Publications

2007

• Singular arcs in the generalized Goddard's Problem
  
  • Bonnans J. Frederic
  • Martinon Pierre
  • Trélat Emmanuel
  , 2007, pp.25. We investigate variants of Goddard's problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this report, performing an analysis based on the Pontryagin Maximum Principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle with the problem of nonsmoothness of the optimal control.
• Asymptotic expansion of the optimal control under logarithmic penalty: worked example and open problems
  
  • Alvarez Felipe
  • Bonnans J. Frederic
  • Laurent-Varin Julien
  , 2007. We discuss the problem of expansion of optimal control, state and costate when a logarithmic penalty is applied to constraints. We show that, in a simple case, that the variation of (a regular) junction point, and of the optimal control, state and costate is of order $\eps\log \eps$, where $\eps$ is the penalty parameter.
• Using switching detection and variational equations for the shooting method
  
  • Martinon Pierre
  • Gergaud Joseph
  Optimal Control Applications and Methods, Wiley, 2007, 28 (2), pp.95-116. We study in this paper the resolution by single shooting of an optimal control problem with a bang-bang control involving a large number of commutations. We focus on the handling of these commutations regarding the precise computation of the shooting function and its Jacobian. We first observe the impact of a switching detection algorithm on the shooting method results. Then, we study the computation of the Jacobian of the shooting function, by comparing classical finite differences to a formulation using the variational equations. We consider as an application a low thrust orbital transfer with payload maximization. This kind of problem presents a discontinuous optimal control, and involves up to 1800 commutations for the lowest thrust. Copyright c 2000 John Wiley & Sons, Ltd.
• Global weak solutions to a ferrofluid flow model
  
  • Amirat Youcef
  • Hamdache Kamel
  Mathematical Methods in the Applied Sciences, Wiley, 2007, 31 (2), pp.123-151.
• The Korteweg-de Vries equation with multiplicative homogeneous noise
  
  • de Bouard Anne
  • Debussche Arnaud
  , 2007, pp.113-133.