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We explore order reduction techniques for solving the algebraic Riccatiequation (ARE), and investigating the numerical solution of thelinear-quadratic regulator problem (LQR). A classical approach is to build asurrogate low dimensional model of the dynamical system, for instance by meansof balanced truncation, and then solve the corresponding ARE. Alternatively,iterative methods can be used to directly solve the ARE and use its approximatesolution to estimate quantities associated with the LQR. We propose a class ofPetrov-Galerkin strategies that simultaneously reduce the dynamical systemwhile approximately solving the ARE by projection. This methodologysignificantly generalizes a recently developed Galerkin method by using a pairof projection spaces, as it is often done in model order reduction of dynamicalsystems. Numerical experiments illustrate the advantages of the new class ofmethods over classical approaches when dealing with large matrices.

Order reduction approaches for the algebraic Riccati equation and the LQR problem