The averaged control system of fast oscillating control systems

THE A VERA GED CONTR OL SYSTEM OF F AST OSCILLA TING CONTR OL SYSTEMS ∗ ALEX BOMBRUN † AND JEAN-BAPTISTE POMET ‡ Abstract. F or con trol systems that either ha ve a fast explicit p eriodic dep endence on time and bounded con trols or hav e perio dic solutions and small controls, we define an aver age c ontr ol system that ta kes in to a ccoun t all possibl e v ariations of the cont rol, and pro ve that i ts solutions appro ximate all solutions of the oscillating s ystem as oscill ations go f aster. The dimension of i ts ve lo cit y set i s cha racterize d geometrically . Whe n it is maximum the av erage system defines a Finsler metric, not t wice differen tiable i n general. F or mini mum time contr ol, this a ve rage system allows one to giv e a rigoro us pro of that a v eraging the Hamiltonian give n by the maximum principle is a v alid appro ximation. Key words. A v erag ing, contr ol systems, small contr ol, optimal cont rol, Finsler geometry . AMS sub ject cl assifications. 34C29, 34H05, 49J15, 93B11, 93C15, 93C70, 53B40 1. In tro duction. W e consider either a “fast-o s cillating control system” (1): ˙ x = u 1 X 1 ( t ε , x ) + · · · + u m X m ( t ε , x ) , k u k ≤ 1 , where all X i ’s ar e 2 π -p erio dic with r espect to t/ε , or a “Kepler control sys tem” (46): ˙ ξ = f 0 ( ξ ) + v 1 f 1 ( ξ ) + · · · + v m f m ( ξ ) , k v k ≤ ε where all so lutions of ˙ ξ = f 0 ( ξ ) a r e per iodic. A veraging techniques for conserv ativ e —p erio dic or not— o r dinary differential equations (ODEs) date back at least to H. P oincaré; see [2, §5 2] o r [23] for rec e nt exp ositions. Roughly sp eaking, o n a fixed int erv al, the solutions of ˙ x = F ( t/ε, x ) differ fro m those of ˙ x = F ( x ) by a term of o rder ε , with F t he av erage of F with resp ect to its fir st a rgument . If u o r v ab ov e is assig ned to be a fix e d function o f state and time (or computed from a dditional s ta te v ariables as in u = α ( p, x ) , ˙ p = g ( p, x ) ), then these techniques for O DEs ca n b e applied to give an approximation a t first or der with r espect to small ε of the mov emen t of the slow v ariables. A veraging is us ually used in this wa y in co ntrol theory: in vibra tio nal control [19], fast o scillating controls a re desig ned and av eraging tech niques a llo ws analysis and pro of of stability; in the same way , it solves stabilit y and path planning questions in co n trol of mec hanical systems, see for instance [8]; in [12, §5 ], high frequency control is used to approa c h a non-fla t system by a fla t one; one may also ment ion many applications to control [2 1, 18, 20] of the work [17] that mimics Lie brackets by highly os cillatory controls along the orig inal vector fields. A commo n feature to these refer ences is that the use o f oscilla tions “creates ” new independent co ntrols used for design. The use of av eraging in optimal co n trol of oscillating systems [10, 13, 1 4, 7] is similar in spirit to the ab ov e, but clos er to the framework of this pap er be c a use oscillatio ns are pr esen t in the system instead of b eing in tro duced by the control. V ery int eresting results are obtained applying av eraging to ∗ This work was partly supp orted by Thales Alenia Sp ac e , in 2004-2007. Submitte d to SIAM J. Control. Optim., Dec. 5, 2011. R evised Sept . 10 and Dec. 7, 2012. † Most of this work was done when this author was with INRIA. Alex.Bombru n@gmail.c om . ‡ INRIA, B. P . 93, 06902 Sophia Antipolis cedex, F rance. Email: Jean-Bapti ste.Pomet @inria.fr . 1 2 A. BOMBRUN AND J.-B . POMET the Hamiltonia n eq uations arising from Pon tryagin Maximum principle. F or instance, in [7], the authors ha ve s tudied in this wa y the problem o f minimal energy transfer betw een t wo elliptic orbits; e xtremals ar e the sa me as those giving the geo desics of a Riemannian metric. Again, av eraging introduces “new indep endent controls”: Riemannian geo desics are minimizers o f a problem where all v elo cit y directions are allow ed whereas the velocity set of the orig inal system at ea c h p oint had po sitiv e co dimension. The same av eraging co mputation ma y be applied to the Hamiltonian different ial equation obtained for minimu m time, but, since this differential equatio n is discontin uous, there is no theoretical justification for averaging in that case. Our contribution is to introduce a different way of averaging that takes in to account all p ossible v ar iations of the control —hence the control strategy can b e decided after per forming av eraging — and to pr o ve that it has s a tisfying reg ularit y prop erties and is a g oo d first o rder approximation o f the ab ov e s y stems as ε → 0 . This gives, as a side result, a justification of the use o f av eraging fo r minimum time in [13, 14]. This pr oce dur e also “ creates new independent control”, i.e. increa ses the dimension of the velocity s e t, that we characterize in terms of the origina l v ector fields. When this dimension is maximum, the av erage sys tem defines a Finsler metric [3 ] on the manifold, whose geo desics are the limits of minim um time tra jectories for the original systems a s ε → 0 . This Finsler metric is in genera l not t wice differentiable (hence it is not a Finsler metric in the sens e of [3], indeed); w e howev er prov e that, at least in the less degenerate ca s e, the Hamiltonian system governing extremals, although it is not lo cally Lipschit z, generates a flow on the cota ngen t bundle. Low thrust planar o rbit tr ansfer b elongs to this less degener ate cas e. The average control system may b e used for other purp oses than optimal control, for instance [4] designs a Lyapuno v function for feedback control in the average system and uses it for the osc illa ting systems; indeed the present work was dev elop ed out of comparing feedback control based on a priori ch osen Lyapuno v functions with minim um time control for low thrust o rbital tra ns fer . Preliminary versions of this pap er ca n b e found in [5, 4 ]. It is orga nized as follows: the co nstruction and results a re developed for “fast-o scillating control system” in §3 and then transferr ed in §4 to “Kepler control sy stems”, and applied to minimum time orbit transfer in the plana r 2-b o dy pro blem in §5. 2. Notations and con ven tio ns. 2.1. M is a smo oth co nnected manifold o f dimension n ; its tangent and co tangen t bundles are denoted by T M a nd T ∗ M . One may as sume fo r simplicit y M = R n , T M = R n × R n , T ∗ M = R n × ( R n ) ∗ , and, for x ∈ M , T x M = R n , T ∗ x M = ( R n ) ∗ . F or v ∈ T x M , p ∈ T ∗ x M (or any v , p taken in a vector space and its dua l), we denote by h p, v i (rather than p ( v ) ) their duality pro duct. 2.2. If E is a subset of a vector space V , then E ⊥ is its annihilator, the vector subspace of its dual V ∗ made of all p ’s such that h p, v i = 0 for all v in E . 2.3. W e assume that M is endowed with a n a r bitrary Riemannian distance d . If M = R n , just ch o ose the cano nical Euclidean distance. In lo cal co ordinates, k . k and ( . | . ) stand for the canonica l Euclidean norm a nd scalar pro duct. O n a compact c o or dinate ch art, k 1 k x − y k ≤ d ( x, y ) ≤ k 2 k x − y k for some p ositive k 1 , k 2 (Lipschi tz equiv alence). W e also denote op erator nor ms by k . k . 2.4. S 1 is R / 2 π Z . F or θ in S 1 (an a ngle), we denote b y µ ( θ ) the unique r eal n umber in [0 , 2 π ) such that µ ( θ ) ≡ θ mo d 2 π . F or a real n um ber s ∈ R , w e denote the angle it represents by s mo d 2 π ; it b elongs to the quotient S 1 . Maps S 1 → E (ar bitrary s et) are iden tified with 2 π -p erio dic maps R → E . F or A VERAGED CO N TR OL SYSTEM 3 instance, if f is such a map S 1 → E and τ ∈ R , we write f ( τ ) instead of f ( τ mo d 2 π ) ; the av erage of f is denoted by 1 2 π R 2 π 0 f ( θ )d θ , or 1 2 π R π − π f ( θ )d θ , or 1 2 π R θ ∈ S 1 f ( θ )d θ ; one iden tifies L p ( S 1 , R m ) with the subset of L p ( R , R m ) made of 2 π -p erio dic functions. 2.5. The Euclidean norm in R m or ( R m ) ∗ is deno ted by k . k , and the ball of radius one ce ntered at the origin b y B m . W e view an element o f R m as m × 1 matrix (co lumn) o f rea l num ber s and an ele ment of ( R m ) ∗ as a 1 × m matrix (line); transp osition, deno ted . ⊤ , sends R m to ( R m ) ∗ and vice- v ersa. 3. F ast oscillating con trol syste m s. W e call fast oscil lating c ontr ol system on M a family o f non- autonomous systems, linear in the control u ∈ R m : ˙ x = G ( t ε , x ) u = m X i =1 G i ( t ε , x ) u i , k u k ≤ 1 (1) indexed by a positive num ber ε . Each G i is a s mo oth “perio dic time-v arying” vector field: G i ∈ C ∞ ( S 1 × M , T M ) . An admissible control is a measura ble u ( . ) : [0 , T ] → B m for some T > 0 . F or a given control u ( . ) and initial condition x (0) , there is a unique solution x ( . ) , defined either o n [0 , T ] or o nly on a maxima l interv al [0 , T ′ ) , T ′ < T . R emark 3.1. Apar t from b eing a notation defined b y the double equality in (1 ), G ( θ , x ) defines a linear map R m → T x M that sends ( u 1 , . . . , u m ) ⊤ to P m i =1 G i ( t ε , x ) u i . 