Premium
The role of symplectic integrators in optimal control
Author(s) -
Chyba Monique,
Hairer Ernst,
Vilmart Gilles
Publication year - 2008
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.855
Subject(s) - symplectic geometry , symplectic integrator , variational integrator , hamiltonian (control theory) , mathematics , hamiltonian system , optimal control , hamiltonian mechanics , symplectomorphism , integrator , symplectic manifold , boundary value problem , mathematical analysis , mathematical optimization , physics , phase space , quantum mechanics , voltage
For general optimal control problems, Pontryagin's maximum principle gives necessary optimality conditions, which are in the form of a Hamiltonian differential equation. For its numerical integration, symplectic methods are a natural choice. This article investigates to which extent the excellent performance of symplectic integrators for long‐time integrations in astronomy and molecular dynamics carries over to problems in optimal control. Numerical experiments supported by a backward error analysis show that for problems in low dimension close to a critical value of the Hamiltonian, symplectic integrators have a clear advantage. This is illustrated using the Martinet case in sub‐Riemannian geometry. For problems like the orbital transfer of a spacecraft or the control of a submerged rigid body, such an advantage cannot be observed. The Hamiltonian system is a boundary value problem and the time interval is in general not large enough so that symplectic integrators could benefit from their structure preservation of the flow. Copyright © 2008 John Wiley & Sons, Ltd.