  This paper presents a real-time optimization method for nonlinear model
predictive control (NMPC) of systems governed by partial differential equations
(PDEs). The NMPC problem to be solved is formulated by discretizing the PDE
system in space and time by using the finite difference method. The proposed
method is called the double-layer Jacobi method, which exploits both the
spatial and temporal sparsities of the PDE-constrained NMPC problem. In the
upper layer, the NMPC problem is solved by ignoring the temporal couplings of
either the state or costate (Lagrange multiplier corresponding to the state
equation) equations so that the spatial sparsity is preserved. The lower-layer
Jacobi method is a linear solver dedicated to PDE-constrained NMPC problems by
exploiting the spatial sparsity. Convergence analysis indicates that the
convergence of the proposed method is related to the prediction horizon and
regularization. Results of a numerical experiment of controlling a heat
transfer process show that the proposed method is two orders of magnitude
faster than the conventional structure-exploiting Newton's method.
