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

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 $u_{opt}(t,cdot) = P x(t,cdot)$, 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 + P A - P B B^{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 integro-differential equation (IDE) 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.