Abstractions of Hamiltonian control systems

IONS OF HAMILTONIAN CONTROL SYSTEMS PAULO TABUADA AND GEORGE J. PAPPAS Abstra t. Given a ontrol system and a desired property, an abstra ted system is a redu ed system that preserves the property of interest while ignoring modeling detail. In previous work, we onsidered abstra tions of linear and analyti ontrol systems while preserving rea hability properties. In this paper we onsider the abstra tion problem for Hamiltonian ontrol systems, and abstra t systems while preserving their Hamiltonian stru ture. We show how the me hani al stru ture of Hamiltonian ontrol systems an be exploited in the abstra tion pro ess. We then fo us on lo al a essibility preserving abstra tions and provide onditions under whi h lo al a essibility properties of the abstra ted Hamiltonian system are equivalent to the a essibility properties of the original Hamiltonian ontrol system. CDC PAPER ID : CDC01-REG1620 1. Introdu tion Abstra tions of ontrol systems are important for redu ing the omplexity of their analysis or design. From an analysis perspe tive, given a large-s ale ontrol system and a property to be veri ed, one extra ts a smaller abstra ted system with equivalent properties. Che king the property on the abstra tion is then equivalent to he king the property on the original system. From a design perspe tive, rather than designing a ontroller for the original large s ale system, one designs a ontroller for the smaller abstra ted system, and then re nes the design to the original system while in orporating modeling detail. A formal approa h to a modeling framework of abstra tion riti ally depends on whether we are able to onstru t hierar hies of abstra tions as well as hara terize onditions under whi h various properties propagate from the original to the abstra ted system and vi e versa. In [8℄, hierar hi al abstra tions of linear ontrol systems were extra ted using omputationally eÆ ient onstru tions, and onditions under whi h ontrollability of the abstra ted system implied ontrollability of the original system were obtained. This led to extremely eÆ ient hierar hi al ontrollability algorithms. In the same spirit, abstra tions of analyti ontrol systems were onsidered in [13℄. The anoni al onstru tion for linear systems was generalized to analyti systems, yielding a anoni al onstru tion for extra ting abstra tions of nonlinear ontrol systems. The ondition under whi h lo al a essibility of the abstra ted system is equivalent to the lo al a essibility of the original system aptured the linear ondition of [12℄. In this paper, we pro eed in the spirit of [13℄ and onsider abstra tions of Hamiltonian ontrol systems. In Hamiltonian ontrol systems are ompletely spe i ed by ontrolled Hamiltonians. This additional stru ture allow a simpli ation of the abstra tion pro ess sin e the relevant information that must be aptured by the This work was performed while the rst author was visiting the University of Pennsylvania. This resear h is partially supported by DARPA under grant F33615-00-C-1707, DARPA Grant N66001-99-C-8510, the University of Pennsylvania Resear h Foundation, and by Funda ~ ao para a Ciên ia e Te nologia under grant PRAXIS XXI/BD/18149/98. 1 2 PAULO TABUADA AND GEORGE J. PAPPAS abstra ted system is simply the ontrolled Hamiltonian. On the other hand, to be able to relate the dynami s indu ed by the ontrolled Hamiltonians we need to restri t the lass of abstra ting maps to those that preserve the Hamiltonian stru ture. Given a Hamiltonian ontrol system on Poisson manifold M , and a (quotient) Poisson map : M ! N , we present a anoni al onstru tion that extra ts an abstra ted Hamiltonian system on N . This anoni al onstru tion is dual to the onstru tion of [13℄. We then hara terize abstra ting maps for whi h the original and abstra ted system are equivalent from a lo al a essibility point of view. Redu tion of me hani al ontrol systems is a very ri h and mature area [5, 9, 7, 10℄. The approa h presented in this paper is quite di erent from these established notions of redu tion for me hani al systems. When performing an abstra tion one is interested in ignoring irrelevant modeling details. In this spirit one quotients the original model by groups a tions that do not ne essarily represent symmetries. This extra freedom in performing redu tion is balan ed by the fa t that information about the system is lost when performing an abstra tion, whereas when redu