Convergence Rate for a Radau hp Collocation Method Applied to Constrained Optimal Control
Provides theoretical convergence guarantees for hp collocation methods in optimal control, addressing a gap for constrained problems where global polynomial methods lack guarantees.
The paper establishes a local convergence rate for an hp collocation method at Radau points applied to constrained optimal control problems, showing exponential convergence with polynomial degree and polynomial convergence with mesh refinement.
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.