We present convergence rates for the error between the direct transcription solution and the true solution of an unconstrained optimal control problem. The problem is discretized using collocation at Radau points (aka Gauss-Radau or Legendre-Gauss-Radau quadrature). The precision of Radau quadrature is the highest after Gauss (aka Legendre-Gauss) quadrature, and it has the added advantage that the end point is one of the abscissas where the function, to be integrated, is evaluated. We analyze convergence from a Nonlinear Programming (NLP)/matrix algebra perspective. This enables us to predict the norms of various constituents of a matrix that is "close" to the KKT matrix of the discretized problem. We present the convergence rates for the various components, for a sufficiently small discretization size, as functions of the discretization size and the number of collocation points. We illustrate this using several test examples. This also leads to an adjoint estimation procedure, given the Lagrange multipliers for the large scale NLP.

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It is very common in chemical engineering applications to find optimal control problems whose optimality conditions do not provide information about the control over an interval. This type of problems is called partially singular, as the control switches between nonsingular and singular arcs. When direct transcription is applied, the resulting nonlinear programming problem is ill conditioned. Some mesh refinement and rigorous iterative methods have been developed to determine the control profile and switching points. This work presents a practical alternative that quickly produces accurate state and control profiles without adding nonconvex terms. The problem is first solved with a large number of equally spaced finite elements. Then, unnecessary elements are removed while keeping the solution structure. Finally, direct and indirect approaches are combined to apply a regularization scheme only to the singular part. Seven examples were solved to test our strategy. Results provide good approximations to the analytical switching points.