ing using symmetries no essential information is lost. However abstra ting a ontrol system and in parti ular an Hamiltonian one is always possible therefore leading to a more general notion of redu tion. The stru ture of this paper is as follows : In Se tion 2 we review Poisson geometry in order to establish notation. In Se tion 3 we present a global de nition of Hamiltonian ontrol systems, and in Se tion 4 we de ne abstra tions of Hamiltonian ontrol systems. In Se tion 5 we obtain a anoni al onstru tion for abstra ting Hamiltonian ontrol systems, and hara terize lo al a essibility equivalen e between the original and the abstra ted system. Se tion 6 illustrated our results on a spheri al pendulum example, and Se tion 7 points to interesting future resear h.2. Mathemati al Preliminaries In this se tion we review some basi fa ts from di erential and Poisson geometry as well as ontrol theory and Hamiltonian ontrol systems, in order to establish onsistent notation. The reader may whish to onsult numerous books on these subje ts su h as [1, 2, 11, 6℄. 2.1. Di erential Geometry. Let M be a di erentiable manifold and TxM its tangent spa e at x 2M . The tangent bundle of M is denoted by TM = [x2MTxM and is the anoni al proje tion map : TM ! M taking a tangent ve tor X(x) 2 TxM TM to the base point x 2M . Dually we de ne the otangent bundle as T M = [x2MT xM , where T xM is the otangent spa e of M at x. Now let M and N be smooth manifolds and : M ! N a smooth map. Given a map : M ! N , we denote by Tx : TxM ! T (x)N the indu ed tangent whi h maps tangent ve tors from TxM to tangent ve tors at T (x)N . A ber bundle is a tuple (B;M; B ; U; fOigi2I), where B, M and U are smooth manifolds alled the total spa e, the base spa e and standard ber respe tively. The map B : B ! M is a surje tive submersion and fOigi2I is an open over of M su h that for every i 2 I there exists a di eomorphism i : 1 B (Oi) ! Oi U satisfying o Æ i = B , where o is the proje tion from Oi U to Oi. The submanifold 1(x) is alled the ber at x 2M . ABSTRACTIONS OF HAMILTONIAN CONTROL SYSTEMS 3 2.2. Poisson Geometry. For the purposes of this paper, it will be more natural to work with Poisson manifolds, rather than symple ti manifolds 1. A Poisson stru ture on manifold M is a bilinear map from C1(M) C1(M) to C1(M) alled Poisson bra ket, denoted by ff; ggM or simply ff; gg, satisfying the following identities ff; gg = fg; fg skew-symmetry (2.1) ff; fg; hgg+ fg; fh; fgg+ fh; ff; ggg = 0 Ja obi identity (2.2) ff; ghg = ff; ggh+ gff; hg Leibnitz rule (2.3) A Poisson manifold (M; f; gM ) is a smooth manifold M equipped with a Poisson stru ture f; gM . Given a smooth fun tion h : M ! R, the Poisson bra ket allows us to obtain a Hamiltonian ve tor eld Xh with Hamiltonian h using LXhf = ff; hg 8f 2 C1(M) (2.4) where LXhf is the Lie derivative of f along Xh. Note that the ve tor eld Xh is well de ned sin e the Poisson bra ket veri es the Leibnitz rule and therefore de nes a derivation on C1(M) ( [10℄). Furthermore C1(M) equipped with f; g is a Lie algebra, also alled a Poisson algebra. Asso iated with the Poisson bra ket there is a ontravariant anti-symmetri two-tensor B : T M T M ! R (2.5) su h that B(x)(df; dg) = ff; gg(x) (2.6) We say that the Poisson stru ture is non-degenerate if the map B# : T M ! TM de ned by dg(B#(df)) = B(df; dg) is an isomorphism for every x 2M . Given a map : (M; f; gM ) ! (N; f; gN) between Poisson manifolds, we say that preserves the Poisson stru ture or that is a Poisson map i ff Æ ; g Æ gM = ff; ggN Æ (2.7) for every f; g 2 C1(N). The lassi al Hamilton equations an be re overed using the Poisson bra ket. Let N be any manifold of dimension n, then M = T N is a Poisson manifold of dimension 2n with natural oordinates given by (qi; pi). The anoni al Poisson bra ket is ff; gg = f qi g pi f pi g qi (2.8) Given a smooth map h :M ! R the Hamiltonian ve tor eld Xh is given in the natural oordinates by: d dtqi = LXhqi = fqi; hg = h pi (2.9) d dtpi = LXhpi = fpi; hg = h qi (2.10) whi h is just the usual form of Hamilton equations. 