An a posteriori error estimate for Symplectic Euler approximation of optimal control problems

TL;DR

This paper develops an a posteriori error estimate for the Symplectic Euler method applied to optimal control problems, enabling adaptive time stepping based on computable error densities.

Contribution

It introduces a new error representation for Symplectic Euler solutions of Hamiltonian systems in optimal control, facilitating adaptive algorithms.

Findings

Error density is computable from the numerical solution.

The remainder term is of higher order under certain convergence assumptions.

Numerical tests demonstrate the effectiveness of the adaptive method.

Abstract

This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns Symplectic Euler solutions of the Hamiltonian system connected with the optimal control problem. The error representation has a leading order term consisting of an error density that is computable from Symplectic Euler solutions. Under an assumption of the pathwise convergence of the approximate dual function as the maximum time step goes to zero, we prove that the remainder is of higher order than the leading error density part in the error representation. With the error representation, it is possible to perform adaptive time stepping. We apply an adaptive algorithm originally developed for ordinary differential equations. The performance is illustrated by numerical tests.