3.1. A v erage control system of fast oscillating contr ol systems . Define the map G : M × L ∞ ([0 , 2 π ] , R m ) → T M by G ( x, U ) = 1 2 π Z 2 π 0 G ( θ , x ) U ( θ ) d θ . (2) It allows o ne to define, for all x ∈ M , the subset E ( x ) ⊂ T x M by E ( x ) =  G ( x, U ) , U ∈ L ∞ ([0 , 2 π ] , R m ) , k U k ∞ ≤ 1  ⊂ T x M , (3) and the aver age c ontr ol system of (1) as follows 1 . Definition 3.2. The aver age c ont ro l system of (1) is the differ ential inclusion ˙ x ∈ E ( x ) . (4) A solution of (4) is an absolutely c ontinuous x ( . ) : [0 , T ] → M su ch that ˙ x ( t ) ∈ E ( x ( t )) for almost al l t . Proposition 3.3. F or al l x in M , E ( x ) is c onvex, c omp act and symmetric with r esp e ct t o the origin. Pr o of . It is close d, conv ex and symmetric b ecause it is the imag e of the unit ball of L ∞  S 1 , R m  b y a linear ma p; it is co mpact beca use G ( x, . ) is b ounded on S 1 . F urther characterizations of E ( x ) use the map H : T ∗ M → [0 , + ∞ ) defined by H ( x, p ) = 1 2 π Z 2 π 0 kh p, G ( θ, x ) ik d θ (5) where the Euclidean norm is used acco r ding to §2.5 and, for ea c h ( θ , x ) , h p, G ( θ , x ) i = ( h p, G 1 ( θ, x ) i , . . . , h p, G m ( θ, x ) i ) ∈ ( R m ) ∗ . (6) 1 Its relation to the limit case of (1) as ε → 0 is discussed i n the next section. 4 A. BOMBRUN AND J.-B . POMET Proposition 3.4. F or al l ( x, p ) ∈ T ∗ M , one has, with H define d in (5) , E ( x ) = n v ∈ T x M , sup p ∈ T ∗ x M H ( x,p ) ≤ 1 h p, v i ≤ 1 o , (7) H ( x, p ) = sup v ∈ E ( x ) h p, v i = sup U ∈ L ∞ ( S 1 , R m ) , k U k ∞ ≤ 1 h p, G ( x, U ) i = h p, G ( x, U ∗ p,x ) i , (8) with U ∗ p,x ∈ L ∞  S 1 , R m  define d by: U ∗ p,x ( θ ) = ( 0 if h p, G ( θ, x ) i = 0 , h p, G ( θ ,x ) i ⊤ kh p, G ( θ ,x ) ik if h p, G ( θ, x ) i 6 = 0 . (9) Pr o of . The last equality in (8) is a straig h tforward ma x imization, the second one comes from the definition (3) of E ( x ) and a simple computation yields H ( x, p ) = h p, G ( x, U ∗ p,x ) i ; this prov es (8 ). Being closed and conv ex, E ( x ) is the in tersection of all its suppor ting half-spa ces [24, Coro llary 1.3.5]; ac c o rding to (8), this yields the following relation, eq uiv alent to (7): E ( x ) = T p ∈ T ∗ x M { v ∈ T x M , h p, v i ≤ H ( x, p ) } . A c onvenient char acteriza tion of solutions of (4) . According to Definition 3.2, a solution x ( . ) is such that, for almo st a ll t , there is U ( t ) ∈ L ∞ ([0 , 2 π ] , R m ) such that ˙ x ( t ) = G ( x ( t ) , U ( t )) ; the map ( t, θ ) 7→ U ( t )( θ ) is measurable with resp ect to θ only . It turns out that it may alwa ys be chosen jointly measurable with resp ect to ( t, θ ) according to the following “meas urable selec tion” res ult: Proposition 3.5. A map x : [0 , T ] → R n is a solution of the differ ential inclu- sion (4) if and only if ther e ex ists b u ∈ L ∞ ([0 , T ] × S 1 , R m ) , k b u k ∞ ≤ 1 su ch t hat ˙ x ( t ) = 1 2 π Z 2 π 0 G ( θ , x ( t )) b u ( t, θ ) d θ (10) for almost al l t in [0 , T ] . Pr o of . After poss ibly partitioning [0 , T ] into interv als where ˙ x ( t ) remains in the same co or dinate chart, we work in co or dinates and use a Euclidean nor m when useful. Sufficiency is clear: fr o m F ubini theorem, θ 7→ b u ( t, θ ) is measur able for a lmost all t , hence x ( . ) is a solution of (4 ). Conv ersely , let x ( . ) be a solution of (4): ˙ x ( . ) is measurable and, for almost all t , there ex is ts ˜ u t ∈ L ∞ ( S 1 , R m ) , k ˜ u t k ∞ ≤ 1 such that ˙ x ( t ) = G ( x ( t ) , ˜ u t ) = 1 2 π Z 2 π 0 G ( s 1 , x ( t )) ˜ u t ( s 1 )d s 1 . (11) Let φ : L ∞  [0 , T ] × S 1 , R m  → L 2 ([0 , T ] , R n ) b e the linear map defined by φ ( u )( t ) = G ( x ( t ) , u ( t, . )) = 1 2 π Z 2 π 0 G ( s 1 , x ( t )) u ( t, s 1 ) d s 1 and I the image b y φ of the unit ba ll of L ∞ ([0 , T ] × S 1 , R m ) . Since, by (11), ˙ x ( . ) is essentially b ounded, it is in L 2 ([0 , T ] , R n ) ; since I is clos ed and conv ex in that Hilb e r t space, the distance from ˙ x to I is reached for a unique element ¯ ξ ∈ I : ¯ ξ = φ ( ¯ u ) , ¯ u ∈ L ∞  [0 , T ] × S 1 , R m  , k ¯ u k L ∞ ≤ 1 . Let us pr o ve by contradiction that ¯ ξ = ˙ x , i.e. ˙ x ( . ) ∈ I ; this will end the pro of. A VERAGED CO N TR OL SYSTEM 5 If ˙ x 6 = ¯ ξ , o ne ha s, for all u in the unit ball of L ∞  [0 , T ] × S 1 , R m  ,  ˙ x − ¯ ξ   φ ( u ) − φ ( ¯ u )  L 2 ≤ 0 (12) with equality only if φ ( u ) = φ ( ¯ u ) . Define b u by b u ( t, s ) = U ∗ ( ˙ x ( t ) − ¯ ξ ( t ) ) ⊤ , x ( t ) ( s ) with U ∗ p,x defined b y (9); clearly , b u is in the unit ball of L ∞  [0 , T ] × S 1 , R m  , and, for all ( t, s ) ∈ [0 , T ] × S 1 and all u ∈ R m , k u k ≤ 1 ⇒  ˙ x ( t ) − ¯ ξ ( t )  ⊤ G ( s, x ( t )) ( b u ( t, s ) − u ) ≥ 0 , (13) hence  ˙ x ( t ) − ¯ ξ ( t )  ⊤ G ( s 1 , x ( t )) ( b u ( t, s 1 ) − ¯ u ( t, s 1 )) is non- negative for almos t all ( t, s 1 ) and, since it is the integrand of the left-hand s ide of (1 2), it m ust b e zero; hence ¯ ξ = φ ( ¯ u ) = φ ( b u ) and ¯ ξ ( t ) = G ( x ( t ) , b u ( t, . )) for almost a ll t . In (11), ˜ u t satisfies k ˜ u t ( s 1 ) k ≤ 1 for a lmost all s 1 , hence, a ccording to (13),  ˙ x ( t ) − ¯ ξ ( t )  ⊤ G ( s 1 , x ( t )) ( b u ( t, s 1 ) − ˜ u t ( s 1 )) ≥ 0 . Since ˙ x ( t ) = G ( x ( t ) , ˜ u t ) , ¯ ξ ( t ) = G ( x ( t ) , b u ( t, . )) , the in tegration with resp ect to the v ar ia ble s 1 yields −k ˙ x ( t ) − ¯ ξ ( t ) k 2 ≥ 0 for almost a ll t ; this c on tradicts ˙ x 6 = ¯ ξ . R emark 3.6. The differential inclusion (4) is equiv alent to the “co n trol system” ˙ x = G ( x, U ) , U ∈ L ∞ ( S 1 , R m ) , k U k ∞ ≤ 1 where, by Pro position 3 .5, admiss ible controls a r e ma ps t 7→ U ( t ) such that b u : ( t, θ ) 7→ U ( t )( θ ) is mea surable with resp ect to ( t, θ ) . Since this “control” is infinite dimensional, and we could not find a repr e s en tation of the type ˙ x = f ( x, v ) , v ∈ U ⊂ R r , r finite, we stay with the differential inclusion (4), with E ( x ) describ ed b y (8) and (5). 3.2. Con v ergence the o rem. The following res ult relates solutions of the fast oscillating systems as ε tends to zero to solutions of the av erage system. T o our knowledge, this kind of theorem where the control is not chosen prior to a veraging has never b een sta ted in the litera ture. Theorem 3.7 ( Con vergence for fast- o scillating co n trol systems ). 1. L et x 0 ( . ) : [0 , T ] → M b e an arbitr ary s olut ion of (4) . Ther e exist a family of me asur able fun ctions u ε ( . ) : [0 , T ] → B m , indexe d by ε > 0 , and p ositive c onstants c, ε 0 , su ch that, c al ling x ε ( . ) the solution of (1) with c ontr ol u = u ε ( t ) and initial c ondition x ε (0) = x 0 (0) , one has: x ε ( . ) is define d on [0 , T ] for al l ε smal ler than ε 0 and c onver ges to x 0 ( . ) as ε → 0 , with an err or of uniform or der ε : d ( x ε ( t ) , x 0 ( t )) < c ε , t ∈ [0 , T ] , 0 < ε < ε 0 . (14) 2. L et K b e a c omp act subset of M , ( ε n ) n ∈ N a de cr e asing se quenc e of p ositive r e al n u mb ers c onver ging to zer o, and, for e ach n , x n ( . ) : [0 , T ] → K a solution of (1) with ε = ε n and for some c ontr ol u = u n ( t ) , u n ( . ) ∈ L ∞ ([0 , T ] , R m ) , k u n ( . ) k ∞ ≤ 1 . Then the se quenc e  x n ( . )  n ∈ N is c omp act for the top olo gy of uniform c onver genc e on [0 , T ] and any ac cumulation p oint is a solution of the aver age system (4) . The statement is mo re complex than the one for O DEs, e.g. [2, §52.C], due to underdetermination (ch oice of control in (1), multi-v alued rig h t-hand side in (4 )). Informal ly, “ 1” states that an y so lution of the av erage system is the limit o f solutions of fast oscillating systems with w ell chosen controls and “ 2 ” states that, conv ersely , a n y limit of solutions of os c illating systems, with arbitrary co n trols, is a solution of the av erage control system. There is an estimate on the erro r in “ 1” but not in “ 2” b ecause so me sequences may conv erge slower than others . 6 A. BOMBRUN AND J.-B . POMET R emark 3.8. One ma y consider systems that are affine instead of linear in the control by a dding a dr ift vector field G 0 ( t/ε, x ) to (1). Then, in the av erage control system, E ( x ) is replaced b y G 0 ( x ) + E ( x ) , with G 0 ( x ) = 1 2 π R 2 π 0 G 0 ( θ, x )d θ . By a straightforw ard extension, co n vergence do es hold for these systems to o. In the pro of o f Theorem 3.7, the following tec hnical lemma is needed. Lemma 3.9. L et ε > 0 and a < b b e r e al nu mb ers and b u : [ a − 2 π ε, b ] × S 1 → R m b e me asur able. One has t he fol lowing identity (se e §2.4 for the notation µ ( . ) ) : Z Z θ ∈ S 1 a ≤ s ≤ b G ( θ , x ( s )) b u ( s, θ ) d θ d s = Z Z θ ∈ S 1 a ≤ s ≤ b G ( s ε , x ( s )) b u ( s + εµ ( θ ) , s ε ) d θ d s + ∆ ε (15) with ∆ ε = Z Z T a ε G ( s ε , x ( s + εµ ( θ ))) b u ( s + εµ ( θ ) , s ε ) d θ d s − Z Z T b ε G ( s ε , x ( s + εµ ( θ ))) b u ( s + εµ ( θ ) , s ε ) d θ d s + Z Z θ ∈ S 1 a ≤ s ≤ b h G ( s ε , x ( s + εµ ( θ ))) − G ( s ε , x ( s )) i b u ( s + εµ ( θ ) , s ε ) d θ d s (16 ) and the set T a ε define d by T a ε = { ( s, θ ) , θ ∈ S 1 , a − ε µ ( θ ) ≤ s ≤ a } and T b ε ac c or dingly. Pr o of . Thanks to the change o f v ariables θ = τ /ε mo d 2 π , s = τ + ε µ ( φ ) , with µ ( θ ) as defined in §2 .4, the left-hand side of (15) is equa l to Z Z φ ∈ S 1 a − εφ ≤ τ ≤ b − εµ ( φ ) G ( τ ε , x ( τ + εµ ( φ ))) b u ( τ + εµ ( φ ) , τ ε ) d τ d φ . Keeping the names ( s, θ ) instead of ( τ , φ ) , one gets (15), the c orrecting term ∆ ε coming from the mo dified domain o f integration and a rgument o f x . Pr o of of The or em 3.7, p oint 1 . Consider a s olution x 0 : [0 , T ] → M n of (4). By Prop osition 3 .5 there exists b u 0 ∈ L ∞ ([0 , T ] × S 1 , R m ) , k b u 0 k ∞ ≤ 1 sa tisfying (10). F o r ε > 0 , define u ε ( . ) ∈ L ∞ ([0 , T ] , R m ) by (see §2.4 for notations): u