1More detailed expositions on symple ti and Poisson geometry an be found in [10, 3℄IONS OF HAMILTONIAN CONTROL SYSTEMS 3 2.2. Poisson Geometry. For the purposes of this paper, it will be more natural to work with Poisson manifolds, rather than symple ti manifolds 1. A Poisson stru ture on manifold M is a bilinear map from C1(M) C1(M) to C1(M) alled Poisson bra ket, denoted by ff; ggM or simply ff; gg, satisfying the following identities ff; gg = fg; fg skew-symmetry (2.1) ff; fg; hgg+ fg; fh; fgg+ fh; ff; ggg = 0 Ja obi identity (2.2) ff; ghg = ff; ggh+ gff; hg Leibnitz rule (2.3) A Poisson manifold (M; f; gM ) is a smooth manifold M equipped with a Poisson stru ture f; gM . Given a smooth fun tion h : M ! R, the Poisson bra ket allows us to obtain a Hamiltonian ve tor eld Xh with Hamiltonian h using LXhf = ff; hg 8f 2 C1(M) (2.4) where LXhf is the Lie derivative of f along Xh. Note that the ve tor eld Xh is well de ned sin e the Poisson bra ket veri es the Leibnitz rule and therefore de nes a derivation on C1(M) ( [10℄). Furthermore C1(M) equipped with f; g is a Lie algebra, also alled a Poisson algebra. Asso iated with the Poisson bra ket there is a ontravariant anti-symmetri two-tensor B : T M T M ! R (2.5) su h that B(x)(df; dg) = ff; gg(x) (2.6) We say that the Poisson stru ture is non-degenerate if the map B# : T M ! TM de ned by dg(B#(df)) = B(df; dg) is an isomorphism for every x 2M . Given a map : (M; f; gM ) ! (N; f; gN) between Poisson manifolds, we say that preserves the Poisson stru ture or that is a Poisson map i ff Æ ; g Æ gM = ff; ggN Æ (2.7) for every f; g 2 C1(N). The lassi al Hamilton equations an be re overed using the Poisson bra ket. Let N be any manifold of dimension n, then M = T N is a Poisson manifold of dimension 2n with natural oordinates given by (qi; pi). The anoni al Poisson bra ket is ff; gg = f qi g pi f pi g qi (2.8) Given a smooth map h :M ! R the Hamiltonian ve tor eld Xh is given in the natural oordinates by: d dtqi = LXhqi = fqi; hg = h pi (2.9) d dtpi = LXhpi = fpi; hg = h qi (2.10) whi h is just the usual form of Hamilton equations. 1More detailed expositions on symple ti and Poisson geometry an be found in [10, 3℄ 4 PAULO TABUADA AND GEORGE J. PAPPAS 3. Hamiltonian Control Systems Before de ning Hamiltonian ontrol systems, we present a global de nition of a ontrol systems [11℄. De nition 3.1 (Control System). A ontrol system S = (U; F ) onsists of a ber bundle : U !M alled the ontrol bundle and a smooth map F : U ! TM whi h is ber preserving, that is 0 Æ F = where 0 : TM ! M is the tangent bundle proje tion. Given a ontrol system S = (U; F ), the ontrol distribution D of ontrol system S, is naturally de ned pointwise by D(x) = F ( 1(x)) for all x 2M . The ontrol spa e U is modeled as a ber bundle sin e in general the ontrol inputs available may depend on the urrent state of the system. In lo al oordinates, De nition 3.1 redu es to the familiar _ x = f(x; u) with u 2 1(x). Using this de nition of ontrol systems, the on ept of traje tories of ontrol systems be omes as follows. De nition 3.2 (Traje tories of Control Systems). A urve : I ! M , I R+0 is alled an traje tory of ontrol system S = (U; F ), if there exists a urve U : I ! U satisfying: Æ U = d dt (t) = F ( U ) Again in lo al oordinates, the above de nition says that x(t) is a traje tory of a ontrol system if there exists an input u(t) su h that x(t) satis es _ x(t) = f(x(t); u(t)) and u(t) 2 U(x(t)) = 1(x(t)) for all t 2 I . Hamiltonian ontrol systems are ontrol systems endowed with additional stru ture. The extra stru ture omes from the fa t that they model me hani al systems so they are essentially a olle tion of Hamiltonian ve tor elds parameterized by the ontrol input. The following global and oordinate free des ription of Hamiltonian ontrol systems is inspired from [14℄. De nition 3.3 (Hamiltonian Control Systems). A Hamiltonian ontrol system SH = (U;H) onsists of a ontrol bundle : U ! M over a Poisson manifold (M; f; g) with non-degenerate Poisson bra ket, and a smooth fun tion H : U ! R. With the Hamiltonian ontrol system SH = (U;H) we asso iate the olle tion of Hamiltonian H as the olle tion of all smooth fun tions satisfying H(x) = H( 1(x)) for all x 2 M . This family indu es the ontrol distribution DH de ned pointwise by DH(x) = XH(x), where for all x 2M , XH(x) satis es LXH(x)f = ff;Hg(x), that is LXH(x)f = ff; hg(x) for all h 2 H(x), f 2 C1(M). The map H should be thought of as a ontrolled Hamiltonian sin e it assigns a Hamiltonian fun tion to ea h ontrol input. Note that the ontrol bundle, and the ontrolled Hamiltonian ompletely spe ify the Hamiltonian ontrol system. In parti ular, by xing the ontrol input, one obtains a Hamiltonian ve tor eld. 4. Abstra tions of Hamiltonian Control Systems Given a Hamiltonian ontrol system2 SHM de ned on a Poisson manifold (M; f; gM) our