A local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as the number of collocation points increases, the discrete solution convergences exponentially fast in the sup-norm to the continuous solution. This is the first convergence rate result for an orthogonal collocation method based on global polynomials applied to an optimal control problem.

For control problems with control constraints, a local convergence rate is established for an hp-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the hp-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.

Abstract. Estimates are obtained for the Lebesgue constants associated with the Gauss quadrature points on (−1, +1) augmented by the point −1 and with the Radau quadrature points on either (−1, +1] or [−1, +1). It is shown that the Lebesgue constants are O( √ N ), where N is the number of quadrature points. These point sets arise in the estimation of the residual associated with recently developed orthogonal collocation schemes for optimal control problems. For problems with smooth solutions, the estimates for the Lebesgue constants can imply an exponential decay of the residual in the collocated problem as a function of the number of quadrature points.Key words. Lebesgue constants, Gauss quadrature, Radau quadrature, collocation methods Recently,in [3,4,8,9,10,11,20], a class of methods was developed for solving optimal control problems using collocation at either Gauss or Radau quadrature points. In [14] and [15] an exponential convergence rate is established for these schemes. The analysis is based on a bound for the inverse of a linearized operator associated with the discretized problem, and an estimate for the residual one gets when substituting the solution to the continuous problem into the discretized problem. This paper focuses on the estimation of the residual. We show that the residual in the sup-norm is bounded by the sup-norm distance between the derivative of the solution to the continuous problem and the derivative of the interpolant of the solution. By Markov's inequality [18], this distance can be bounded in terms of the Lebesgue constant for the point set and the error in best polynomial approximation. A classic result of Jackson [17] gives an estimate for the error in best approximation. The Lebesgue constant that we need to analyze corresponds to the roots of a Jacobi polynomial on (−1, +1) augmented by either τ = +1 or τ = −1. The effects of the added endpoints were analyzed by Vértesi in [24]. For either the Gauss quadrature points on (−1, +1) augmented by τ = +1 or the Radau quadrature points on (−1, +1] or on [−1, +1), the bound given in [24, Thm. 2.1] for the Lebesgue constants is O(log(N ) √ N ), where N is the number of quadrature points. We sharpen this bound to O( √ N ). To motivate the relevance of the Lebesgue constant to collocation methods, let us consider the scalar first-order differential equatioṅ Introduction.

A convergence theory is presented for approximations of continuous-time optimal control problems based on a Gauss pseudospectral discretization. Under assumptions of coercivity and smoothness, the Gauss pseudospectral method has a local minimizer that converges exponentially fast in the sup-norm to a local minimizer of the continuous-time optimal control problem. The convergence theorem is presented and an example is given that illustrates the exponential convergence.