## Analysis of Krylov subspace approximation to large-scale differential Riccati equations

Antti Koskela and Hermann Mena

### Abstract

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 projecting the 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 preserves two important properties of the exact flow, namely the positivity of the exact flow and also the property of monotonicity. We provide a theoretical a priori error analysis that shows superlinear convergence of the method. Moreover, we derive an a posteriori error estimate that is shown to be effective in numerical examples.

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### Key words

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

### AMS subject classifications

65F10, 65F60, 65L20, 65M22, 93A15, 93C05

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