goal is to onstru t a map :M ! N , the abstra tion map or aggregation map that will indu e a new Hamiltonian ontrol system SHN on the lower dimensional Poisson manifold (N; f; gN) having as traje tories ( M ), where M are SHM 2From now on, SHM = (UM ;HM ) or simply SHM denotes a Hamiltonian ontrol system on Poisson manifold (M; f; gM ). ABSTRACTIONS OF HAMILTONIAN CONTROL SYSTEMS 5 traje tories. The on ept of abstra tion map for ontinuous, not ne essarily Hamiltonian, ontrol systems is de ned in [8℄. De nition 4.1 (Abstra ting Maps). Let SM and SN be two ontrol systems on manifolds M and N , respe tively. A smooth surje tive submersion : M ! N is alled an abstra tion or aggregation map i for every traje tory M of SM , ( M ) is a traje tory of SN . Control system SN is alled a -abstra tion of SM . From the above de nition it is lear that an abstra tion aptures all the traje tories of the original system, but may also ontain redundant traje tories. These redundant traje tories are not feasible by the original system and are therefore undesired. Clearly, it is diÆ ult to determine whether a ontrol system is an abstra tion of another at the level of traje tories. One is then interested in a hara terization of abstra tions whi h is equivalent to De nition 4.1 but he kable. To leads to the notion of -related ontrol systems. De nition 4.2 ( -related ontrol systems [8℄). Let SM and SN be two ontrol systems de ned on manifolds M and N , respe tively. Let : M ! N be a surje tive submersion. Then ontrol systems SM and SN are -related i for every x 2M : Tx (DM (x)) DN ( (x)) (4.1) The notion of -related ontrol system is a generalization of the notion of -related ve tor elds ommonly found in di erential geometry. It is also evident that given a ontrol system SM , there is a minimal -related ontrol system SN , up to ontrol parameterization. The relationship between -abstra tions and -related ontrol systems is now given. Theorem 4.3 ([12℄). Let SM and SN be ontrol systems on manifolds M and N , respe tively, and :M ! N a smooth map. Then SM and SN are -related ontrol systems if and only if SN is a -abstra tion of SM . We now onsider these notions for Hamiltonian ontrol systems. Sin e Hamiltonian ontrol systems are uniquely determined by their ontrolled Hamiltonian, the notion of -related ontrol systems spe ializes to Hamiltonian ontrol systems as follows: De nition 4.4 ( -related Hamiltonian ontrol systems). Let SHM and SHN be two Hamiltonian ontrol systems de ned on Poisson manifolds (M; f; gM ) and (N; f; gN), respe tively. Let :M ! N be a surje tive Poisson submersion, and let B be de ned by B = (B# N ) 1 Æ T Æ B# M . Then Hamiltonian ontrol systems SHM and SHN are -related i for all x 2M , B(dHM (x)) dHN ( (x)) (4.2) Although the above de nition is stated in terms of the exterior derivative of the family of Hamiltonian de ning the ontrol systems, a anoni al onstru tion to be presented at se tion 4.8 will allows us to ompute HN dire tly from from HM . The relation between -related Hamiltonian ontrol systems and -abstra tions parallels the general ase. Proposition 4.5. Let SHM and SHN be Hamiltonian ontrol systems on Poisson manifolds (M; f; gM ) and (N; f; gN), respe tively, and : M ! N a smooth Poisson map. Then SHM and SHN are -related if and only if SHN is a -abstra tion of SHM .IONS OF HAMILTONIAN CONTROL SYSTEMS 5 traje tories. The on ept of abstra tion map for ontinuous, not ne essarily Hamiltonian, ontrol systems is de ned in [8℄. De nition 4.1 (Abstra ting Maps). Let SM and SN be two ontrol systems on manifolds M and N , respe tively. A smooth surje tive submersion : M ! N is alled an abstra tion or aggregation map i for every traje tory M of SM , ( M ) is a traje tory of SN . Control system SN is alled a -abstra tion of SM . From the above de nition it is lear that an abstra tion aptures all the traje tories of the original system, but may also ontain redundant traje tories. These redundant traje tories are not feasible by the original system and are therefore undesired. Clearly, it is diÆ ult to determine whether a ontrol system is an abstra tion of another at the level of traje tories. One is then interested in a hara terization of abstra tions whi h is equivalent to De nition 4.1 but he kable. To leads to the notion of -related ontrol systems. De nition 4.2 ( -related ontrol systems [8℄). Let SM and SN be two ontrol systems de ned on manifolds M and N , respe tively. Let : M ! N