namely the positivity of the exact flowand also the property of monotonicity. We provide a theoretical a priori error analysis thatshows superlinear convergence of the method.Moreover, we derive an a posteriori error estimate that is shown to be effective in numerical examples.
Keywords: differential Riccati equations, LQR optimal control problems, large-scale ordinary differential equations, Krylov subspace methods, matrix exponential, exponential integrators, model order reduction, low-rank approximation