Joseph Gergaud

Low-Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach

We describe in this paper the study of an earth orbital transfer with a low thrust (typically electro-ionic) propulsion system. The objective is the maximization of the final mass, which leads to a discontinuous control with a huge number of thrust arcs. The resolution method is based on single shooting, combined to a homotopic approach in order to cope with the problem of the initial guess, which is actually critical for non-trivial problems. An important aspect of this choice is that we make no assumptions on the control structure, and in particular do not set the number of thrust arcs. This strategy allowed us to solve our problem (a transfer from Low Earth Orbit to Geosynchronous Equatorial Orbit, for a spacecraft with mass of 1500 kgs, either with or without a rendezvous) for thrusts as low as 0.1N, which corresponds to a one-year transfer involving several hundreds of revolutions and thrust arcs. The numerical results obtained also revealed strong regularity in the optimal control structure, as well as some practically interesting empiric laws concerning the dependency of the final mass with respect to the transfer time and maximal thrust.

Differential continuation for regular optimal control problems

Regular control problems in the sense of the Legendre condition are defined, and second-order necessary and sufficient optimality conditions in this class are reviewed. Adding a scalar homotopy parameter, differential pathfollowing is introduced. The previous sufficient conditions ensure the definiteness and regularity of the path. The role of AD for the implementation of this approach is discussed, and two examples excerpted from quantum and space mechanics are detailed.

On Some Riemannian Aspects of Two and Three-Body Controlled Problems*

The flow of the Kepler problem (motion of two mutually attracting bodies) is known to be geodesic after the work of Moser [21], extended by Belbruno and Osipov [2, 22]: Trajectories are reparameterizations of minimum length curves for some Riemannian metric. This is not true anymore in the case of the three-body problem, and there are topological obstructions as observed by McCord et al. [20]. The controlled formulations of these two problems are considered so as to model the motion of a spacecraft within the influence of one or two planets. The averaged flow of the (energy minimum) controlled Kepler problem with two controls is shown to remain geodesic. The same holds true in the case of only one control provided one allows singularities in the metric. Some numerical insight into the control of the circular restricted three-body problem is also given.

Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft: Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft

In this article, the minimum time and fuel consumption of an aircraft in its climbing phase is studied. The controls are the thrust and the lift coefficient and state constraints are taken into account: air slope and speed limitations. The application of the Maximum Principle leads to parameterize the optimal control and the multipliers associated to the state constraints with the state and the costate and leads to describe a Multi- Point Boundary Value Problem which is solved by multiple shooting. This indirect method is the numerical implementation of the Maximum Principle with state-constraints and it is initialized by the direct method, both to determine the optimal structure and to obtain a satisfying initial guess. The solutions of the boundary value problems we define give extremals which satisfy necessary conditions of optimality with at most two boundary arcs. Note that the aircraft dynamics has a singular perturbation but no reduction is performed.