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

Bonnans, J. Frédéric; Laurent-Varin, Julien

doi:10.1007/s00211-005-0661-y

Cited by 64 publications

(71 citation statements)

References 11 publications

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“…This is for instance the case for linear quadratic problems. We refer also the reader to [105] for further comments on this result and for other considerations on symplectic integrators. The class of Legendre pseudospectral methods is up to now the other one for which the commutation issues have been proved (see [87,88,89,106] and see [107] for a detailed discussion on the commutation properties).…”

Section: An Open Problemmentioning

confidence: 95%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

189

179

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This article surveys the classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the classical tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 80's and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric re-entry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low cost interplanetary space missions. The article ends with open problems and perspectives.

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Section: An Open Problemmentioning

confidence: 95%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

189

179

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“…Also, order conditions on the coefficients of the adjoint scheme are identified up to order 4. In [9], these are extended up to order 7.…”

Section: The Adjoint Systemsmentioning

confidence: 99%

Discrete mechanics and optimal control: An analysis

Ober‐Blöbaum¹,

Junge²,

Marsden³

2010

ESAIM: COCV

145

120

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Abstract. The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.Mathematics Subject Classification. 49M25, 49N99, 65K10.

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“…Furthermore, a series of papers [2,3,4,5] develops variational integrators for optimal control problems and emphasizes their symplecticity. The work of [6,7] shows that, for partitioned Runge-Kutta discretizations based on symplectic pairs, the direct and indirect approaches are equivalent.…”

Section: Introductionmentioning

confidence: 99%

The role of symplectic integrators in optimal control

Chyba¹,

Hairer²,

Vilmart³

2008

Optim Control Appl Methods

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SUMMARYFor 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.

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

Cited by 64 publications

References 11 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

Discrete mechanics and optimal control: An analysis

The role of symplectic integrators in optimal control

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