On state constraint representations and mesh-dependent gradient projection convergence rates for optimal control problems

Abstract

This note briefly summarizes the main results in a SIAM article (Dunn, 2000) with the same title. In the SIAM paper, two distinct nonlinear programming formulations are investigated for ODE optimal control problems with pointwise state and control constraints. The first formulation treats the differential equations of state as an equality constraint in the conventional manner. The second formulation employs a different equality constraint entailing the integrated state transition map. Related convergence rate estimates are developed for augmented gradient projection methods and discrete-time approximations to a large representative class of ODE control problems. In the first formulation, the rate estimates are mesh-dependent, and the predicted number of inner loop gradient projection iterations needed to achieve a fixed small deviation from the optimal value of the augmented Lagrangian is inversely proportional to the square of the mesh width. In the second formulation, the convergence rates and predicted iteration counts are mesh-invariant. The computational costs-per-iteration in the two formulations are comparable.