Convergence rate for a Gauss collocation method applied to constrained optimal control
Provides theoretical convergence guarantees for a widely used numerical method in optimal control, addressing a gap in rigorous analysis for problems with control constraints.
The paper establishes a local convergence rate for a Gauss collocation method applied to constrained optimal control problems, proving convergence when the optimal state and costate have two square integrable derivatives and the Hamiltonian is strongly convex. The theory is validated with a numerical example.
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.