Convergence rates for direct transcription of optimal control problems using Second Derivative Methods

In this paper, Second Derivative Method (SDM) of numerical discretization is applied to optimal control problems. Convergence rates for the error between the discretized solution of SDM and the corresponding analytical solution of optimal control problems are analyzed. Illustrative examples are included to demonstrate the applicability and benefits of SDM. The comparison of the convergence rates of SDM with implicit Runge-Kutta methods (third order, 2-stage RadauIIA and fourth order, 3-stage LobattoIIIA) is also presented. Using SDM, for optimal control problems with non-stiff type of state equations, the fourth order convergence for states and second order convergence for controls is observed, while for certain stiff/oscillatory equations, it results in reduced order of convergence as observed in other approaches. Depending on the choice of optimization algorithms/platforms used, the proposed method is found to be comparable to other approaches and for certain cases, more efficient.