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, .) = Px(t, .), where x(t, .) 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*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.

Automated summary

The paper investigates the sparse grid approximation of the Riccati operator P in closed loop parabolic control problems, focusing on linear quadratic regulator (LQR) problems. By expressing P in terms of an integral kernel p, the weak form of the algebraic Riccati equation leads to a nonlinear partial integro-differential equation for the kernel p.