# 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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## Abstract

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)
$AP + PA - PBB^\star P + Q = 0,$
where A, B, and Q are linear operators associated to the LQR problem. By expressing P in terms of an integral kernel p, the weak form of the ARE leads to a non-linear partial integro-differential equation for the kernel p – the Riccati-IDE. We represent the kernel function as an element of a sparse grid space, which considerably reduces the cost to solve the Riccati IDE. Numerical results are given to validate the approach.
Faculties and Departments: 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)12 Special Collections > Preprints Fachbereich Mathematik Harbrecht, Helmut and Kalmykov, Ilja Preprint Universität Basel English 31 May 2019 10:07 31 May 2019 10:07

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