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__Numerical Solution of Large-Scale Differential Riccati Equations__

H. Mena

**Abstract**

The numerical treatment of linear quadratic regulator/gaussian design
problems for parabolic partial differential equations requires solving
large-scale Riccati equations. In the finite time horizon case, the differential Riccati equation (DRE) arises. Typically, the coefficient matrices of the resulting DRE have a given structure [1] (e.g. sparse, symmetric or low rank). Methods based on a low-rank approximation of the solution and a
matrix-valued implementation of the usual ODE methods exploit efficiently this structure [2]. Here, we discuss several variants of the available methods, which allow to have a fast computation. In particular, we discussed
the Rosenbrock type methods, BDF methods and different ways for solving the resulting algebraic Riccati equation. The performance of each of these methods is tested in numerical experiments.

**Bibliography**

[1]
P. Benner, J.R. Li and T. Penzl,
Numerical Solution of Large Lyapunov equations, Riccati Equations, and Linear-Quadratic Control Problems, Numerical Linear Algebra with Applications, Vol. 15, No. 9 (2008), pp. 755-777.

[2]
P. Benner and H. Mena,
Numerical solution of the Infinite-Dimensional LQR-Problem and the associated Differential Riccati Equations, MPI Magdeburg Preprint, MPIMD/12-13 (2012).