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

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.