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. * PhD student,

Abstract. The numerical resolution of the low thrust orbital transfer problem around the Earth with the maximization of the final mass or minimization of the consumption is investigated. This problem is difficult to solve by shooting method because the optimal control is discontinuous and a homotopic method is proposed to deal with these difficulties for which convergence properties are established. For a thrust of 0.1 Newton and a final time 50% greater than the minimum one, we obtain 1786 switching times.Mathematics Subject Classification. 49M05, 65H20, 70Q05. IntroductionThe minimum time orbit transfer of a satellite around the Earth with low thrust has previously been investigated and solved by Caillau and Noailles [4]. Here, we are interested in the same problem with the final time fixed in three dimensions but with the maximization of the final mass, or the minimization of the consumption. To solve such a problem by shooting methods it is assumed, because the optimal control is discontinuous, that the structure of the optimal control is known [10]. The aim of this study is to construct a method to obtain the result without any a priori information on the optimal control.The basic idea of the method we propose is to define a set of optimal control problems which depend on a parameter λ ∈ [0, 1] which connect the regular optimal control problem with minimization of the energy (square of L 2 norm of the control) for λ = 0 to our problem with minimization of the consumption (L 1 norm of the control) for λ = 1. The Pontryagin Maximum Principle then provides a family of Boundary Value Problems and the shooting function associated to this family of (BV P ) λ gives us a homotopy S(z, λ). Following the path of the homotopy zeros will generate a solution of the problem. In particular, the number of switching times and their localization are found. This paper is organized as follows: the optimal control formulation of the orbital transfer problem is described in Section 1. Sections 2 and 3 are devoted to the necessary condition and to the existence of solution. In Section 4, the differentiability properties of the shooting function are studied and the numerical difficulties for solving this problem using shooting methods are analyzed. Section 5 explains the homotopy method and studies the convergence theorem. In Section 6 the Predictor-Corrector algorithm is presented. Finally, the numerical results are given in Section 7.

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International audienceThis article is devoted to the introduction and study of a photoacoustic tomography model, an imaging technique based on the reconstruction of an internal photoacoustic source distribution from measurements acquired by scanning ultrasound detectors over a surface that encloses the body containing the source under study. In a nutshell, the inverse problem consists in determining absorption and diffusion coefficients in a system coupling a hyperbolic equation (acoustic pressure wave) with a parabolic equation (diffusion of the fluence rate), from boundary measurements of the photoacoustic pressure. Since such kinds of inverse problems are known to be generically ill-posed, we propose here an optimal control approach, introducing a penalized functional with a regularizing term in order to deal with such difficulties. The coefficients we want to recover stand for the control variable. We provide a mathematical analysis of this problem, showing that this approach makes sense. We finally write necessary first order optimality conditions and give preliminary numerical results