  The direct shooting method is a classic approach for the solution of Optimal
Control Problems (OCPs). It parameterizes the control variables and transforms
the OCP to the Nonlinear Programming (NLP) problem to solve. This method is
easy to use and it often introduces less parameters compared with all-variable
parameterization method like the Pseudo-spectral (PS) method. However, it is
long believed that its solution is not guaranteed to satisfy the optimality
conditions of the OCP and the costates are not available in using this method.
In this paper, we show that the direct shooting method may also provide the
costate information, and it is proved that both the state and the costate
solutions converge to the optimal as long as the control variable tends to the
optimal, while the parameterized control may approach the optimal control with
reasonable parameterization. This gives us the credit for the optimal control
computation when employing the direct shooting method.
