Constrained LQR problems in elliptic distributed control systems with point observations

In this paper, we study (bound constrained) LQR problems in distributed control systems governed by the elliptic equation with point observations, which are motivated by problems in corrosion engineering and contemporary ''smart materials.'' Several regularity and characterization theorems have been established. In particular, three decomposition formulas are obtained to characterize the optimal control and optimal layer density, and are used to direct the numerical computations. These results cannot be obtained by the traditional Galerkin variational method. In the process, several useful lemmas are established, which are of independent interest. We point out that the classical Lagrangian multiplier method (LMM) may fail to provide a reliable numerical algorithm. Based on our characterization results and the boundary element method, two algorithms are proposed to carry out numerical computations. It has been shown by our numerical experiments that both algorithms are efficient and insensitive to the partition number of the boundary. An adaptive local refinement scheme has also been designed to handle the rough behavior of the optimal solution around sensor locations.