Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

Abstract

Abstract:The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit schemes should be used in this situation. In this paper, we derive efficient numerical methods for solving RDEs capable of exploiting this structure, which are based on a matrix-valued implementation of the BDF and Rosenbrock methods. We show that these methods are suitable for large-scale problems by working only on approximate low-rank factors of the solutions. We also incorporate step size and order control in our numerical algorithms for solving RDEs. In addition, we show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. Numerical experiments show the performance of the proposed methods.

Numerical solution of the finite horizon stochastic linear quadratic control problem

“…The application of the Rosenbrock method of order two proposed in two studies, as a matrix‐valued algorithm, to the SRE yields Denoting and rewriting Equations and similar to Equation , we can write the method as …”

“…The application of the Rosenbrock method of order two proposed in two studies, 24,31 as a matrix-valued algorithm, to the SRE 5 yields…”

“…21 Considering the general setting described in the work of Hafizoglu et al, 21 an approximation scheme for solving the control problem and the associated Riccati equation has been proposed in the work of Levajković et al 22 The numerical solution of the SLQR relies on solving efficiently the associated Riccati equation. For the deterministic LQR problem, most of the methods for solving differential Riccati equations, for example, the previous studies [23][24][25] are based on a low-rank approximation of the solution, and their performance relies on the rapid decay of the singular values. This phenomenon is observed in singular estimate control systems in applications and has been studied in detail by, for example, the previous studies.…”

Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data

“…For matrix-valued Riccati equations, the backward Euler scheme and BDF methods have been studied e.g. in [2,5,6]. Many further aspects related to matrix-valued Riccati equations can be found in [25].…”

A Numerical Comparison of Different Solvers for Large-Scale, Continuous-Time Algebraic Riccati Equations and LQR Problems

“…GCAREs appear in various areas related to control theory, for instance, linear-quadratic optimal regulator problems [70,78], H 2 and H ∞ controller design, nonlinear controller design via state-dependent Riccati equations [37], and balancing-related model reduction [3,18,64]. Solving differential Riccati equations by implicit integration schemes [28,38,44,71] can also lead to GCAREs.…”