be a surje tive submersion. Then ontrol systems SM and SN are -related i for every x 2M : Tx (DM (x)) DN ( (x)) (4.1) The notion of -related ontrol system is a generalization of the notion of -related ve tor elds ommonly found in di erential geometry. It is also evident that given a ontrol system SM , there is a minimal -related ontrol system SN , up to ontrol parameterization. The relationship between -abstra tions and -related ontrol systems is now given. Theorem 4.3 ([12℄). Let SM and SN be ontrol systems on manifolds M and N , respe tively, and :M ! N a smooth map. Then SM and SN are -related ontrol systems if and only if SN is a -abstra tion of SM . We now onsider these notions for Hamiltonian ontrol systems. Sin e Hamiltonian ontrol systems are uniquely determined by their ontrolled Hamiltonian, the notion of -related ontrol systems spe ializes to Hamiltonian ontrol systems as follows: De nition 4.4 ( -related Hamiltonian ontrol systems). Let SHM and SHN be two Hamiltonian ontrol systems de ned on Poisson manifolds (M; f; gM ) and (N; f; gN), respe tively. Let :M ! N be a surje tive Poisson submersion, and let B be de ned by B = (B# N ) 1 Æ T Æ B# M . Then Hamiltonian ontrol systems SHM and SHN are -related i for all x 2M , B(dHM (x)) dHN ( (x)) (4.2) Although the above de nition is stated in terms of the exterior derivative of the family of Hamiltonian de ning the ontrol systems, a anoni al onstru tion to be presented at se tion 4.8 will allows us to ompute HN dire tly from from HM . The relation between -related Hamiltonian ontrol systems and -abstra tions parallels the general ase. Proposition 4.5. Let SHM and SHN be Hamiltonian ontrol systems on Poisson manifolds (M; f; gM ) and (N; f; gN), respe tively, and : M ! N a smooth Poisson map. Then SHM and SHN are -related if and only if SHN is a -abstra tion of SHM . 6 PAULO TABUADA AND GEORGE J. PAPPAS Proof. It is enough to show that if is a Poissonmap then De nitions 4.2 and 4.4 are equivalent for Hamiltonian ontrol systems. The result then follows from Theorem 4.3. De nition 4.4 is equivalent to: B(dHM (x)) dHN ( (x)) , Tx (B# M (dHM (x))) B# N (dHN ( (x))) , Tx (DHM (x)) DHN ( (x)) whi h is just De nition 4.2. Proposition 4.5 tell us that the abstra ting pro ess an be hara terized at the level of the ontrolled Hamiltonians. This result should be expe ted sin e the ontrolled Hamiltonians ompletely spe ify the dynami s of Hamiltonian ontrol systems given a Poisson stru ture. 4.1. Constru ting Poisson maps. In order to extra t a Hamiltonian abstra tion from an Hamiltonian ontrol system SHM on a Poisson manifold (M; f; gM ), one needs a Poisson map :M ! N that will indu e the abstra tion on N . In many ases, however, one only knows whi h variables are unimportant and whi h should be ignored. How should this information should be assembled to de ne an abstra ting Poisson map? We must ensure that (1) is a Poisson map, and (2) the Poisson bra ket f; gN is non-degenerate. Even if is Poisson and dim(N) is even it is not true, in general, that the bra ket in N is non-degenerate as the following example shows. Let M = T R3 with the anoni al bra ket, that is ff; ggM = f qi g pi f pi g qi . Denote a point in M by x = (q1; q2; q3; p1; p2; p3) and let (q1; q2; q3; p1; p2; p3) = (q1; q2; q3; p1). The map is Poisson as an easily be veri ed but the bra ket indu ed on N and given by ff; ggN = f q1 g p1 f p1 g q1 (4.3) is degenerate sin e its rank is only 2. This example also shows that to avoid these problems one must make sure that the dire tions ollapsed by are onjugate. More pre isely we have the following well known result, Proposition 4.6. Let (M; f; gM ) be a Poisson manifold with non-degenerate Poisson bra ket and :M ! N an abstra ting map. If for every X 2 Ker(T ), X is Hamiltonian with Hamiltonian fun tion h 2 C1(M) and there exists a g 2 C1(M) su h that fh; ggM 6= 0 and Xg 2 Ker(T ) then is Poisson and indu es a non-degenerate Poisson bra ket on N by: ff1; f2gN Æ = ff1 Æ ; f2 Æ gM (4.4) Proof. Sin e the map is a surje tive submersion it de nes a regular equivalen e relation = by de laring two points x and x0 to be on the same equivalen e lass i (x) = (x0). The equivalen e lasses of this relation are des ribed by the orbits of Ker(T ). Sin e every element of Ker(T ) is Hamiltonian the orbit of Ker(T ) is an Hamiltonian a tion of Rk with k = dim(Ker(T )). The quotient manifold M= = whi h is di eomorphi to N inherits a Poisson stru ture de