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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. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh inter- val. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The 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. Fur- thermore, 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. 

A new method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method developed in this paper employs a modified form of the Legendre-Gauss-Radau (LGR) orthogonal direct collocation method in which an additional variable and two additional constraints are included at the end of a mesh interval. The additional variable is the switch time where a discontinuity occurs. The two additional constraints are a collocation condition on each differential equation that is a function of control along with a control constraint at the endpoint of the mesh interval that defines the location of the nonsmoothness. These additional constraints modify the search space of the NLP in a manner such that an accurate approximation to the location of the nonsmoothness is obtained. An example with a nonsmooth solution is used throughout the paper to illustrate the improvement of the method over the standard Legendre-Gauss-Radau collocation method. 

A modified Legendre-Gauss-Radau collocation method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method includes an additional variable that defines the location of nonsmoothness. In addition, collocation constraints are added at the end of a mesh interval that defines the location of nonsmoothness in the solution on each differential equation that is a function of control along with a control constraint at the endpoint of this same mesh interval. The transformed adjoint system for the modified Legendre-Gauss-Radau collocation method along with a relationship between the Lagrange multipliers of the nonlinear programming problem and a discrete approximation of the costate of the optimal control problem is then derived. Finally, it is shown via example that the new method provides an accurate approximation of the costate.