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Introduction Shooting method Orbit transfer Three-body problem Everything is under control Optimal control and applications to aerospace problems E. Tr´ elat Univ. Paris 6 (Labo. J.-L. Lions) and Institut Universitaire de France Roma, March 2014 E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem What is control theory? Controllability Steer a system from an initial configuration to a final configuration. Optimal control Moreover, minimize a given criterion. Stabilization A trajectory being planned, stabilize it in order to make it robust, insensitive to perturbations. Observability Reconstruct the full state of the system from partial data. E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Application fields are numerous: Control ¡theory ¡and ¡applica0ons ¡ Applica0on ¡domains ¡of ¡control ¡theory: ¡ Mechanics ¡ Biology, ¡medicine ¡ Vehicles ¡(guidance, ¡dampers, ¡ABS, ¡ESP, ¡…), ¡ Aeronau<cs, ¡aerospace ¡(shu=le, ¡satellites), ¡robo<cs ¡ ¡ Predator-­‑prey ¡systems, ¡bioreactors, ¡epidemiology, ¡ medicine ¡(peacemakers, ¡laser ¡surgery) ¡ ¡ Electricity, ¡electronics ¡ RLC ¡circuits, ¡thermostats, ¡regula<on, ¡refrigera<on, ¡computers, ¡internet ¡ and ¡telecommunica<ons ¡in ¡general, ¡photography ¡and ¡digital ¡video ¡ Economics ¡ Gain ¡op<miza<on, ¡control ¡of ¡financial ¡flux, ¡ Market ¡prevision ¡ Chemistry ¡ Chemical ¡kine<cs, ¡engineering ¡process, ¡petroleum, ¡dis<lla<on, ¡petrochemical ¡industry ¡ E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Here we focus on applications of control theory to problems of aerospace. 2 0 q 3 −2 30 20 40 10 20 0 −10 0 −20 −20 −30 −40 −40 q 2 q 1 5 20 q 2 0 q 3 0 −20 −40 −5 −60 −40 −20 0 20 40 −50 0 50 q 1 q 2 E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem The orbit transfer problem with low thrust Controlled Kepler equation r 3 + F q = − q µ ¨ m R 3 : position, r = | q | , F : thrust, m mass: q ∈ I ˙ m = − β | F | Maximal thrust constraint Orbit transfer 3 ) 1 / 2 ≤ F max ≃ 0 . 1 N | F | = ( u 2 1 + u 2 2 + u 2 from an initial orbit to a given final orbit. Controllability properties studied in B. Bonnard, J.-B. Caillau, E. Tr´ elat, Geometric optimal control of elliptic Keplerian orbits , Discrete Contin. Dyn. Syst. Ser. B 5 , 4 (2005), 929–956. B. Bonnard, L. Faubourg, E. Tr´ elat, M´ ecanique c´ eleste et contrˆ ole de syst` emes spatiaux , Math. & Appl. 51 , Springer Verlag (2006), XIV, 276 pages. E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem The orbit transfer problem with low thrust Controlled Kepler equation r 3 + F q = − q µ ¨ m R 3 : position, r = | q | , F : thrust, m mass: q ∈ I ˙ m = − β | F | Maximal thrust constraint Orbit transfer 3 ) 1 / 2 ≤ F max ≃ 0 . 1 N | F | = ( u 2 1 + u 2 2 + u 2 from an initial orbit to a given final orbit. Controllability properties studied in B. Bonnard, J.-B. Caillau, E. Tr´ elat, Geometric optimal control of elliptic Keplerian orbits , Discrete Contin. Dyn. Syst. Ser. B 5 , 4 (2005), 929–956. B. Bonnard, L. Faubourg, E. Tr´ elat, M´ ecanique c´ eleste et contrˆ ole de syst` emes spatiaux , Math. & Appl. 51 , Springer Verlag (2006), XIV, 276 pages. E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Modelization in terms of an optimal control problem „ q ( t ) « State: x ( t ) = q ( t ) ˙ Control: u ( t ) = F ( t ) Optimal control problem R n , R m , ˙ x ( t ) = f ( x ( t ) , u ( t )) , x ( t ) ∈ I u ( t ) ∈ Ω ⊂ I x ( 0 ) = x 0 , x ( T ) = x 1 , Z T f 0 ( x ( t ) , u ( t )) dt min C ( T , u ) , where C ( T , u ) = 0 E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Pontryagin Maximum Principle Optimal control problem R n , R m , ˙ x ( t ) = f ( x ( t ) , u ( t )) , x ( 0 ) = x 0 ∈ I u ( t ) ∈ Ω ⊂ I Z T f 0 ( x ( t ) , u ( t )) dt . x ( T ) = x 1 , min C ( T , u ) , where C ( T , u ) = 0 Pontryagin Maximum Principle Every minimizing trajectory x ( · ) is the projection of an extremal ( x ( · ) , p ( · ) , p 0 , u ( · )) solution of x = ∂ H p = − ∂ H H ( x , p , p 0 , u ) = max v ∈ Ω H ( x , p , p 0 , v ) , ˙ ˙ ∂ p , ∂ x , where H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) . An extremal is said normal whenever p 0 � = 0, and abnormal whenever p 0 = 0. E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Pontryagin Maximum Principle H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) . Pontryagin Maximum Principle Every minimizing trajectory x ( · ) is the projection of an extremal ( x ( · ) , p ( · ) , p 0 , u ( · )) solution of x = ∂ H p = − ∂ H H ( x , p , p 0 , u ) = max v ∈ Ω H ( x , p , p 0 , v ) . ˙ ˙ ∂ p , ∂ x , ւ u ( t ) = u ( x ( t ) , p ( t )) “ locally, e.g. under the strict Legendre assumption: ∂ 2 H ” ∂ u 2 ( x , p , u ) negative definite E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Pontryagin Maximum Principle H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) . Pontryagin Maximum Principle Every minimizing trajectory x ( · ) is the projection of an extremal ( x ( · ) , p ( · ) , p 0 , u ( · )) solution of x = ∂ H p = − ∂ H H ( x , p , p 0 , u ) = max v ∈ Ω H ( x , p , p 0 , v ) . ˙ ˙ ∂ p , ∂ x , տ ւ u ( t ) = u ( x ( t ) , p ( t )) “ locally, e.g. under the strict Legendre assumption: ∂ 2 H ” ∂ u 2 ( x , p , u ) negative definite E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Shooting method: Extremals ( x , p ) are solutions of x = ∂ H Exponential mapping ˙ ∂ p ( x , p ) , x ( 0 ) = x 0 , ( x ( T ) = x 1 ) , exp x 0 ( t , p 0 ) = x ( t , x 0 , p 0 ) , p = − ∂ H ˙ ∂ x ( x , p ) , p ( 0 ) = p 0 , (extremal flow) where the optimal control maximizes the Hamiltonian. − → Shooting method: determine p 0 s.t. exp x 0 ( t , p 0 ) = x 1 Remark - PMP = first-order necessary condition for optimality. - Necessary / sufficient (local) second-order conditions: conjugate points . → test if exp x 0 ( t , · ) is an immersion at p 0 . E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem There exist other numerical approaches to solve optimal control problems: direct methods: discretize the whole problem ⇒ finite-dimensional nonlinear optimization problem with constraints Hamilton-Jacobi methods. The shooting method is called an indirect method. In the present aerospace applications, the use of shooting methods is priviledged in general because of their very good numerical accuracy. BUT: difficult to make converge... (Newton method) To improve their performances and widen their domain of applicability, optimal control tools must be combined with other techniques: geometric tools ⇒ geometric optimal control continuation or homotopy methods dynamical systems theory E. Tr´ elat, Optimal control and applications to aerospace: some results and challenges , J. Optim. Theory Appl. (2012). E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Orbit transfer, minimal time Maximum Principle ⇒ the extremals ( x , p ) are solutions of x = ∂ H p = − ∂ H ˙ ∂ p , x ( 0 ) = x 0 , x ( T ) = x 1 , ˙ ∂ x , p ( 0 ) = p 0 , with an optimal control saturating the constraint: � u ( t ) � = F max . − → Shooting method: determine p 0 s.t. x ( T ) = x 1 , combined with a homotopy on F max �→ p 0 ( F max ) Heuristic on t f : t f ( F max ) · F max ≃ cste . (the optimal trajectories are ”straight lines”, Bonnard-Caillau 2009) (Caillau, Gergaud, Haberkorn, Martinon, Noailles, ...) E. Tr´ elat Optimal control and applications to aerospace problems

Introduction Shooting method Orbit transfer Three-body problem Orbit transfer, minimal time P 0 = 11625 km, | e 0 | = 0 . 75, i 0 = 7 o , P f = 42165 km F max = 6 Newton ! 4 x 10 1 2 0 q 3 −2 arcsh det( ! x) 30 0 40 20 10 20 0 0 −10 ! 1 0 100 200 300 400 500 −20 −20 t −30 −40 ! 3 −40 x 10 q 2 6 q 1 5 5 4 " n ! 1 3 20 2 1 0 q 2 q 3 0 0 0 100 200 300 400 500 −20 t −40 −5 −60 −40 −20 0 20 40 −50 0 50 q 1 q 2 Minimal time: 141.6 hours ( ≃ 6 days). First conjugate time: 522.07 hours. E. Tr´ elat Optimal control and applications to aerospace problems