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 assumptions, 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 nonlinear partial integrodifferential equation for the kernel p – the RiccatiIDE. 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.
\[
AP + PA  PBB^\star P + Q = 0,
\]
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 nonlinear partial integrodifferential equation for the kernel p – the RiccatiIDE. 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 

UniBasel Contributors:  Harbrecht, Helmut and Kalmykov, Ilja 
Item Type:  Preprint 
Publisher:  Universität Basel 
Language:  English 
edoc DOI:  
Last Modified:  31 May 2019 10:07 
Deposited On:  31 May 2019 10:07 
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