ε ( t ) = 1 2 π Z 2 π 0 b u 0 ( t + ε µ ( θ ) , t ε ) d θ , (17) where b u 0 is pro longed by zero outside [0 , T ] : b u 0 ( t + ε µ ( θ ) , t ε ) = 0 if t + ε µ ( θ ) > T . Let us pr o ve that this co nstruction of u ε satisfies the t wo a nno unced pr o perties. Step 1. L et u s fi rst assume that M is an op en su bset of R n and G is zer o outside a c omp act su bset of M . Then G ( θ, x ) is a n × m matrix for a ll ( θ , x ) and, denoting b y k . k the Euclidean norm for vectors and the op erator norm for matrices, there are global constants Lip G a nd sup G such that, for a ll x, x ′ , θ in M × M × S 1 , kG ( θ, x ) − G ( θ , x ′ ) k ≤  Lip G  k x − x ′ k , kG ( θ , x ) k ≤  sup G  . (18) Let b b e a no n-negative constant and consider, for each ε > 0 , a solution x ε ( . ) of (1) with control u = u ε ( t ) and initial co ndition x ε (0) such that k x ε (0) − x 0 (0) k ≤ b ε . (19) A VERAGED CO N TR OL SYSTEM 7 In fact b = 0 in the theorem itself, but we need a nonzero b in step 2. By definition, expanding u ε ( s ) as in (17) and using Lemma 3.9, o ne ha s x ε ( t ) = x ε (0) + 1 2 π Z t 0 Z 2 π 0 G ( s ε , x ε ( s )) b u 0 ( s − ε µ ( θ ) , s ε ) d θ d s , = x ε (0) + 1 2 π  Z t 0 Z 2 π 0 G ( θ , x ε ( s )) b u 0 ( s, θ ) d θ d s − ∆ ε  (20) with ∆ ε given b y (16), that satisfies k ∆ ε k ≤ 4 π 2  Lip G  (1 + T sup G ) ε b ecause, in particular, k b u 0 k ≤ 1 , | εµ ( θ ) | < 2 π ε and k  G ( s ε , x ε ( s )) − G ( s ε , x ε ( s + ε µ ( θ )))  b u 0 ( s + ε µ ( θ ) , s ε ) k ≤ 2 π  Lip G   sup G  ε . Using (19), (20) the b ound on k ∆ ε k and the relation x 0 ( t ) = x 0 (0) + 1 2 π Z t 0 Z 2 π 0 G ( θ , x 0 ( s )) b u 0 ( s, θ ) d θ d s , one gets k x ε ( t ) − x 0 ( t ) k ≤  b + 2 π  Lip G   1 + T sup G  ε +  Lip G  Z t 0 k x ε ( s ) − x 0 ( s ) k d s for all t in [0 , T ] , and finally , by Gronw all lemma, k x ε ( t ) − x 0 ( t ) k ≤  b + 2 π  Lip G  (1 + T sup G )  e T Li p G ε (2 1) for all t in [0 , T ] and ε in [0 , ε 0 ] . This pro ves the theorem if M is an open subset o f R n and G is zero outside a co mpact s ubset, with a n e x plicit constant c co r resp onding to the distance d defined from the Euclidean norm and with ε 0 = + ∞ . Step 2. Gener al c ase. Let x ε ( . ) b e the solution of (1) with control u = u ε ( t ) defined in (17) from b u 0 and with initial condition x ε (0) = x 0 (0) ; it is not necessar ily defined on [0 , T ] but may hav e a maximum in terv a l o f definition [0 , T ε ) with T ε < T . Let e T ∈ [0 , T ] be the supremum of the set of num bers τ ∈ [0 , T ] such that, for s ome ε 0 and s o me c , that may dep end on τ , the s o lution x ε ( . ) is defined o n [0 , τ ] and satisfies d ( x ε ( t ) , x 0 (0)) < c ε for all t ∈ [0 , τ ] a nd ε ∈ [0 , ε 0 ] . Let us prov e b y contradiction that e T = T . This will end the pro of o f Theorem 3.7, p oint 1. Assume e T < T , a nd let - O b e a co ordinate neighborho o d of x 0 ( e T ) , - α > 0 b e such that 0 < e T − α < e T + α ≤ T and x 0 ([ e T − α, e T + α ]) ⊂ O , - c > 0 , ε 0 > 0 be such that d ( x ε ( t ) , x 0 ( t )) < c ε for a ll t ∈ [0 , e T − α ] and ε ∈ [0 , ε 0 ] . T a k ing ε 0 po ssibly smaller, one a lso has x ε ( e T − α ) ∈ O for ε < ε 0 . Let K b e a compact neighborho o d of x 0 ([ e T − α, e T + α ]) contained in O , K ′ a compact neighborho o d of K contained in O , and ρ : M → [0 , 1] a smo oth map, zero outside K ′ and c o nstan t eq ual to 1 in K . Defining G ρ b y G ρ ( θ, x ) = ρ ( x ) G ( θ , x ) , let us apply Step 1 in co ordinates in O , with G ρ instead of G and [ e T − α, e T + α ] instead of [0 , T ] . Call x ρ 0 (resp. x ρ ε , ε > 0 ) the solution of (10 ) (resp. of (1 ) with con trol u = u ε ( t ) ), replacing G by G ρ , with initial co ndition x ρ ε ( e T − α ) = x ε ( e T − α ) , ε ≥ 0 . O ne c lea rly has, as in (19), k x ρ ε ( e T − α ) − x ρ 0 ( e T − α ) k < b ε with b deduced fro m c via Lipschitz equiv alence of 8 A. BOMBRUN AND J.-B . POMET the distance d and the Euclidean nor m in co ordinates (see §2 .3); then Step 1 provides ε ′ 0 > 0 such that, by (21), the inequality k x ρ ε ( t ) − x ρ 0 ( t ) k ≤  b + 2 π  Lip G ρ  (1 + 2 α sup G ρ )  e 2 α Lip G ρ ε (22) is v alid for t ∈ [ e T − α, e T + α ] and ε ∈ [0 , ε ′ 0 ] . P ossibly choos ing a smaller ε ′ 0 , this implies tha t x ε ([ e T − α, e T + α ]) ⊂ K for ε < ε ′ 0 ; since G coincides with G ρ in K , the conclusion holds for x ε and G a s well a s for x ρ ε and G ρ if ε is no larger than ε ′ 0 . W e hav e shown that, for all ε < ε ′ 0 , the solution x ε is defined on [0 , e T + α ] and satisfies d ( x ε ( t ) − x 0 ( t )) ≤ c ′ ε for t in [0 , e T + α ] where c ′ is larger than c and than a b ound deduced from (22) a nd from Lips chitz equiv alence of d with the Euclidean distance. This contradicts the definition of e T . Pr o of of The or em 3.7, p oint 2 . Since G is bo unded on S 1 × K (one may cov er K with a finite num ber of co or dina te c harts and define this b ound in co ordinates), the ma ps x n ( . ) ha ve a common Lipsc hitz co nstan t and the se q uence ( x n ( . )) is equi- contin uo us, hence co mpact by Asco li-Arzela Theorem: o ne may extract a uniformly conv ergent sub-sequence. Still denoting by ( x n ( . )) n ∈ N such a conv erging sub-sequence and by x ∗ ( . ) its (uniform) limit, we need to prove that this limit is a so lution o f (4). Define, for ea c h n , b u n : [0 , T ] × S 1 → R m b y b u n ( t, θ ) = u n ( β n ( t, θ ) ) , (23) where u n ( . ) ∈ L ∞ ([0 , T ] , R m ) is asso ciated to x n ( . ) acc o rding to the assumption of the theorem and where the map β n : [0 , T ] × S 1 → R is defined by t − 2 πε n < β n ( t, θ ) ≤ t , β n ( t, θ ) ε n ≡ θ mo d 2 π . (24) Clearly b u n is in L ∞ ([0 , T ] × S 1 , R m ) and k b u n k ∞ ≤ 1 . Hence, a fter p ossibly extracting a sub-sequence, ( b u n ) conv erges in the weak- ∗ top ology to some b u ∗ . Let us prove that, for almost all t ∈ [0 , T ] , ˙ x ∗ ( t ) = 1 2 π Z 2 π 0 G ( θ , x ∗ ( t )) b u ∗ ( t, θ )d θ . (25) Let e T ∈ [0 , T ] b e the supremum of the s et of num ber s τ ∈ [0 , T ] such that this is true for almost all t in [0 , τ ] , and let us pr o ve by contradiction that e T = T . Assume e T < T , a nd let O be a co ordinate neig h bor hoo d of x 0 ( e T ) a nd α b e such that 0 < e T − α < e T + α ≤ T and x 0 ([ e T − α, e T + α ]) ⊂ O . Uniform conv ergence implies x n ([ e T − α, e T + α ]) ⊂ O for n large enough and then, in co ordina tes , for t ∈ [ e T − α, e T + α ] , x n ( t ) − x n ( e T − α ) = Z t e T − α G ( s ε n , x n ( s )) u n ( s ) d s . (26) F ro m (2 4), o ne has β n ( s + ε n θ, s ε n ) = s , hence, fro m (2 3 ), b u n ( s + ε n θ, s ε n ) = u n ( s ) for all θ ∈ S 1 , s ∈ R ; using this in Lemma 3.9, one has 1 2 π Z Z e T − α ≤ s ≤ t θ ∈ S 1 G ( θ , x n ( s )) b u n ( s, θ ) d θ d s = 1 2 π Z Z e T − α ≤ s ≤ t θ ∈ S 1 G ( s ε n , x n ( s )) u n ( s ) d θ d s + ∆ ε n . A VERAGED CO N TR OL SYSTEM 9 Since the in tegral in the right-hand side —whose integrand do es not depend on θ — is equal to the right-hand side of (26), one gets, using uniform conv ergence of x n to x ∗ , weak co n vergence of b u n to b u ∗ and co n vergence o f ∆ ε n to zer o, x ∗ ( t ) − x ∗ ( e T − α ) = 1 2 π Z t e T − α Z 2 π 0 G ( θ , x ∗ ( s )) b u ∗ ( s, θ ) d θ d s , for t in [ e T − α, e T + α ] , and finally tha t (25) hold for a lmo st all t in [0 , e T + α ] , thus contradicting the definition of e T . 3.3. Dimension of the v elo cit y set E ( x ) . Recall that, for a conv ex subset C of a linear space, co n taining the origin, its linear h ull is the smallest linear subspace that contains C , the in terior of C in its line ar hul l is alwa ys nonempty , and dim C is the dimension o f this linear hull. Viewing ∂ j G ∂ θ j ( θ, x ) as a linear ma p R m → T x M (see Rema rk 3.1), and denoting b y Σ a sum of linear s ubspaces o f T x M , define the integer r ( θ , x ) by: r ( θ , x ) = dim  X j ∈ N Range ∂ j G ∂ θ j ( θ, x )  . (27) It is a lso the rank o f the collection of vectors ∂ j G i ∂ θ j ( θ, x ) ∈ T x M , 1 ≤ i ≤ m , j ≥ 0 . In the following pro position, and it is the so le place where this prop erty is use d, “system (1) is real ana lytic with resp ect to θ ” means that the vector fields G i are real analytic with resp ect to θ for fixed x (while b eing smo oth with resp ect to ( θ , x ) ). Proposition 3.10. 1. The line ar hul l of E ( x ) satisfies the fol lowing two pr op erties for al l x in M , wher e t he inclusion (29) is an e quality if (1 ) is r e al analytic with r esp e ct to θ : Linear hull E ( x ) = X θ ∈ S 1 Range G ( θ , x ) , (28) Linear hull E ( x ) ⊃ X j ∈ N Range ∂ j G ∂ θ j ( θ, x ) for al l θ ∈ S 1 . (29) 2. If r ( θ, x ) = n for at le ast one θ in S 1 , then E ( x ) has a nonempty interior in T x M , i.e. dim E ( x ) = n . 3. If the system (1) is r e al analytic with r esp e ct t o θ , then r ( θ, x ) do es n ot dep end on θ and r ( θ, x ) = dim E ( x ) . Pr o of . If p is in Range G ( θ , x ) ⊥ for all θ , then an y v = G ( x, U ) in E ( x ) sat- isfies h p, v i = 0 b ecause h p, G ( θ , x ) U ( θ ) i is iden tically zero on [0 , 2 π ] . Co n versely , let p b e in E ( x ) ⊥ , a nd consider v = G ( x, U ∗ p,x ) ∈ E ( x ) ; then h p, v i = 0 implies h p, G ( θ , x ) i = 0 , i.e. p ∈ Range G ( θ, x ) ⊥ for a ll θ . W e hav e prov ed the iden tit y E ( x ) ⊥ = T θ ∈ S 1 (Range G ( θ, x )) ⊥ , hence (28). If p is in Range G ( φ, x ) ⊥ for all φ , differentiating h p , G ( φ, x ) i = 0 with respe c t to φ yields  p , ∂ j G / ∂ φ j ( x, φ )  = 0 , j ∈ N ; we hav e prov ed, taking φ = θ , the inclusion T φ ∈ S 1 (Range G ( φ, x )) ⊥ ⊂ T j ∈ N  Range ∂ j G ∂ θ j ( θ, x )  ⊥ , hence (29). T o prov e the reverse inclusion in the rea l ana lytic ca se, fix θ ∈ S 1 and p ∈ T j ∈ N ∂ j G ∂ θ j ( θ, x ) ⊥ , and consider the real ana lytic mapping S 1 → ( R m ) ∗ , φ 7→ h p , G ( φ, x ) i ; the as s umption on p implies that this map v anishes for φ = θ , a s well as its der iv atives at all orders , hence it is iden tically zero: p ∈ T φ ∈ S 1 (Range G ( φ, x )) ⊥ . This ends the pro of o f Poin t 1. Poin t 2 is an eas y co nsequence a nd P oint 3 is class ical. 10 A. BOMBRUN AND J.