Taming Heterogeneity and Complexity of Embedded Control

Smooth Approximations of Single-Input Optimal Orbital Transfer using Continuation and Averaging Techniques

 

Bernard BONNARD¹ — Jean-Baptiste CAILLAU² — Romain DUJOL²

¹ Institut de Mathématiques de Bourgogne F-21078 Dijon, FRANCE

² ENSEEIHT-IRIT F-31071 Toulouse, FRANCE

ABSTRACT. This article deals with the optimal transfer of a satellite between Keplerian orbits when the control is oriented along the tangential direction. We consider the time-optimal control problem and the energy minimization problem. The optimal controls laws have discontinuities and continuations and averaging techniques are applied to smooth the discontinuities. The smooth approximations of the solutions are computed using a shooting method taking into account second-order optimality conditions.

KEYWORDS: Optimal control, shooting method, second-order optimality condition, single-input orbit transfer, continuation method, averaging

1. Introduction

The orbital transfer is described by the controlled KEPLER equation

(1)

and the mass variation equation

(2)

where the gravitation constant μ is normalized to 1, q ∈ R³ is the position of the satellite in a fixed frame whose origin is the Earth center, m is the mass, β > 0 and u ∈ R³ represents the thrust, satisfying |u| ≤ε , and the maximal thrust is small for ...