ned by (4.4), see for example [9, 10℄. We will only show that f; gN is non-degenerate. By Lie-Weinstein theorem [15℄ there is a lo al oordinate transformation 'M : M !M su h that in the new anoni al oordinates (q1; q2; : : : ; qm; p1; p2; : : : ; pm; 1; 2 : : : ; v) the following holds fqi; qjgM = fpi; pjgM = fqi; jgM = fpi; jgM = f i; jgM = 0 and fqi; pjgM = Æij . Sin e ABSTRACTIONS OF HAMILTONIAN CONTROL SYSTEMS 7 f; gM is non degenerate the oordinates redu e to (q1; q2; : : : ; qm; p1; p2; : : : ; pm). By assumption for every h su h that Xh 2 Ker(T ) there is an g su h that fh; ggM 6= 0, and we an write fh Æ ' 1 M ; g Æ ' 1 M gM = fh; ggM Æ ' 1 M . Sin e fh; ggM 6= 0 it follows that fh Æ ' 1 M ; g Æ ' 1 M gM 6= 0 meaning that h = qi and g = pi for some i. We an assume without loss of generality, that the new anoni al oordinates q1; q2; : : : ; qn and p1; p2; : : : ; pn are su h that Xqi and Xpi belong to Ker(T ) for i = 1 : : : n. Consider then the map = ' 1 M Æ Æ'N :M ! N de ned on a open set around the point x 2M . This map sends (q1; : : : ; qm; p1; : : : ; pm) to (qn+1; qn+2; : : : ; qm; pn+1; pn+2; : : : ; pm) and therefore f; gN is nondegenerate at x. Sin e this holds for any x 2M N is non degenerate Poisson manifold. To use Theorem 4.3 the Hamiltonian maps for Ker(T ) need to belong to P(HM ) so we have the following onstru tion to build an abstra ting map . Pi k a olle tion of maps h1; h2; : : : ; hn 2 C1(M) su h that hi 2 P(HM ) and determine the onjugate to hi, that is a map h i su h that fhi; h igM 6= 0. If h i also belongs to P(HM ) then any map su h that Ker(T ) = SpanfXh1 ; Xh 1 ; Xh2 ; Xh 2 ; : : : ; Xhn ; Xh ng veri es the onditions of Proposition 4.6. A illustration of this onstru tion an be found in Se tion 6. 4.2. Canoni al Constru tion. Given a Poisson map, De nition 4.4 provides us with a geometri de nition for Hamiltonian abstra tions whi h is useful on eptually but not omputationally. We now present a anoni al onstru tion that will allow us to obtain an abstra tion SHN from an Hamiltonian ontrol system SHM and a Poisson map :M ! N . Our onstru tion is inspired from the anoni al onstru tion of [13℄, even though the onstru tion presented here uses odistributions as opposed to distributions. This is natural for Hamiltonian systems sin e the di erentials of the Hamiltonians apture all system information. De nition 4.4 and, in parti ular, ondition (4.2) require the union of all the values of dHM evaluated at any x 2 1(y). A way of onstru ting this union is to de ne another family of maps F su h that dF is onstant along 1(y) and furthermore satis es dHM dF . From this new family it suÆ es to ompute dHN (y) = dF(x) for some x 2 1(y) sin e dF is the same for any x 2 1(y). In other words, we would like to onstru t a family of maps F su h that 1. dHM dF 2. For all x; x0 2M su h that (x) = (x0), dF(x) = dF(x0). Let K be the integrable distribution Ker(T ). Then the leaves of the foliation K orrespond to points on M that have the same image under . In this setting, we would like to design the family F so that the resulting odistribution dF is invariant with respe t to the ve tor elds in K. This idea is aptured in the following proposition. Proposition 4.7 (Invariant Codistributions). A olle tion F of smooth fun tions satis es dF(x) = dF(x0) for all x; x0 2M su h that (x) = (x0) if and only if LKdf 2 dF for all K 2 K and all maps f 2 F . Proof. Instead of working with the one-forms df1, df2, : : : , dfv, spanning dF we an asso iate the v-form = df1 ^ df2 ^ : : : ^ dfv with the ve tor spa e spanned by these forms sin e any other set of one-forms f 1; 2; : : : ; vg spanning the same ve tor spa e verify 1 ^ 2 ^ : : : ^ v = for some smooth fun tion . We rst show that if dF(x) = dF(x0) for (x) = (x0) then LKdf 2 dF . Let t(x) be the integral urve of some ve tor eld K belonging to K satisfying 0(x) = x. The equality dF(x) = dF(x0) an be repla ed byIONS OF HAMILTONIAN CONTROL SYSTEMS 7 f; gM is non degenerate the oordinates redu e to (q1; q2; : : : ; qm; p1; p2; : : : ; pm). By assumption for every h su h that Xh 2 Ker(T ) there is an g su h that fh; ggM 6= 0, and we an write fh Æ ' 1 M ; g Æ ' 1 M gM = fh; ggM Æ ' 1 M . Sin e fh; ggM 6= 0 it follows that fh Æ ' 1 M ; g Æ ' 1 M gM 6= 0 meaning that h = qi and g = pi for some i. We an assume without loss of generality, that the new anoni al oordinates q1; q2; : : : ; qn and p1; p2; : : : ; pn are su h that Xqi and Xpi belong to Ker(T ) for i = 1 : : : n. Consider then the map = ' 1 M Æ Æ'N :M ! N de ned on a open set around the point x 2M . This map sends (q1; : : : ; qm; p1; : : : ; pm) to (qn+1; qn+2; : : : ; qm; pn+1; pn+2; : : : ; pm) and therefore f; gN is nondegenerate at x. Sin e this holds for any x 2M N is non degenerate Poisson manifold. To use Theorem 4.3 the Hamiltonian maps for Ker(T ) need to belong to P(HM ) so we have the following onstru tion to build an abstra ting map . Pi k a olle tion of maps h1; h2; : : : ; hn 2 C1(M) su h that hi 2 P(HM ) and determine the onjugate to hi, that is a map h i su h that fhi; h igM 6= 0. If h i also belongs to P(HM ) then any map su h that Ker(T ) = SpanfXh1 ; Xh 1 ; Xh2 ; Xh 2 ; : : : ; Xhn ; Xh ng veri es the onditions of Proposition 4.6. A illustration of this onstru tion an be found in Se tion 6. 4.2. Canoni al Constru tion. Given a Poisson map, De nition 4.4 provides us with a geometri de nition for Hamiltonian abstra tions whi h is useful on eptually but not omputationally. We now present a anoni al onstru tion that will allow us to obtain an abstra tion SHN from an Hamiltonian ontrol system SHM and a Poisson map :M ! N . Our onstru tion is inspired from the anoni al onstru tion of [13℄, even though the onstru tion presented here uses odistributions as opposed to distributions. This is natural for Hamiltonian systems sin e the di erentials of the Hamiltonians apture all system information. De nition 4.4 and, in parti ular, ondition (4.2) require the union of all the values of dHM evaluated at any x 2 1(y). A way of onstru ting this union is to de ne another family of maps F su h that dF is onstant along 1(y) and furthermore satis es dHM dF . From this new family it suÆ es to ompute dHN (y) = dF(x) for some x 2 1(y) sin e dF is the same for any x 2 1(y). In other words, we would like to onstru t a family of maps F su h that 1. dHM dF 2. For all x; x0 2M su h that (x) = (x0), dF(x) = dF(x0). Let K be the integrable distribution Ker(T ). Then the leaves of the foliation K orrespond to points on M that have the same image under . In this setting, we would like to design the family F so that the resulting odistribution dF is invariant with respe t to the ve tor elds in K. This idea is aptured in the following proposition. Proposition 4.7 (Invariant Codistributions). A olle tion F of smooth fun tions satis es dF(x) = dF(x0) for all x; x0 2M su h that (x) = (x0) if and only if LKdf 2 dF for all K 2 K and all maps f 2 F . Proof. Instead of working with the one-forms df1, df2, : : : , dfv, spanning dF we an asso iate the v-form = df1 ^ df2 ^ : : : ^ dfv with the ve tor spa e spanned by these forms sin e any other set of one-forms f 1; 2; : : : ; vg spanning the same ve tor spa e verify 1 ^ 2 ^ : : : ^ v = for some smooth fun tion . We rst show that if dF(x) = dF(x0) for (x) = (x0) then LKdf 2 dF . Let t(x) be the integral urve of some ve tor eld K belonging to K satisfying 0(x) = x. The equality dF(x) = dF(x0) an be repla ed by 8 PAULO TABUADA AND GEORGE J. PAPPAS dF(x) = dF( t(x)) for all t 2 R. Di erentiating with respe t to time we get LK = 0 whi h is equal to LK = d({K ) + {K(dd ) = d (K) Computing d (K) in oordinates and equating to zero we get that d(dfi(K)) = 0 or d(dfi(K)) = ajdfj (4.5) whi h simply means that dLKfi = LKdfi = ajdfj from whi h we on lude that LKdf 2 dF for every f 2 F . For the onverse assume that LKdf 2 dF . We want to show that (x) = ( t) for all t 2 R (4.6) But the derivative of (4.6) is true by assumption and (4.6) holds for t = 0. Proposition 4.7 motivates a anoni al onstru tive pro edure to obtain the abstra ted hamiltonian ontrol system SHN given an hamiltonian ontrol system SHM and an abstra ting Poisson map :M ! N . If we denote the anihilating odistribution of K by Ko, that is Ko = f 2 T M j (K) = 0 8K 2 Kg we an onstru t a olle tion of Hamiltonians HN based on HM as follows: De nition 4.8 (Canoni al onstru tion). Let : (M; f; gM ) ! (N; f; gN) be a Poisson map between manifolds with non-degenerate Poisson bra kets, and let HM be a olle tion of Hamiltonians on M . Denote by HM the following family of smooth maps: HM = HM [ LKHM [ LKLKHM [ : : : (4.7) for all K 2 K. The olle tion of Hamiltonians HN de ned by HN = HM Æ i (4.8) for any embedding i : N ,!M su h that B# M (Ko) T i(TN) is alled anoni ally -related to HM . The olle tion of Hamiltonians obtained by onstru tion of De nition 4.8 is anoni al in the following sense. Proposition 