-B . POMET 3.4. F urther prop ertie s in the full rank case. W e now assume that the mapping G in (1) is such that the rank r ( θ, x ) defined by (27) is max imal: r ( θ , x ) = n for a ll x in M and θ in S 1 . (30) 3.4.1. Co n trol labilit y . Condition (30) is stro ngly r elated to controllabilit y of the linear a pproximation of (1) around e q uilibria, i.e. around s olutions wher e x is constant and u is iden tically zero . Indeed, the linear a pproximation of the time- v ar y ing nonlinear sys tem (take ε = 1 in (1)): ˙ x = G ( t, x ) u (31) around the equilibrium x = x 1 is the time-v arying linear system ˙ ξ = G ( t, x 1 ) u ; ac- cording to [16, p.614], it is “controllable with impulsiv e controls at an y time” if and only if r ( t, x 1 ) = n for all t . If this is tr ue at all p oints x 1 then all end-p oin t mappings are s ubmer sions a round zero co ntrols; we shall need the following more precise res ult: Proposition 3.11. Assume that (30) holds. 1. F or al l x 1 ∈ M and T > 0 , ther e ex ist a c o or dinate neighb orho o d W of x 1 (the b al l B b elow r efers to t he Euclide an norm in these c o or dinates), p ositive c onstants α 0 , c 3 , and, for al l y ∈ W , a smo oth map χ y : B ( y , α 0 ) → L ∞ ([0 , T ] , R m ) with Lipschitz c onstant c 3 , which is a right inverse of the end-p oint mapping of (31 ) on [0 , T ] starting fr om y , i.e. for al l y f ∈ B ( y , α 0 ) , the c ontr ol χ y ( y f ) : [0 , T ] → R m is such that the solut ion of ˙ x = G ( t, x ) χ y ( y f )( t ) , x (0) = y satisfies x ( T ) = y f . 2. F or al l ε > 0 , the system (1) is ful ly c ontr ol lable, i.e. t her e exists, for any ε > 0 and any two p oint x 0 , x 1 in M , a time T and a me asur able c ontr ol u : [0 , T ] → B m such that the solut ion of (1) with x (0) = x 0 satisfies x ( T ) = x 1 . Pr o of . Let E y : L ∞ ([0 , T ] , R m ) → M be the end-p oint mapping with starting po in t y . Condition (30) implies tha t the der iv ative of E x 1 at the zero control has r a nk n ; hence there exists an n -dimensional subspace V o f L ∞ ([0 , T ] , R m ) such that the restriction of E x 1 , and hence of E y for y clo se enough, to V is a lo cal diffeomorphism at zero ; the χ y ’s are the lo cal inv erses of these loca l diffeomorphisms; they depend smo othly on y , hence the co mmon α 0 and c 3 in Poin t 1 . This implies that the reachable set from any p oin t at any p ositive or nega tiv e time contains a neighborho o d o f this point; a classic a l argument then tells us that the reachable set fro m a p oin t x 0 is M , a ssumed to b e connected, for it is b oth op en (obvious) a nd closed (if ¯ x is in the closure of the reachable set, some p o in ts in the reachable set ca n b e r eached in negative time, hence ¯ x ca n b e rea ched fro m x 0 ). Let us now turn to the average system (4). F rom H : T ∗ M → [0 , + ∞ ) defined b y (5 ), we define N : T M → [0 , + ∞ ] by N ( x, v ) = max p ∈ T ∗ x M , H ( x,p ) ≤ 1 h p, v i . (32) Proposition 3.12. Assume the r ank c onditio n (30). 1. F or al l x ∈ M , H ( x, . ) defines a norm on the c otangent sp ac e T ∗ x M , its dual norm on the t angent s p ac e T x M is N ( x, . ) , and E ( x ) is t he unit b al l for N ( x, . ) , i.e . E ( x ) = { v ∈ T x M , N ( x, v ) ≤ 1 } . 2. System (4) is ful ly c ontr ol lable, i.e. ther e exists, for any p oints x 0 , x 1 in M , a time T and a solution x ( . ) : [0 , T ] → M of (4) such that x (0) = x 0 , x ( T ) = x 1 . Pr o of . F rom (5), H ( x, p ) = 0 implies h p, G ( θ, x ) i = 0 for all θ and, differentiating with r espect to θ and us ing (30), this implies p = 0 ; this makes p 7→ H ( x, p ) a norm, the other prop erties b eing straightforward. Hence N given by (32) is finite A VERAGED CO N TR OL SYSTEM 11 for an y ( x, v ) a nd it is, by definition, the dual norm of H ( x, . ) ; E ( x ) is its unit ball b y (7) in Propo sition 3 .4. T o prov e Poin t 2, tak e a contin uously differentiable curve γ : [0 , 1] → M such that γ (0) = x 0 and γ (1) = x 1 and σ : [0 , T ] → [0 , 1] for some T > 0 , differentiable, such that t ≥ Z σ ( t ) 0 N ( γ ( s ) , d γ d s ( s ))d s ( N and H are obviously co n tin uous), then t 7→ x ( t ) = γ ( σ ( t )) is a solution of (4) such that x (0) = x 0 and x ( T ) = x 1 . 3.4.2. O n the di fferen tiabilit y of H . It is clear that H , given by (5) , is as smo oth as G o n T ∗ M \ e Z with e Z = { ( x, p ) ∈ T ∗ M , ∃ θ ∈ S 1 , h p, G ( θ , x ) i = 0 } . (33) Unfortunately , e Z is not empty in general: it is generically a 2 n − m + 1 dimensional submanifold of T ∗ M . One how ever has the following result, v alid also at these p oint s. Theorem 3.13. If c onditi on (30) holds, H 2 is c ontinuously differ entiable. It is stated for H 2 : ( x, p ) 7→ H ( x, p ) 2 , b ecause H itself, homogeneous of degr e e 1 with resp ect to p , cannot b e differentiable on { p = 0 } , that co incides with { H ( x, p ) = 0 } by Pro pos itio n 3.12 item 1. The map H fails in gener al to b e twice different iable on e Z . W e hav e the following estimate of the of the mo dulus o f contin uit y of its first deriv ative, that ev en fails to be Lipschitz co n tin uous. Its main consequence is Theorem 3.2 1. Theorem 3.14. Assume that t he ra nk c ondi tion (30) holds and that (i) for ( x, p ) ∈ T ∗ M , p 6 = 0 , ther e is at most one θ ∈ S 1 such that h p, G ( θ , x ) i = 0 , and h p, ∂ G ∂ θ ( θ, x ) i do es n ot vanish at the same p oint, (ii) for al l ( θ , x ) ∈ S 1 × M , one has rank G ( θ , x ) = m , then any p oint ( ¯ x, ¯ p ) has a c onstant c and a c o or dinate n eighb orho o d in T ∗ M such that for al l X and Y in R 2 n , c o or dinates of p oints in the neighb orho o d, k d H ( Y ) − d H ( X ) k ≤ c k X − Y k ln 1 k X − Y k . (34) In the left-hand side, k . k stands for the o pera tor nor m in co ordinates, see §2 .3. R emark 3.15 (Finsler ge ometry). If H 2 was an least t wic e contin uo usly differe n- tiable, with a p ositiv e definite Hessia n with r espect to p , so would b e N 2 (see (32)), and it would define a (reversible) Finsler metric [3] on M . The lack of different iability calls for further developmen ts. Before proving these theore ms , we state a more generic result, whose notations are totally indep endent from the r est o f the pap er. Its pro of is in the app endix. Proposition 3.16. L et d b e a p ositive inte ger, O d an op en subset of R d , V : S 1 × O d → R m a smo oth map ( C ∞ ), e Z the subset of O d wher e V vanishes for some θ and H : O d → [0 , + ∞ ) the aver age of the n orm of V : H ( X ) = 1 2 π Z 2 π 0 k V ( θ , X ) k d θ , e Z = { X ∈ O d , ∃ θ ∈ S 1 , V ( θ , X ) = 0 } . (35) 12 A. BOMBRUN AND J.-B . POMET 1. Assume that, for al l X in O d , the set { θ ∈ S 1 , V ( θ, X ) = 0 } has me asur e zer o in S 1 . (36) Then H is c ontinuously differ entiable and, for al l X , d H ( X ) .h = 1 2 π Z 2 π 0  ∂ V ∂ X ( θ, X ) .h     V ( θ , X ) k V ( θ , X ) k  d θ . (37) 2. L et V satisfy the following assumptions: (a) V ( θ , X ) = 0 for al l ( θ , X ) ∈ S 1 × O d such that V ( θ , X ) = 0 , (b) for any X ∈ O d , ther e is at most one θ such that V ( θ , X ) = 0 , (c) V and ∂ V / ∂ θ do not vanish simult ane ously ,    (38) and let ¯ X b e in e Z . Ther e is a neighb orho o d U of ¯ X in O d , and a c onstant K > 0 such that, for al l X , Y in U ,    d H ( X ) − d H ( Y )    ≤ K k X − Y k ln 1 k X − Y k . (39) In the left-hand side of (39), k . k stands for the op erator norm, see §2.3. Pr o of of The or em 3.13. This is a lo cal pr oper t y . W e op erate in co ordina tes and apply Propo s ition 3.1 6 (Poin t 1) with d = 2 n , O d a neigh b orho od of a po in t where p 6 = 0 , X = ( x, p ) ∈ R 2 n and V ( θ, X ) = h p, G ( θ, x ) i . The rank co ndition implies that deriv atives of a ll order s of the map θ 7→ V ( θ, X ) never v anish a t the sa me p oint, so that its zero es a re isolated and the set { θ ∈ S 1 , V ( θ, X ) = 0 } is finite and a fortiori has measure zer o; hence H is contin uously differen tiable outside { p = 0 } . Since 0 ≤ H ( x, p ) ≤ k k p k for some lo cal consta n t k the deriv ativ e o f H 2 is zer o a t all po in ts ( x, 0) and, since (37) implies that the norm o f d H ( x, p ) at neig hbo ring p oin ts where p 6 = 0 is bo unded, the deriv a tiv e of H 2 at these p oints tends to zero as p → 0 . H 2 is therefor e contin uously differentiable everywhere. Pr o of of The or em 3.14. Smo othness outside e Z is o b vious from the expression (5) of H ; inequality (34) is a c o nsequence o f Prop osition 3.16 (P oint 2), a pplied with d = 2 n , X = ( x, p ) ∈ R 2 n , V ( θ , X ) = h p, G ( θ, x ) i and O d a neighbor hoo d o f a po in t of e Z \ { p = 0 } ; it is clear that p oint s (i) and (ii) imply the three conditions (38). 