Improved Numerical Methods for Solving Optimal Control Problems with Nonsmooth and Singular Solutions

This project advances computational methods for solving nonsmooth and singular optimal control problems, in which a control input is selected to maximize or minimize some objective function. Non-smooth problems arise when the optimizing control includes sudden jumps. Singular problems arise when the control input does not directly influence the objective function.

Solutions to optimal control problems are found using numerical approximations that are constructed on a pattern of grid points. This project improves upon existing approximation methods for nonsmooth problems by constructing grid patterns that better capture the jump points, resulting in higher accuracy using less computing power. This innovation also helps better demarcate any singular regions of the problem.

The project further improves the control solution for problems with singular regions by using corrective modifications to the objective function only in those regions. A wide range of problems of national importance may be formulated as nonsmooth or singular optimal control problems. These include control of high-speed vehicles, treatment of diseases, and optimization of manufacturing processes.

The results of this project will confer benefits to the national health and prosperity by offering faster and more accurate solutions to these problems. Graduate students from diverse backgrounds will play a central role in the research including one mathematician and one engineer. The methods that are developed will be implemented in high quality software that will be made widely available.

This research focuses on the development of new collocation methods, called hp methods, and the use of these collocation methods in solving optimal control problems with nonsmooth solutions. The approach is purely computational and does not require any a priori knowledge of the structure of the optimal solution. In addition, the methodology is aimed at solving challenging problems that arise when an optimal control is singular.

The hp collocation methods are developed in a manner that enables accurate identification of the points where the optimal solution is nonsmooth. The methods developed in this research can be employed extremely efficiently using sparse nonlinear optimization techniques and will provide much higher accuracy solutions without a priori knowledge of the solution structure.

With a suitable placement of the variable mesh points, the hp collocation methods converge exponentially fast even though the solution may be nonsmoooth. The approach developed in this research could lead to a very rapid mesh refinement process where a small mesh will be maintained and few mesh refinement iterations would be required to obtain a high-accuracy approximation of the optimal solution.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.