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Convergence rates for direct transcription of optimal control problems using collocation at Radau points. (English) Zbl 1219.49029

Summary: 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 the 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.

MSC:

49M37 Numerical methods based on nonlinear programming
49M25 Discrete approximations in optimal control
34H05 Control problems involving ordinary differential equations
90C30 Nonlinear programming
93A05 Axiomatic systems theory

Software:

Ipopt; COLNEW
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