Anil V. Rao

William W. Hager, Jun Liu, Subhashree Mohapatra et al.

A local convergence rate is established for a Gauss orthogonal collocation method applied to optimal control problems with control constraints. If the Hamiltonian possesses a strong convexity property, then the theory yields convergence for problems whose optimal state and costate possess two square integrable derivatives. The convergence theory is based on a stability result for the sup-norm change in the solution of a variational inequality relative to a 2-norm perturbation, and on a Sobolev space bound for the error in interpolation at the Gauss quadrature points and the additional point -1. The tightness of the convergence theory is examined using a numerical example.

Alexander M. Davies, Sara Pollock, Miriam E. Dennis et al.

An adaptive direct collocation method is developed for solving optimal control problems constrained by parabolic partial differential equations. The partial differential equation is first reformulated in a variational setting, where the spatial domain is discretized using the hp-Galerkin finite element method. To address nonlinearities in the variational form, a Kirchhoff-like integral transformation is applied to linearize the dynamics. In the temporal dimension, an orthogonal collocation scheme, the hp-flipped Legendre-Gauss-Radau method, is employed to fully discretize the problem, yielding a large, sparse nonlinear programming problem. Upon solving the nonlinear programming problem, solution accuracy is assessed through an implicit residual estimation procedure. This approach evaluates the local error by solving auxiliary residual problems over selected subdomains, providing a novel means of error estimation within an orthogonal collocation framework for optimal control. Based on the computed error estimate, the mesh is adaptively refined or coarsened to meet a prescribed error tolerance. Mesh refinement is guided by the estimated regularity of the solution which is determined via the decay rate of the coefficients of a Legendre polynomial expansion. In overcollocated regions, a mesh reduction strategy is adapted from orthogonal collocation methods for application within the finite element framework. Numerical examples demonstrate that the proposed method can reduce the error by up to five orders of magnitude in both spatial and temporal dimensions.

William W. Hager, Hongyan Hou, Subhashree Mohapatra et al.

For unconstrained control problems, 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.

William W. Hager, Hongyan Hou, Anil V. Rao

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 $O(\sqrt{N})$, 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.

William W. Hager, Hongyan Hou, Anil V. Rao

A local convergence rate is established for an orthogonal collocation method based on Radau 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. An earlier paper analyzes an orthogonal collocation method based on Gauss quadrature, where neither end point of the problem domain is a collocation point. For the Radau quadrature scheme, one end point is a collocation point.