3.5. Application to the mini m um ti me problem. Fix tw o p oin ts x 0 , x 1 in M and co nsider the time optimal problem a sso ciated to (1) for ε > 0 : ( P ε ) , ε > 0 : ˙ x ( t ) = G ( t/ε , x ( t )) u ( t ) , u ( t ) ∈ B m , t ∈ [0 , T ] , x (0) = x 0 , x ( T ) = x 1  min T , (40) and the time optimal problem asso ciated to the av erage s ystem: ( P 0 ) : ˙ x ( t ) ∈ E ( x ( t )) , t ∈ [0 , T ] , x (0) = x 0 , x ( T ) = x 1  min T . (41) Call T ε ( x 0 , x 1 ) the minimum time for ( P ε ) , ε > 0 and T 0 ( x 0 , x 1 ) the o ne for ( P 0 ) ; when no c onfusion arises, we wr ite T ε and T 0 . Let us develop (40)– (41 ): concerning (4 0), T ε is the infimum o f the set of T ’s s uc h that ther e is an admissible cont rol u ( . ) : [0 , T ] → B m , and x ( . ): [0 , T ] → M satisfying x (0) = x 0 , x ( T ) = x 1 and ˙ x ( t ) = G ( t/ε, x ( t )) u ( t ) for almost all t ; Prop osition 3.11, po in t 2 implies that this set is nonempt y , hence T ε is finite. Concerning (4 1), T 0 is the infimum of the set o f T ’s such that there is x ( . ) : [0 , T ] → M sa tisfying x (0) = x 0 , A VERAGED CO N TR OL SYSTEM 13 x ( T ) = x 1 and ˙ x ( t ) ∈ E ( x ( t )) for almost all t , T 0 is finite from Prop osition 3 .12, p oint 2. A solution to ( P ε ) (resp. to ( P 0 ) ) is x ( . ) , u ( . ) (resp. x ( . ) ) as ab o ve with T = T ε (resp. T = T 0 ). In general, the minimum T ε or T 0 need not b e re a c hed, i.e. there need not b e a solution. Lemma 3.17. Assume the r ank c onditio n (30). 1. Ther e is a n eigh b orho o d W of any x 1 and two c onstants α 0 > 0 and C 3 > 0 such that, for al l y in W , T ε ( y , x 1 ) ≤ 2 πε + C 3 d ( x 1 , y ) . 2. F or any x 0 , x ′ 1 , x 1 in M , one has T ε ( x 0 , x 1 ) ≤ T ε ( x 0 , x ′ 1 ) + T ε ( x ′ 1 , x 1 ) + 2 π ε . Pr o of . Apply Prop osition 3.11, p oint 1 with T = 2 π , using as a distance in W the Euclidean no rm in so me co ordinates: for a ny t wo points y , y ′ in W such that k y − y ′ k ≤ α 0 , there is a control defined on [0 , 2 π ] , with L ∞ norm s ma ller than c 3 k y − y ′ k that bring s y ′ at time 0 to y at time 2 π for sys tem (31); rescaling time a nd control by ε yields, if c 3 k y − y ′ k ≤ ε , a control with L ∞ norm less than 1 that brings y ′ at time 0 to y at time 2 π ε for system (1) and hence, by conca tenating controls and using p erio dicit y of G , for any p ositiv e integer k , a co n trol with L ∞ norm less than 1 that brings y ′ at time 0 to y at time 2 k π ε for system (1) if c 3 k y − y ′ k ≤ k ε . In other words, T ε ( y ′ , y ) ≤ 2 π ( ε + c 3 k y − y ′ k ) . T ake y ′ = x 1 and 2 π c 3 /C 3 the r atio betw een the Euclidean nor m and the dis ta nce d ; this prov es p oin t 1. Poin t 2 follows from using p erio dicit y o f G a nd co ncatenating controls while inserting a ze r o control betw een time T ε ( y ′ , y ) and the next m ultiple of 2 π . Theorem 3.18 ( limit of minimum time ). Assume the r ank c ondition (30). 1. T ε is b ounde d as ε → 0 and lim s up ε → 0 T ε ≤ T 0 . 2. If, for ε > 0 smal l enough, e ach ( P ε ) has a solution x ε : [0 , T ε ] → M and ther e exists a c omp act K ⊂ M such that x ε ([0 , T ε ]) ⊂ K for al l ε > 0 smal l enough, then al l ac cumulation p oints of the c omp act family ( x ε ( . )) ε> 0 in C 0 ([0 , T 0 ] , M ) ar e solutions of ( P 0 ) and lim ε → 0 T ε = T 0 . Pr o of . C o nsider a minimizing sequence for pr oblem ( P 0 ) , i.e. solutions x k : [0 , T 0 + β k ] → M of the average system (4) with ( β k ) a s equence of p ositiv e n umbers that tends to zer o and x k (0) = x 0 , x k ( T 0 + β k ) = x 1 for all k . F or ea c h x k , there is, according to Theorem 3.7, a family ( x k ε ( . ) ) ε> 0 such that each x k ε ( . ) is a solution of (1) with x k ε (0) = x 0 and d ( x k ε ( t ) , x k ( t )) ≤ c 1 ε for all t in [0 , T 0 + β k ] . In particular d ( x k ε ( T 0 + β k ) , x k ( t ) 1 ) ≤ c 1 ε . Now, from Lemma 3 .17, T ε ( x k ε ( T 0 + β k ) , x 1 ) ≤ (2 π + c 1 C 3 ) ε ; hence, from the second point of that lemma (with x ′ 1 = x k ε ( T 0 + β k ) ), one has T ε = T ε ( x 0 , x 1 ) ≤ T 0 + β k + (4 π + c 1 C 3 ) ε a nd, letting k go to infinit y , T ε ≤ T 0 + (4 π + c 1 C 3 ) ε ; this implies Poin t 1. Let us turn to p oint 2. Extend x ε on [0 , T ] , with T an upper bound of T ε , b y taking x ε ( t ) = x 1 for t in [ T ε , T ] . An y sequence ( x ε k ( . )) k ∈ N with lim ε k = 0 is co mpact in C 0 ([0 , T ] , M ) : take a conv ergent s ubsequence such that T ε k also conv erges to s ome T ∗ . The uniform limit go es through x 0 at time 0 and x 1 at time T ∗ and is , by Theorem 3.7, a solution of the av erage sys tem (4), hence T ∗ ≥ T 0 b y definition of T 0 . This, together with P oin t 1, implies P oint 2 b ecause T ∗ can b e any accumulation p oin t of ( T ε ) as ε → 0 . Let us now write the Pon try agin Maximum Principle [22] b oth for ( P ε ) , ε > 0 and for ( P 0 ) and see how they are r elated. The extr emals o f pr oblem ( P ε ) , ε > 0 , a re absolutely contin uous ma ps t 7→ ( x ( t ) , p ( t )) solution to ˙ p = − ∂ H ε ∂ x , ˙ x = ∂ H ε ∂ p with H ε ( t, p, x ) = kh p, G ( t/ε, x ) ik , (42) 14 A. BOMBRUN AND J.-B . POMET whose right-hand side is disco n tin uous o n S ε = { ( x, p , t ) , h p, G ( t/ε, x ) i = 0 } (the “switching surface”), wher e it is in fact not defined. The extr emals of ( P 0 ) ar e absolutely contin uous t 7→ ( x ( t ) , p ( t )) solution to ˙ p = − ∂ H ∂ x , ˙ x = ∂ H ∂ p . (43) with H given by (5). The rig h t-hand sides are co n tin uous acc o rding to Theorem 3 .13. Theorem 3.19. If an absolutely c ontinu ous map t 7→ x ( t ) define d on [0 , T ] is a solution of ( P ε ) , ε > 0 , (r esp. of ( P 0 ) ), then ther e exists t 7→ p ( t ) define d on [0 , T ] such that t 7→ ( p ( t ) , x ( t )) is an extr emal of ( P ε ) , ε > 0 (r esp. of ( P 0 ) ). Pr o of . Problem ( P ε ) , ε > 0 deals with a classical smo oth control system; acco rd- ing to [22, 1], the pseudo-Ha miltonian is h ( t, x, p, u ) = h p, G ( t/ε, x ) u i ; an extremal is a curve on the co -tangent bundle solution, in lo cal co ordinates, of: ˙ p = − ∂ h ∂ x ( t, x, p, u ∗ ) = −h p, ∂ G ∂ x u ∗ i , ˙ x = ∂ h ∂ p ( t, x, p, u ∗ ) = G u ∗ , (44) with u ∗ ( t ) a control that maximizes the pseudo-Ha miltonia n for a lmost all time; it is defined by u ∗ = h p, G i kh p, G ik if h p, G ( t/ε , x ) i 6 = 0 ; the ma x imized Hamiltonian H ε ( t, p, x ) = max u h ( t, x, p , u ) is the one in (42), and (44) is then the differential e quation (42) , whose right-hand s ide is disco n tin uous at p oints where h p, G ( t/ε, x ) i v a nishes. Let us now turn to ( P 0 ) . Since the set of admissible velocities is not a pr iori smo oth with resp ect to the state v a riable we use a no n- smoo th version of the Pon try a- gin maximum principle for differential inclusio ns, that we r ecall fo r self-co n tainedness: Theorem 9.1 in [11, Cha pter 4]: if ˙ x ∈ E ( x ) is a lo c al ly Lipschitz differ ential inclusion and t 7→ x ( t ) is an absolutely c ontinuous function define d on [0 , T ] solution to the pr oblem (41) , then ther e exists t 7→ p ( t ) define d on [0 , T ] such that ( − ˙ p, ˙ x ) ∈ ∂ C H ( x, p ) for almost al l t ∈ [0 , T ] with H ( x, p ) = max v ∈ E ( x ) h p, v i and ∂ C H t he gener alize d gr adient of H . The set-v alued ma p E ( . ) in (3) is indeed lo cally Lipschitz: in loca l co ordinates , for x 1 , x 2 in R n , denoting by δ the Hausdorff dista nce b etw een tw o sets, one has : δ ( E ( x 1 ) , E ( x 2 )) = max ( sup v 1 ∈ E ( x 1 ) inf v 2 ∈ E ( x 2 ) k v 1 − v 2 k , s up v 2 ∈ E ( x 2 ) inf v 1 ∈ E ( x 1 ) k v 1 − v 2 k ) = max k U 1 k ∞ ≤ 1 min k U 2 k ∞ ≤ 1 k G ( x 1 , U 1 ) − G ( x 2 , U 2 ) k ≤ Lip G k x 1 − x 2 k . A ccording to (8), the Ha miltonian H defined in the above quoted theorem coincides with the ma p H defined in (5). R emark 3.20. This result a nd Theo r em 3 .18 have tw o interpretations: 1. They prov e that the o pera tions of aver aging and c omputing the Hamiltonian for the minimum time pr oblem co mm ute. Indeed, the Hamiltonian H w as obtained by applying the maximum pr inciple to problem (4 1), i.e. minimum time for the av erage system (4), but it also the average of the one in (42 ) with r espect to the fast v ariable. 2. They prove indirectly an averaging result fo r the minim um time con trol problem (41); the averaging techniques in [10] do not apply to minim um time for they require smo othness of the Hamiltonian, while averaging is used in [14, 13] for minimum time with only pa rtial theor e tica l justifications but numerical evidence o f efficiency . A VERAGED CO N TR OL SYSTEM 15 Let us no w fo cus on the differen tial equa tio ns (43 ) that govern the extremals of ( P 0 ) . It is of gr eat imp ortance to know whether it defines a Hamiltonia n flow o n T ∗ M , i.e. whether solutions trough all initial conditions are unique or not. Its right -hand side is contin uous b ecause, from Theorem 3.1 3, H is contin uously differentiable; this ensures existence of s olutions. W e saw that H is s mooth ( C ∞ ) on T ∗ M \ e Z (see (3 3)), hence so lutions through p oints outside e Z are alwa ys unique. The following result gives uniqueness of solutions even on e Z in the less degenerated case po ssible. Theorem 3.21 ( Hamiltonian flow for ( P 0 ) ). Assume that the r ank c ondition (30) hold s, as wel l as c onditions (i) and (ii) in The or em 3.14. Then t he differ ential e quation (4 3) has a unique solution fr om any initial c ondition. Pr o of . F or an autonomo us ODE ˙ z = f ( z ) in a finite dimensional space, where f satisfies k f ( z 1 ) − f ( z 2 ) k ≤ ω ( k z 1 − z 2 k ) with ω : [0 , + ∞ ) → [0 , + ∞ ) non-decr easing, Kamke uniqueness Theor em [15, chap. II I, Th. 6.1 ] states that uniqueness of solutions holds if R α 0 d u ω ( u ) = + ∞ for a rbitrarily sma ll α > 0 . F rom Theorem 3.1 4, we are in this case with ω ( u ) = c u ln (1 /u ) , and R d u ω ( u ) = − 1 c ln ln(1 /u ) . Proving existence of a flow for (43) in more general situa tio ns (weaker sufficient condition) than this theorem is an in teresting progr am to b e pursued. Ho w ever, it turns out to b e applicable to the control of or bit trans fer wi th low thrus t, s ee §5. P oint (ii) is v ery mild and o nly states that the control vector fields ar e linearly independent. P oin t (i) is more ar tificial: the fact that h p, G ( θ , x ) i = 0 has at mos t one solution θ has to b e check e d by ha nd, while the fact that ∂ G /∂ θ do es not v anish at the sa me time is equiv alen t to the rank condition rank  G ( θ , x ) , ∂ G ∂ θ ( θ, x )  = dim  Range G ( θ , x ) + Range ∂ G ∂ θ ( θ, x )  = n . (45) It is true for the Kepler problem and used in [9] to s how that the discontin uities in (42) are always “ π -s ing ularities”, i.e. the control u ∗ switches to its opp osite. 