A Reduced Order Iterative Linear Quadratic Regulator (ILQR) Technique for the Optimal Control of Nonlinear Partial Differential Equations

In this paper, we introduce a reduced order Iterative Linear Quadratic Regulator (RO-ILQR) approach for the optimal control of nonlinear Partial Differential Equations (PDE). The approach proposes a novel modification of the ILQR technique: it uses the Method of Snapshots to identify a reduced order Linear Time Varying (LTV) approximation of the nonlinear PDE dynamics around a current estimate of the optimal trajectory, utilizes the identified LTV model to solve a time varying reduced order LQR problem to obtain an improved estimate of the optimal trajectory along with a new reduced basis, and iterates till convergence. The proposed approach is tested on the viscous Burger's equation and two phase field models for microstructure evolution in materials, and the results show that there is a significant reduction in the computational burden over the standard ILQR approach, without sacrificing performance.