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Unformatted text preview: PART 3 Nonlinear Systems Problem 3.1 Minimum time control of the Kepler equation JeanBaptiste Caillau, Joseph Gergaud, and Joseph Noailles ENSEEIHTIRIT, UMR CNRS 5505 2 rue Camichel F31071 Toulouse France { caillau, gergaud, jnoaille } @enseeiht.fr 1 DESCRIPTION OF THE PROBLEM We consider the controlled Kepler equation in three dimensions ¨ r = k r  r  3 + γ (1) where r = ( r 1 ,r 2 ,r 3 ) is the position vector–the double dot denoting the second order time derivative–, k a strictly positive constant,  .  the Euclidean norm in R 3 , and where γ = ( γ 1 ,γ 2 ,γ 3 ) is the control. The minimum time problem is then stated as follows: find a positive time T and a measurable function γ defined on [0 ,T ] such that (1) holds almost everywhere on [0 ,T ] and: T → min r (0) = r , ˙ r (0) = ˙ r (2) h ( r ( T ) , ˙ r ( T )) = 0 (3)  γ  ≤ Γ . (4) In (2), r and ˙ r are the known initial position and speed with:  ˙ r  2 2 k  r  < in order that the uncontrolled initial motion be periodic [1]. In (3) h is a fixed submersion of R 6 onto R l , l ≤ 6, defining a nontrivial endpoint condition. The constraint (4) on the Euclidean norm of the control, with Γ a strictly positive constant, means that almost everywhere on [0 ,T ] γ 2 1 + γ 2 2 + γ 2 3 ≤ Γ 2 . 90 PROBLEM 3.1 Our first concern is uniqueness (see § 3 about existence): Question 1 . Is the optimal control unique? The second point is about regularity, namely: Question 2 . Are there continuous optimal controls? Denoting by T (Γ) the value function that assigns to any strictly positive Γ (parameter involved in (4)) the associated minimum time, our third and last question is: Question 3 . Does the product T (Γ) · Γ have a limit when Γ tends to zero? 2 MOTIVATION This problem originates in the computation of optimal orbit transfers in celestial mechanics for satellites with very low thrust engines [5]. Since the 1990s, low electroionic propulsion is been considered as an alternative to strong chemical propulsion, but the lower the thrust, the longer the transfer time, hence the idea of minimizing the final time. In this context, γ is the ratio u/m of the engine thrust by the mass of the satellite, and one has moreover to take into account the mass variation due to fuel consumption: ˙ m = β  u  . Typical boundary conditions in this case consist in inserting the spacecraft on a high geostationnary orbit, and the terminal condition is defined by:  r ( T )  and  ˙ r ( T )  fixed , r ( T ) · ˙ r ( T ) = 0 , r ( T ) × ˙ r ( T ) × ~ k = 0 where ~ k is the normal vector to the equatorial plane. In contrast with the impulsional manoeuvres performed using the strong classic chemical propulsion, the gradual control by a low thrust engine is sometimes referred to as “continuous?” Thus, question 2 could be rephrased according to: Are “continuous” optimal controls continuous?...
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 Spring '11
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