# Sparse grid approximation of the Riccati operator for closed loop parabolic control problems with Dirichlet boundary control

Harbrecht, Helmut and Kalmykov, Ilja. (2019) Sparse grid approximation of the Riccati operator for closed loop parabolic control problems with Dirichlet boundary control. Preprints Fachbereich Mathematik, 2019 (09).

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Official URL: https://edoc.unibas.ch/70764/

Abstract. We consider the sparse grid approximation of the Riccati operator P arising from closed loop parabolic control problems. In particular, we concentrate on the linear quadratic regulator (LQR) problems, i.e. we are looking for an optimal control $u_{opt}$ in the linear state feedback form $u_{opt}(t,\cdot)=Px(t,\cdot)$, where $x(t,\cdot)$ is the solution of the controlled partial differential equation (PDE) for a time point t. Under sufficient regularity assump-tions, the Riccati operator P fulfills the algebraic Riccati equation (ARE)
$AP + PA - PBB^\star P + Q = 0,$