  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 interval. 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. 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.
