The role of symplectic integrators in optimal control
M. Chyba, E. Hairer, and G. Vilmart
Abstract. For general optimal control problems, Pontryagins 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.
Key Words. symplectic integrator, backward error analysis, sub-Riemannian
geometry, Martinet, abnormal geodesic, orbital transfer, submerged rigid body.