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 inequality state constrained optimal control problems with discretizations which do not converge when used as integrators 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 phenomenon can occur. The difference between solving index 3 differential algebraic equations (DAEs) using the trapezoid method in the context of direct transcription for optimal control problems and a straightforward implicit Runge-Kutta (IRK) 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(3)) in the control. The optimization can use a small sacrifice in the accuracy of the states in the early stages of the trapezoidal method to produce better accuracy in the control, whereas more precise solutions converge to an incorrect solution. Convergence independent of the index of the constraints is then proven for one class of problems. The theoretical results are used to explain computational observations.