4. Kepler con trol systems. W e ca ll Kepler c ontr ol system with small co n trol a family of control system o n S 1 × M o f the form ( K ε )  ˙ θ = ω ( θ , x ) + g ( θ, x ) v ˙ x = G ( θ, x ) v , k v k ≤ ε , (46 ) where G and g can b e v iew ed, with the same conv en tion is in (1), as n × m and 1 × m matrices smo othly dep ending on ( θ, x ) and ω is a smo oth function S 1 × M → R that remains larger than a str ictly p ositive co nstan t: ω ( θ , x ) ≥ k ω > 0 ∀ ( θ , x ) ∈ S 1 × M . (47) In fact, this is an affine control s ystem on S 1 × M ˙ ξ = f 0 ( ξ ) + m X i =1 v i f i ( ξ ) (48) with ξ = ( θ , x ) , f 0 = ω ∂ ∂ θ and, for 1 ≤ i ≤ m , the smo oth vector field f i is represented b y the i th column o f the matrix notations G and g . If, in (48), one only assumes that, all s olutions of ˙ ξ = f 0 ( ξ ) a r e p erio dic, additional conditions ar e needed for the orbits to induce a nice folia tion that splits the state manifold in to a pro duct M × S 1 . 4.1. Relation with fast oscil lating systems. F or a so lution t 7→ ( θ ( t ) , x ( t )) of ( K ε ) in (46), let Θ ( t ) b e the cumulated angle i.e. Θ ( . ) is contin uous [0 , T ] → R with Θ( t ) ≡ θ ( t ) mo d 2 π for all t a nd Θ(0) ∈ [0 , 2 π ) , and define a new “time” λ = R ( t ) ∆ = ε (Θ( t ) − Θ(0)) . (49) 16 A. BOMBRUN AND J.-B . POMET T a k ing ε 0 small enough so that | ω ( θ , x ) + ε g ( θ , x ) u | > k ω / 2 for x ∈ K , k u k ≤ 1 , ε < ε 0 , one has d R / d t > ε k ω / 2 hence R is strictly increa sing and o ne-to-one, and k ω 2 ε t ≤ R ( t ) ≤ k ω ε t with k ω = sup S 1 × K ω + ε 0 sup S 1 × K k g k . (50) Then λ 7→ e x ( λ ) = x ( R − 1 ( λ )) is a so lution o f the system ( e Σ θ 0 ,ε ) d e x d λ = G ( θ 0 + λ ε , e x ) b u ω ( θ 0 + λ ε , e x ) + ε g ( θ 0 + λ ε , e x ) b u , k b u k ≤ 1 , (51) asso ciated with the control λ 7→ b u ( λ ) = v ( R − 1 ( λ )) /ε . Except for the term εg b u in the denominator, this is a fas t o s cillating system (1) with G = G/ω . W e now apply §3. 4.2. A v erage con trol system. The definition uses ω defined by 1 ω ( x ) = 1 2 π Z 2 π 0 d θ ω ( θ , x ) . (52) Definition 4.1 ( A v erag e control system of Kepler control systems ). The aver age c ontro l system of the Kepler c ontr ol system (46 ) is the differ ential inclusion ˙ x ∈ E ( x ) (53) with E define d by (3) u s ing G : M × L ∞ ( S 1 , R m ) → T M define d by G ( x, U ) = ω ( x ) 1 2 π Z 2 π 0 G ( θ, x ) ω ( θ , x ) U ( θ )d θ (54) inste ad of (2) . Solutions ar e define d as in Definition 3.2. R emark 4.2. This is almost Definition 3 .2 a pplied to (51), which is equiv alent to (46) via time changes, exc ept : (i) the term εg b u in the denominator of (51) ha s b een discarded, (ii) the r ig h t-hand side has been multiplied by ω ( x ) . 4.3. Con v ergence Theorem. The counterpart of Theorem 3 .7 is: Theorem 4.3 ( Con vergence for K e pler control systems ). 1. L et x 0 ( . ) : [0 , T ] → M b e an arbitr ary solution of (53 ) and θ 0 ∈ S 1 . Ther e exist a family of me asur able functions u ε ( . ) : [0 , T ] → B m , indexe d by ε > 0 , and p ositive c onstants c, ε 0 , su ch that, if t 7→ ( θ ε ( t ) , x ε ( t )) is t he solution of (46) with c ontro l u = u ε ( t ) and initial c ondition ( θ ε (0) , x ε (0)) = ( θ 0 , x 0 (0)) , it is define d on [0 , T /ε ] for ε sm al ler than ε 0 and d ( x ε ( t ) , x 0 ( εt ) ) < c ε , t ∈ [0 , T ε ] , 0 < ε < ε 0 , (55) thus τ 7→ x ε ( τ /ε ) c onver ges un ifo rmly on [0 , T ] to τ 7→ x 0 ( τ ) when ε t ends to zer o. 2. L et K b e a c omp act subset of M , ( ε n ) n ∈ N a de cr e asing se quenc e of p ositive r e al numb ers c onver ging to zer o, and  θ n ( . ) , x n ( . )  : [0 , T /ε n ] → S 1 × K a solu- tion of syst em (46) for e ach n , with ε = ε n and some c ontr ol u = u n ( t ) , u n ( . ) ∈ L ∞ ([0 , T /ε n ] , R m ) , k u n ( . ) k ∞ ≤ 1 . Then the se quenc e  τ 7→ ( x n ( τ /ε n ))  n ∈ N is c om- p act for t he top olo gy of uniform c onver genc e on [0 , T ] and the limit of any c onver ging sub-se quenc e is a solution x ∗ ( . ) of t he aver age differ ential inclusion (53) . A VERAGED CO N TR OL SYSTEM 17 Pr o of . W e a s sume that M is R n , d the Euclidean distance and all vector fields hav e a common compact supp ort, hence all maps shar e a global Lipschitz co nstan t and a global b ound; by “ a co nstan t”, we mea n a n um ber that dep ends o nly o n these bo unds a nd L ips c hitz constants. It is left to the r e a der to chec k that, as for the pro of of Theorem 3 .7, the pres en t pro of extends to M with any distance d des cribed in §2.3. Let τ 7→ x 0 ( τ ) b e a solution of (53) on [0 , T ] . Define P ( . ) by P ( τ ) = Z τ 0 ω ( x 0 ( t )) d t ( 56) and b x 0 ( . ) by b x 0 ( λ ) = x 0 ( P − 1 ( λ )) . The la tter is a solution on [0 , P ( T )] of d b x 0 d λ ∈ 1 ω ( b x 0 ) E ( b x 0 ) (57) with E defined by (3) and (54). This is the average system (in the sens e of Definition 3.2) of the fast o scillating co n trol s ystem ( b Σ θ 0 ,ε ) d b x d λ = G ( θ 0 + λ ε , b x ) b u ω ( θ 0 + λ ε , b x ) , k b u k ≤ 1 . (58) Theorem 3.7 (P o in t 1 ) yields a fa mily of c o n trols b u ε such that the solutions b x ε ( . ) o f ( b Σ θ 0 ,ε ) with initial condition b x 0 (0) and c on trol b u ε conv erge to b x 0 ( . ) unifor mly: d ( b x ε ( λ ) , b x 0 ( λ )) ≤ c ′ ε for a ll λ ∈ [0 , P ( T )] (59) for some constant c ′ . F or each ε , let e x ε ( . ) b e the so lution of ( e Σ θ 0 ,ε ) — see (51) — with same initial condition a nd sa me co n trol. Since (51 ) ca n b e re - written as d e x d λ =  1 − ε g ( θ 0 + λ ε , e x ) b u ω ( θ 0 + λ ε , e x ) + ε g ( θ 0 + λ ε , e x ) b u  G ( θ 0 + λ ε , e x ) b u ω ( θ 0 + λ ε , e x ) , (60) the norm of the difference b etw een the right-hand sides of ( e Σ θ 0 ,ε ) and ( b Σ θ 0 ,ε ) is bo unded by k ε for so me constant k > 0 ; classic a l theorems on smo oth dep endence of solutions on “ parameters” yield so me c onstan t c ′′ such that d ( e x ε ( λ ) , b x ε ( λ )) ≤ c ′′ ε for a ll λ ∈ [0 , P ( T )] . (61) Then define t = T ( λ ) = 1 ε Z λ 0 d ℓ ω ( θ 0 + ℓ ε , e x ε ( ℓ )) + ε g ( θ 0 + ℓ ε , e x ε ( ℓ )) b u ε ( ℓ ) (62) and the controls t 7→ u ε ( t ) by b u ε ( λ ) = u ε ( T ( λ )) ; the solutions x ε ( . ) of (46) with these controls are given by e x ε ( λ ) = x ε ( T ( λ )) , and one therefore has d ( x ε ( T ◦ P ( τ )) , x 0 ( τ )) < ( c ′ + c ′′ ) ε, τ ∈ [0 , T ] . (63) Now, on the one hand, (62) yields T ( P ( τ )) = 1 ε Z P ( τ ) 0 d ℓ ω ( θ 0 + ℓ ε , b x ε ( ℓ )) + ρ (64) with ρ = 1 ε Z P ( τ ) 0  b ω ( ℓ ) − e ω ( ℓ ) b ω ( ℓ ) + ε g ( ℓ ) b u ε ( ℓ ) e ω ( ℓ ) + ε g ( ℓ ) b u ε ( ℓ )  d ℓ e ω ( ℓ ) 18 A. BOMBRUN AND J.-B . POMET where e ω ( ℓ ) stands for ω ( θ 0 + ℓ ε , e x ε ( ℓ )) , b ω ( ℓ ) stands for ω ( θ 0 + ℓ ε , b x ε ( ℓ )) , and g ( ℓ ) stands for g ( θ 0 + ℓ ε , e x ε ( ℓ )) ; using (61) a nd Lipschitz contin uit y of ω to bo und the first term in the integral, this implies that | ρ | is b ounded by a co nstan t. On the other hand, one has, ac c ording to (56) , τ = R P ( τ ) 0 d λ/ ω ( b x 0 ( λ )) . Dev eloping ω a ccording to its definition (52), in which we add θ 0 + λ ε to θ without changing the integral due to p erio dicity , τ is a lso equal to 1 2 π RR θ ∈ S 1 , 0 ≤ λ ≤ P ( τ ) d λ d θ /ω ( θ 0 + λ ε + θ , b x 0 ( λ )) . Finally , p e rforming the change of v ariable λ = ℓ − εµ ( θ ) , with µ ( θ ) defined in §2 .4, yields τ = Z P ( τ )+ εµ ( θ ) εµ ( θ ) 1 2 π Z 2 π 0 d θ ω ( θ 0 + ℓ ε , b x 0 ( ℓ − εµ ( θ ))) ! d ℓ Using (59), the fact that | µ ( θ ) | < 2 π and Lipschitz co n tin uit y of b oth b x 0 and ω , we deduce from this and equation (64) that   T ◦ P ( τ ) − τ ε   ≤ | ρ | + k ′ for so me con- stant k ′ and finally , using the fact that x ε ( . ) is Lipschitz contin uous with constant 2 ε sup k G k / k ω , one has d ( x ε ( T ◦ P ( τ )) , x ε ( τ ε )) < c ′′′ ε for some consta n t c ′′′ . This and equation (63) imply implies p oint 1 of the theorem, with c = c ′ + c ′′ + c ′′′ in (55). F or p oint 2, co ns ider  θ n ( . ) , x n ( . )  : [0 , T /ε n ] → S 1 × K a solution o f sys tem (46) with ε = ε n and so me control u = u n ( t ) . F ollowing (49)–(51) and setting λ = R n ( t ) (w e write R n bec a use R in (49) is co nstructed for sy stem ( e Σ θ 0 ,ε n ) a nd th us depends o n n ), one asso ciates to these x and u a control λ 7→ e u n ( λ ) and a so lution λ 7→ e x n ( λ ) of ( e Σ θ 0 ,ε n ) . The s olutions λ 7→ b x n ( λ ) o f ( b Σ θ 0 ,ε n ) with sa me control and initial condition satisfy , for the same rea s ons a s (61), d ( b x n ( λ ) , e x n ( λ )) < c ′′ ε n for some co nstan t c ′′ . By Theorem 3 .7 (Poin t 2), the sequence ( b x n ) is co mpact a nd subsequences conv erge to solutions λ 7→ b x 0 ( λ ) of (57) , hence the same subsequences of ( e x n ) co n verge a s well, and, with τ = Q ( λ ) ∆ = R λ 0 d ℓ ω ( b x ( ℓ )) , the maps τ 7→ e x n ( Q − 1 ( τ )) = x n (( Q ◦ R n ) − 1 ( τ )) conv erge to a so lution τ 7→ x 0 ( τ ) = b x 0 ( Q − 1 ( τ )) of the average system (53 ), with distance les s than c ′ ε n for s ome c o nstan t c ′ . Using the same ar gumen t as in Poin t 1 for T ◦ P ( τ ) , one gets a b ound for | ( Q ◦ R n ) − 1 ( τ ) − τ ε n | and, fo r some co nstan t c ′′′ , d ( x n (( Q ◦ R n ) − 1 ( τ )) , x n ( τ ε )) ≤ c ′′′ ε n . Poin t 2 is prov ed. 4.4. Dimension of E ( x ) . In §3.3, and in particular in Pro positio n 3 .10, G can simply be replaced with G . It is how ever interesting to g iv e a mo re intrinsic charac- terization of r ( θ , x ) and thus of dim E ( x ) . Proposition 4.4. If (46 ) and (48) r epr esent t he same c ontro l system, t hen dim  X j ∈ N Range ∂ j G ∂ θ j ( θ, x )  = − 1 + rank  { f 0 ( θ, x ) } ∪ n ad j f 0 f k ( θ, x ) , j ∈ N , 1 ≤ k ≤ m o . (65) Pr o of . Straig h tforward computation using the fact that f 0 = ∂ /∂ θ . Note that the right-hand side is r ( θ , x ) . Propo sition 3.1 0 a pplies, with this defi- nition of r . In particular , the “full rank c a se” b ecomes: Proposition 4.5. If the ve ctor fields f 0 and ad j f 0 f k , 1 ≤ k ≤ m , j ∈ N sp an the whole tangent sp ac e of S 1 × M , then E ( x ) has a nonempty int erior for al l x . 4.5. The function H ( x, p ) . Instead of (5), H has to b e taken a s follows, with ω defined in (52): H ( x, p ) = ω ( x ) 1 2 π Z 2 π 0    p, G ( θ, x ) ω ( θ , x )    d θ . (66) A VERAGED CO N TR OL SYSTEM 19 The characterization of E ( x ) in Prop osition 3 .4 is unchanged. In the “full r ank case”, the results from §3.4 on the deg r ee of differentiabilit y a pply without a change. 4.6. Application to the mini m um time probl em. As in §3.5, but for the Kepler sy stem (46), let x 0 , x 1 be fixed, call T ε the minimum time such that, from some θ 0 , θ 1 , ( θ 1 , x 1 ) ca n b e r eac hed from ( θ 0 , x 0 ) in system ( K ε ) (obviously T ε → + ∞ as ε → 0 ) and T 0 the minim um time such that x 1 can b e reached fro m x 0 in the av erage s y stem (53). The equiv a len t of Theorem 3.1 8, with a similar pr oo f, using Theorem 4.3, is: Theorem 4.6. In the ful l r ank c ase, one has lim sup ε → 0 εT ε ≤ T 0 (henc e ε T ε is b ounde d as ε → 0 ). If, for al l ε > 0 smal l enough, ther e is a minimizing solution ( θ ε , x ε ) : [0 , T ε ] → S 1 × M and they al l r emain in a c ommon c omp act subset of M , then al l ac cumulation p oints (as ε → 0 ) of t he c omp act family ( τ → x ε ( τ ε )) ε> 0 in C 0 ([0 , T 0 ] , M ) ar e minimizi ng for t he aver age system and lim ε → 0 ε T ε = T 0 . The Hamiltonian for minimum time for the average system is g iv en by (66); one has to p erform the time scaling desc r ibed in §4.1 to hav e a r esult like Theorem 3.2 1 and the simple “commutation b et ween averaging and wr iting Hamiltonian” noted in Remark 3.20. Let us translate in terms of (46) the sufficient condition for existence of a Hamiltonian flow given b y Theore m 3.21: Theorem 4.7. In t he ful l r ank c ase, assume t hat h p, G ( θ , x ) i and h p, ∂ G/∂ θ ( θ , x ) i do not vanish simultane ously outside { p = 0 } , that θ 7→ h p, G ( θ , x ) i vanishes at most onc e for e ach ( x, p ) ∈ T ∗ M , p 6 = 0 , and that rank G ( θ, x ) = m for e ach ( θ , x ) ∈ S 1 × M . Then (43) , with H given by (66) , has a u nique solution for any initial c ondition. The discussion that follows Theor em 3.2 1 a lso applies to the ab ov e; le t us mention that, o nce it has b een check ed that, for each ( x, p ) , h p, G ( θ , x ) i v anishes for at most one θ , the other conditions a re guaranteed if (45) ho lds with G r eplaced b y G o r , in terms of the vector fields in (48), if, for all ξ = ( θ , x ) , rank { f 0 ( ξ ) , f 1 ( ξ ) , . . . , f m ( ξ ) , ad f 0 f 1 ( ξ ) , . . . , ad f 0 f m ( ξ ) } = n + 1 . (67) W e prov e in the nex t section that the a bov e conditions are true for the planar control 2-b o dy problem. 5. Application to the con trolled 2-b o dy system. In this section we study some prop erties of the planar co n trol system and demonstrate that it satisfies the condition of Theor em 3 .21 o n the doma in o f non-deg enerated e lliptic orbits. 5.1. Planar con trol 2-b o dy system. It is class ic a lly des cribed b y some first in tegrals of the free mov ement —here the semi-ma jor axis a and the eccentricit y vector ( e x , e y ) — and one angle L followin g the dynamics; we restrict to the set of non-degenera ted elliptic o rbits ro tating in the dir ect sense, i.e. the state space is S 1 × M with M = { ( a, e x , e y ) ∈ R 3 , a > 0 a nd e x 2 + e y 2 < 1 } . The co n trol u = ( u t , u n ) is expressed in the tangential-normal fra me a nd the sy stem r eads: d d t     a e x e y L     = 1 a 3 / 2     0 0 0 w ( e x , e y , L )     + √ a     2 a a a ( e x , e y , L ) 0 2 a x ( e x , e y , L ) b x ( e x , e y , L ) 2 a y ( e x , e y , L ) b y ( e x , e y , L ) 0 0      u t u n  (68) with w ( e x , e y , L ) = (1 + e x cos L + e y sin L ) 2 (1 − e 2 ) 3 / 2 , 20 A. BOMBRUN AND J.-B . POMET a a ( e x , e y , L ) = p 1 + e 2 + 2 e x cos L + 2 e y sin L √ 1 − e 2 , a x ( e x , e y , L ) = √ 1 − e 2 p 1 + e 2 + 2 e x cos L + 2 e y sin L ( e x + cos L ) , a y ( e x , e y , L ) = √ 1 − e 2 p 1 + e 2 + 2 e x cos L + 2 e y sin L ( e x + cos L ) , b x ( e x , e y , L ) = √ 1 − e 2 p 1 + e 2 + 2 e x cos L + 2 e y sin L × − 2 e y + ( e 2 x − e 2 y − 1) sin L − 2 e x e y cos L 1 + 2 e x cos L + 2 e y sin L , b y ( e x , e y , L ) = √ 1 − e 2 p 1 + e 2 + 2 e x cos L + 2 e y sin L × 2 e x + ( e 2 x − e 2 y + 1) cos L + 2 e x e y sin L 1 + 2 e x cos L + 2 e y sin L . The eccentricit y e is the norm of the eccentricit y vector, e = q e 2 x + e 2 y . Low thrust translates into k u k ≤ ε for a small ε . R emark 5.1. This is indeed a “Kepler control system” o f the type (46) except that ω = w /a 3 / 2 is, althoug h strictly po sitiv e, not b o unded fro m b elow by a p ositive con- stant on S 1 × M . There is s uch a low erb ound if one replaces M b y M ¯ c = { ( a, e x , e y ) ∈ R 3 , a > 0 and e x 2 + e y 2 < ¯ c } with ¯ c < 1 . Str ictly sp eaking, the res ults of the pap er hav e to b e applied in M ¯ c , ¯ c < 1 . How ever, Theorems 4.3 or 4.7, for instance, may be applied in M b ecause ea ch statement may ultimately be restricted to a compact subset of M , itself included in some M ¯ c , ¯ c < 1 . The Ha miltonian that b oth defines the av erage s ystem acco rding to (6) and yields the Hamiltonian s ystem gov erning extrema ls for minimum time is given by (66). Since R 2 π 0 d L/ w ( e x , e y , L ) = 2 π , it ca n be expres sed as H ( a, e x , e y , p a , p e x , p e y ) = √ a H ( e x , e y , ap a , p e x , p e y ) with H ( e x , e y , A, X, Y ) = 1 2 π Z 2 π 0 k ( A X Y ) G ( e x , e y , L ) k , G ( e x , e y , L ) =   2 a a / w 0 2 a x / w b x / w 2 a y / w b y / w   . 5.2. Hamiltonian flo w. Theorem 4.7 applies to this system. Indeed: Proposition 5.2. F or e e ach ( e x , e y , a ) with e x 2 + e y 2 < 1 and a > 0 , and e ach ( A, X , Y ) 6 = (0 , 0 , 0) , the ve ctor ( A X Y ) G ( e x , e y , L ) vanishes for at most one angle L . Pr o of . Removing denominator s, the equations A a a + X a x + Y a y = 0 and X b x + Y b y = 0 can b e written:  2 e x A + 2 (1 − e 2 ) X  cos L +  2 e y A +2 (1 − e 2 ) Y  sin L = − (1 + e 2 ) A − 2 e x (1 − e 2 ) X − 2 e y (1 − e 2 ) Y  − 2 e x e y X + ( e 2 x − e 2 y + 1) Y  cos L +  ( e 2 x − e 2 y − 1) X + 2 e x e y Y  sin L = 2 e y X − 2 e x Y . A VERAGED CO N TR OL SYSTEM 21 If ∆ =     2 e x A + 2(1 − e 2 ) X 2 e y A + 2 (1 − e 2 ) Y − 2 e x e y X + ( e 2 x − e 2 y + 1) Y ( e 2 x − e 2 y − 1) X + 2 e x e y Y     is nonzer o, there is clear ly at most one solution L . If ∆ = 0 , there exists λ 6 = 0 s uc h that 2 e x A + 2 (1 − e 2 ) X = λ  − 2 e x e y X + ( e 2 x − e 2 y + 1) Y  , 2 e y A + 2(1 − e 2 ) Y = λ  ( e 2 x − e 2 y − 1) X + 2 e x e y Y  , and there may b e a solution to the system a bove o nly if (1 + e 2 ) A + 2 e x (1 − e 2 ) X + 2 e y (1 − e 2 ) Y = − 2 λ ( e y X − e x Y ) These three equa tions fo rms a linear system in ( A, X , Y ) , M ( A, X , Y ) T = 0 with M =   2 e x 2(1 − e 2 + λe x e y ) − λ ( e 2 x − e 2 y + 1) 2 e y − λ ( e 2 x − e 2 y − 1) 2(1 − e 2 − λe x e y ) (1 + e 2 ) 2  e x (1 − e 2 ) + λe y  2  e y (1 − e 2 ) − λe x    . A brief computation gives det M = (1 − e ) 3 (1 + e ) 3 ( λ 2 + 4) , strictly p ositive when 0 ≤ e < 1 . Hence M ( A, X , Y ) T = 0 implies ( A, X , Y ) = 0 . Since the rank o f G is obviously e q ual to 2 a nd the rank o f { G , ∂ G /∂ L } equal to 3 for a n y ( e x , e y , L ) , the hypotheses of Theorem 4.7 are s a tisfied by the planar control 2-b o dy system, and it guarantees existence of a flo w for the Hamiltonia n system gov erning the extremals of minim um time fo r its average s ystem. 