Modified Legendre-Gauss-Radau Collocation Method for Solving Optimal Control Problems with Nonsmooth Solutions

Joseph D. Eide, William W. Hager, Anil V. Rao

Published 2019-09-07Version 1

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre-Gauss-Radau orthogonal direct collocation method. This modified Legendre-Gauss-Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre-Gauss-Radau collocation method. These new variables are the time at the intersection of each mesh interval and the control that at the end of each mesh interval. The two additional constraints are a collocation condition on each differential equation that is a function of control and an inequality constraint on the control at the endpoint of each mesh interval. These additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre-Gauss-Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass-Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.