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

Bibliography:

J. Karlsson, S. Larsson, M. Sandberg, A. Szepessy, R. Tempone, An a posteriori error estimate for symplectic Euler approximation of optimal control problems, SIAM J. Sci. Comput. 37 (2015), A946-A969. [doi:10.1137/140959481].

Authors:

J. Karlsson, S. Larsson, M. Sandberg, A. Szepessy, R. Tempone

Keywords:

Optimal Control, Error Estimates, Adaptivity, Error Control

Year:

2015

Abstract:

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 di erential equations. The performance is illustrated by numerical tests.

ISSN:

1064-8275; 1095-7197; DOI:10.1137/140959481