The Direct Shooting Method is a Complete Method

TL;DR

This paper demonstrates that the direct shooting method for solving optimal control problems can reliably provide both state and costate solutions, ensuring convergence to the true optimal as control parameters improve.

Contribution

It proves that the direct shooting method can yield costate information and guarantees convergence to the optimal solution, challenging previous beliefs.

Findings

The method provides costate information.

Both state and costate solutions converge to the optimal.

Control parameterization approaches the optimal control.

Abstract

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.