Convergence of Nonconvergent IRK Discretizations of Optimal Control Problems with State Inequality Constraints

It has been observed that optimization codes are sometimes able to solve state constrained optimal control problems with discretizations which do not converge when used as an integrator on the constrained dynamics. Understanding this phenomenon could lead to a more robust design for direct transcription codes as well as better test problems. This paper examines how this phenomena can occur. The difference between solving index three Differential Algebraic Equations (DAEs) using the trapezoid method in the context of direct transcription for optimal control problems and a straight forward Implicit Runge-Kutta formulation of the same trapezoidal discretization is analyzed. It is shown through numerical experience and theory that the two can differ as much as O(1/h) in the control. Moreover, a small sacrifice in the accuracy of the states in the early stages of the trapezoidal method allows better accuracy in the control, where more precise solutions converge to an incorrect solution. The theoretical results are used to explain computational observations.