Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control

J. Frederic Bonnans (1) , Julien Laurent-Varin

1 SYDOCO - Dynamic systems, optimisation and optimal command

We discuss the derivation of order conditions for the discretization of (unconstrained) optimal control problems, when the scheme for the state equation is of Runge-Kutta type. This problem appears to be essentially the one of checking order conditions for symplectic partitioned Runge-Kutta schemes. We show that the the computations using bi-coloured trees are naturally expressed in this case in terms of oriented free tree. This gives a way to compute them by an appropriate computer program. Our software is able to compute conditions up to order 7 (we display them up to order 6). The results are in accordance with those of Hager (where they were computed for order up to 4) as well as those of Murua where the number of conditions up to order 7 is stated.

Mots clés: OPTIMAL CONTROL; HAMILTONIAN SYSTEMS; PARTITIONED RUNGE-KUTTA METHODS; SYMPLECTIC SCHEMES; ORDER CONDITIONS; P-SERIES; H-TREES; ORIENTED FREE-TREES

Domaines: Autre [cs.OH]; Optimisation et contrôle [math.OC]

J. Frederic Bonnans, Julien Laurent-Varin. Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control. [Research Report] RR-5398, INRIA. 2004, pp.18. ⟨inria-00070605⟩