6. Conclusion. A ttempting to for m ulate a control theo ry equiv alen t to the av- eraging theorems for ODEs naturally lea ds to, and justifies, the notion o f av erage control sys tem introduced in this paper . It has a co nce ptual imp ortance as well a s, for instance, a pplications to approximation o f minimum time control. Besides its definition and description, we gav e res ults on its r e g ularity and on the dimension o f its velocity s e t (“n um b er of inputs”). These ar e ho wev er mostly a starting p oin t. The reg ularity of H has to b e further explored when the conditions o f Theorem 3.21 do not hold, see the last para graph of §3. It has already allowed us to g iv e (with r e strictions on the eccentricities, s ee Re- mark 5.1) a pr oo f [6] that the minimum time b et ween 2 ellipses grows like 1 /ε fo r the planar 2-b o dy problem. Here also, prog r ess must b e made. Explicit computation of the av erage sy stem a nd its extre mals for the 2-b o dy problem ha s to b e co nducted. A c knowledgemen ts. The authors ar e indebted to Jana Němcová from Institute of Chemical T echnology , Prag ue, for a careful pr oo f-reading of the dra ft manuscript, and to tw o ano n ymous referees from this journal for extremely co nstructiv e re views that make this pap er considera bly easier to read than the or iginal s ubmiss io n. App endix. Pro of of Propo sition 3.16. Pr o of of Point 1. The in tegral in (37) is w ell defined (its integrand is b ounded) and, b y (36 ) and Lebesg ue conv ergence theorem, it is contin uous with r espect to X and h . Let us pr o ve that this d H is the deriv ativ e of H . Since V is smo oth, one ha s k V ( θ , X + h ) − V ( θ , X ) − ∂ V ∂ X ( θ, X ) .h k ≤ k k h k 2 , (69) where ∂ V ∂ X ( θ, X ) is smo oth with resp ect to ( θ , X ) a nd k is some lo cal constant. No w, assuming V ( θ , X ) 6 = 0 , one ha s 22 A. BOMBRUN AND J.-B . POMET k V ( θ , X + h ) k − k V ( θ, X ) k =  V ( θ , X + h ) − V ( θ , X )     V ( θ , X ) k V ( θ , X ) k  + a ( θ , X , h ) k V ( θ , X + h ) − V ( θ , X ) k 2 k V ( θ , X ) k + k V ( θ , X + h ) k with | a ( θ , X , h ) | ≤ 2 . Hence, from (69) and (37), one has , for some lo cal co nstan t k ′ , k H ( X + h ) − H ( X ) − d H ( X ) .h k k h k ≤ k ′ 2 π Z 2 π 0  k h k + k V ( θ , X + h ) − V ( θ , X ) k k V ( θ , X ) k + k V ( θ , X + h ) k  d θ for k h k sma ll enough. F or fixed X and h → 0 , the in tegrand in the right-hand s ide is bo unded by 1 + k h k and c o n verges to zero for θ outside the set { θ ∈ S 1 , V ( θ, X ) = 0 } : b y (36) a nd Leb esgue c o n vergence theorem, the right-hand si de tends to zero. Let us now state tw o lemmas that are needed in the pro of of Poin t 2. Lemma A.1. Ass u me t hat ¯ X ∈ e Z and (38) is satisfie d. Ther e is a neighb orho o d U of ¯ X in O d and a smo oth map b χ : U → S 1 such that, for ( θ , X ) ∈ U , one has V ( θ , X ) = 0 only if θ = b χ ( X ) , and  ∂ V ∂ θ ( b χ ( X ) , X )     V ( b χ ( X ) , X )  = 0 , X ∈ U . (70) Pr o of . F ro m (38 .a), Z = { ( θ , X ) ∈ S 1 × O d , V ( θ , X ) = 0 } is a s mooth submani- fold o f S 1 × O d and fr om (38.c), e Z g iv en by (35) a s mooth submanifold of O d , b oth of dimension d + 1 − m , and the pro jection π : S 1 × O d → O d induces a diffeomor phism Z → e Z w ho se in verse is of the for m x 7→ ( χ ( x ) , x ) with χ a smo oth map e Z → S 1 that satisfies, for all X ∈ e Z : V ( θ , X ) = 0 if an only if θ = χ ( x ) . Consider the map T : S 1 × O d → R given by T ( θ , X ) =  ∂ V ∂ θ ( θ, X )   V ( θ , X )  . Let ¯ X b e in e Z ; since V ( χ ( ¯ X ) , ¯ X ) = 0 , o ne has T ( χ ( ¯ X ) , ¯ X ) = 0 a nd ∂ T /∂ θ ( χ ( ¯ X ) , ¯ X ) = k ∂ V ∂ θ ( χ ( ¯ X ) , ¯ X ) k 2 , nonzero from ass umption (38.b): the implicit function theorem yields a unique map b χ fro m a neighborho o d U o f ¯ X in O d to a neighbor hoo d of χ ( ¯ X ) in S 1 such that θ = b χ ( X ) solves T ( θ , X ) = 0 ; it must therefore coincide with χ in U ∩ e Z and satisfies the lemma. Lemma A .2. Assume t hat ¯ X ∈ e Z and (38) is satisfie d. Ther e exist a neighb or- ho o d U of ¯ X in O d , lo c al c o or dinates x 1 , . . . , x d define d on U , and smo oth maps P : U → S O ( m ) , α : U → R , and W : S 1 × U → R m such that, with X I =     x 1 . . . x m − 1     , V ( θ , X ) = P ( X ) h  X I α ( X ) ( θ − b χ ( X ))  + ( θ − b χ ( X )) 2 W ( θ, X ) i (71) = P ( X ) h   X I 0   + ( θ − b χ ( X )) W 1 ( θ, X ) i (72) with W 1 ( θ, X ) =   0 m − 1 α ( X )   + ( θ − b χ ( X )) W ( θ , X ) , (73) in S 1 × U , wher e α is b ounde d fr om b elow: 0 < α 0 < α ( X ) , X ∈ U . F urthermor e, for a c onstant K 3 > 0 , one has, for al l ( θ, X ) ∈ S 1 × U , k V ( θ , X ) k ≥ K 3 q k X I k 2 + α ( X ) 2 ( θ − b χ ( X )) 2 , (74) and X I = 0 ⇒ k W 1 ( θ, X ) k ≥ K 3 . (75) A VERAGED CO N TR OL SYSTEM 23 Pr o of . The map X 7→ ∂ V ∂ θ ( b χ ( X ) , X ) is no nze r o for X = ¯ X , hence it do es not v anish o n a sufficiently s mall neig h bo rho od U of ¯ X , a nd one may write ∂ V ∂ θ ( b χ ( X ) , X ) = P ( X )  0 m − 1 α ( X )  , α ( X ) > α 0 > 0 . (76) Define v 1 , . . . , v m , smo oth maps S 1 × U → R by    v 1 ( θ, X ) . . . v m ( θ, X )    = P − 1 ( X ) V ( θ, X ) . (77) F or i b et ween 1 and m − 1 , ∂ v i ∂ θ ( b χ ( X ) , X ) = 0 from (76), and v i ( b χ ( ¯ X ) , ¯ X ) = 0 from Lemma A.1 and, using(38.a), the ra nk o f the ma p X 7→ ( v 1 ( b χ ( X ) , X ) , . . . , v m − 1 ( b χ ( X ) , X )) is m − 1 at X = ¯ X : o n a p o ssibly smaller neig h bor hoo d U , there are lo cal co ordinates x 1 , . . . , x d such that v i ( θ, X ) = x i + ( θ − b χ ( X )) 2 W i ( θ, X ) for i ≤ m − 1 and for some smo oth W i ; substitutin g (76) a nd (77) in (7 0) implies v m ( b χ ( X ) , X ) = 0 , hence v m ( θ, X ) = α ( X ) ( θ − b χ ( X )) + W m ( θ, X ) ( θ − b χ ( X )) 2 for a smo oth W m ; (71) is proved. P ossibly restricting U to a subset with compact closure, k W ( θ , X ) k is b ounded on S 1 × U ; if | θ − b χ ( X ) | ≤ 1 2 α 0 / max k W k , then (74) ho lds with K 3 = 1 2 according to (71 ); on the set where | θ − b χ ( X ) | ≥ 1 2 α 0 / max k W k , V do es not v anish and hence ( k X I k 2 + α ( X ) 2 ( θ − b χ ( X )) 2 ) 1 / 2 / k V ( θ , X ) k is bo unded from b elow; (74) is prov ed, with K 3 smaller than this bo und and than 1 2 . F ro m (73) , W 1 ( b χ ( ¯ X ) , ¯ X ) 6 = 0 b ecause α do es no t v anish; from assumption (38.b) and (7 2 ) (where X I = 0 if X = ¯ X ), W 1 ( θ, ¯ X ) 6 = 0 if θ 6 = b χ ( ¯ X ) , hence W 1 do es not v anish o n S 1 × { ¯ X } ; it is therefore bo unded from b elow on S 1 × U with U a s ma ll enough neighborho o d of ¯ X : (75) holds with K 3 smaller than this b ound. Pr o of of Pr op osition 3.16 (Point 2) . W e use [ − π, π ] inste ad of [0 , 2 π ] as an int erval of inte gr ation . Let h ∈ R d , w ith k h k = 1 . F r om (37), one has, for so me constant e K using b ounds on the deriv ativ es of the smo oth V , | d H ( X ) .h − d H ( Y ) .h | ≤     1 2 π Z π − π  ∂ V ∂ X ( θ, X ) .h − ∂ V ∂ X ( θ, Y ) .h     V ( θ , X ) k V ( θ , X ) k  d θ     +     1 2 π Z π − π  ∂ V ∂ X ( θ, Y ) .h     V ( θ , X ) k V ( θ , X ) k − V ( θ , Y ) k V ( θ , Y ) k  d θ     ≤ e K k X − Y k + e K 2 π     Z π − π V ( θ , X ) k V ( θ , X ) k d θ − Z π − π V ( θ , Y ) k V ( θ , Y ) k d θ     . Finally , defining b V ( ϕ, X ) = V ( b χ ( X ) + ϕ, X ) , c W 1 ( ϕ, X ) = W 1 ( b χ ( X ) + ϕ, X ) , (78) and making a different change of v a riables in the la st tw o in tegrals , one has k d H ( X ) .h − d H ( Y ) .h k ≤ e K k X − Y k + e K 2 π Z π − π   b V ( ϕ, X ) k b V ( ϕ, X ) k − b V ( ϕ, Y ) k b V ( ϕ, Y ) k   d ϕ ≤ e K k X − Y k + e K π Z π − π k b V ( ϕ, X ) − b V ( ϕ, Y ) k k b V ( ϕ, X ) k d ϕ (79) 24 A. BOMBRUN AND J.-B . POMET where the las t inequality us e s the fact k u k u k − v k v k k ≤ 2 min { k u − v k k u k , k u − v k k v k } , and a lso holds with k b V ( ϕ, Y ) k instead of k b V ( ϕ, X ) k in the denomina to r. Now let us use Lemma A.2, let X = ( x 1 , . . . , x d ) and Y = ( y 1 , . . . , y d ) in these co or dinates; from (72), one has , with c W 1 defined by (78), b V ( ϕ, X ) = P ( X ) h   X I 0   + ϕ c W 1 ( ϕ, X ) i , b V ( ϕ, Y ) = P ( Y ) h   Y I 0   + ϕ c W 1 ( ϕ, Y ) i . (80 ) Hence b V ( ϕ, X ) − b V ( ϕ, Y ) =  P ( X ) − P ( Y )  P ( X ) − 1 b V ( ϕ, X ) + P ( Y ) h ϕ  W 1 ( ϕ, X ) − W 1 ( ϕ, Y )  +   X I − Y I 0   i and finally k b V ( ϕ, X ) − b V ( ϕ, Y ) k k b V ( ϕ, X ) k ≤ k P ( X ) − P ( Y ) k + | ϕ | k W 1 ( ϕ, X ) − W 1 ( ϕ, Y ) k k b V ( ϕ, X ) k + k X I − Y I k k b V ( ϕ, X ) k . (81) T wo ca ses are to be distinguished: (i) If X I = Y I = 0 , then ϕ factors out o f b V ( ϕ, X ) and b V ( ϕ, Y ) in (80) a nd the last term in (81) is zero: according to (75), the integrand in (79) is b ounded by k P ( X ) − P ( Y ) k + k c W 1 ( ϕ, X ) − c W 1 ( ϕ, Y ) k K 3 , and fina lly | d H ( X ) .h − d H ( Y ) .h | ≤ K k X − Y k with a co nstan t K that dep ends only on V , the o p en set U and the co or dinates. (ii) If X I 6 = 0 (or Y I 6 = 0 , int erchanging X and Y ), then (8 1), using (74), implies that the integrand in (79) is b ounded by k P ( X ) − P ( Y ) k + 1 K 3 1 α 0 k W 1 ( ϕ, X ) − W 1 ( ϕ, Y ) k + 1 K 3 s k X I − Y I k 2 k X I k 2 + α ( X ) ϕ 2 , but the sa me is also true repla cing α ( X ) with α ( Y ) a nd k X I k 2 with k Y I k 2 ; hence, since k a − b k 2 ≤ 4 ma x {k a k 2 , k b k 2 } , the last term may be re pla ced by 2 K 3 q k X I − Y I k 2 k X I − Y I k 2 +4 α 0 ϕ 2 , whose int egral betw een − π and π is equal to k X I − Y I k K 3 √ α 0 ln(1 + 4 π √ α 0 k X I − Y I k + 8 π 2 α 0 k X I − Y I k 2 ) , which is less than k X I − Y I k ( k 1 + k 2 ln 1 k X I − Y I k ) for some k 1 , k 2 when, s ay , k X I − Y I k 2 √ α 0 < 1 . Finally , since k X I − Y I k is le s s than k X − Y k a nd u 7→ u ln(1 /u ) is nondecreas ing , less than k X − Y k ( k 1 + k 2 ln 1 k X − Y k ) . Cases (i) and (ii) do imply (3 9), p ossibly restricting U so tha t ln 1 k X − Y k ≥ 1 . 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