4.9 (Minimal Abstra tion). The odistribution dHM is the smallest odistribution satisfying 1. dHM dHM 2. For all x1; x2 2M su h that (x1) = (x2), dHM (x1) = dHM (x2). and the Hamiltonian ontrol system de ned by HN is the smallest Hamiltonian ontrol system -related to HM . Proof. The olle tion of Hamiltonians HM ontains HM by onstru tion so that is follows trivially that it satis es property 1. It also veri es property 2 sin e by onstru tion it satis es the onditions of Proposition 4.7. To show that this odistribution is the smallest one verifying these properties onsider any other olle tion of Hamiltonians GM also verifying properties 1 and 2. Let h be any map su h that h 2 HM , if dh 2 dHM then dh 2 dGM be ause GM veri es property 1. Otherwise dh is obtained from a nite number of Lie derivatives alongK of some one-form in dHM . Sin e LKdf = dLKf , property 2 and Proposition 4.7 imply that dh 2 dGM . The above dis ussion shows that for all dh 2 dHM we have that dh 2 dGM , therefore dHM is ontained in any other odistribution verifying properties 1 and 2. ABSTRACTIONS OF HAMILTONIAN CONTROL SYSTEMS 9 To prove the se ond assertion we need to show rst that the Hamiltonian ontrol system de ned by HN is -related to SHM . To see that this is the ase onsider any hM 2 HM and any point x 2 M . By de nition of HN there is a map hN satisfying hN = hM Æ i, for some i embedding N into M through the point x. The Hamiltonian hN de nes a Hamiltonian inM by hN Æ and sin e is Poisson we have that T XhNÆ = XhN Æ . This means that if we an prove that: Tx XhNÆ (x) = Tx XhM (x) (4.9) then -relatedness of the ontrol systems follows. Rewriting (4.9) as: (LT XhN Æ g)(x) = (LT XhM g)(x) 8g 2 C1(N) , (d(f Æ ) XhNÆ )(x) = (d(f Æ ) XhM )(x) , fhN Æ ; f Æ gM (x) = fhM ; f Æ gM (x) , ff Æ ; hN Æ gM (x) = ff Æ ; hMgM (x) , (d(hN Æ hM ) XfÆ )(x) = 0 Sin e f Æ is onstant on the leaves of K it follows that XfÆ 2 B# M (Ko) T i(TN). Therefore (d(hN Æ hM ) XfÆ )(x) = 0 sin e hN equals hM on the integral manifold (whi h is just N) of T i(TN) passing through the point x. Sin e the argument does not depend on the point x the on lusion holds for all x 2M . To show that the Hamiltonian ontrol system indu ed by HN is the smallest one -related to SHM we will onsider any other family of Hamiltonians GN indu ing a ontrol system -related to SHM . This family de nes a odistribution on M verifying properties 1 and 2 by d(GN Æ ). However sin e dHM is ontained in any other su h odistribution it follows that dHM d(GN Æ ) whi h is equivalent to B(dHM ) B(dGN Æ ) and lead to the desired in lusion dHN dGN . As asserted by Proposition 4.9 the abstra tion obtained by the anoni al onstru tion is the smallest Hamiltonian ontrol system -related to SHM , therefore we are always able to ompute the minimal -abstra tion of any Hamiltonian ontrol system given an abstra ting Poisson map . 5. Lo al A essibility Equivalen e In addition to propagating traje tories and Hamiltonians from the original Hamiltonian ontrol system to the abstra ted Hamiltonian system, we will investigate how a essibility properties an be preserved in the abstra tion pro ess. We rst review several (lo al) a essibility properties for ontrol systems [4, 6, 11℄. De nition 5.1 (Rea hable sets [6℄). Let SM be a ontrol system on a smooth manifold M . For ea h T > 0 and ea h x 2 M , the set of points rea hable from x at time T , denoted by Rea h(x; T ), is equal to the set of terminal points M (T ) of SM traje tories that originate at x. The set of points rea hable from x in T or fewer units of time, denoted by Rea h(x; T ) is given by Rea h(x; T ) = [t TRea h(x; T ). De nition 5.2. A ontrol system SM is said to be Lo ally a essible from x there is a neighborhood V of x su h that Rea h(x; T ) ontains a non-empty open set of M for all T > 0 and Rea h(x; T ) V . Lo ally a essible if it is lo ally a essible from all x 2M . Controllable if for all x 2M , Rea h(x; T ) =M for some T .IONS OF HAMILTONIAN CONTROL SYSTEMS 9 To prove the se ond assertion we need to show rst that the Hamiltonian ontrol system de ned by HN is -related to SHM . To see that this is the ase onsider any hM 2 HM and any point x 2 M . By de nition of HN there is a map hN satisfying hN = hM Æ i, for some i embedding N into M through the point x. The Hamiltonian hN de nes a Hamiltonian inM by hN Æ and sin e is Poisson we have that T XhNÆ = XhN Æ . This means that if we an prove that: Tx XhNÆ (x) = Tx XhM (x) (4.9) then -relatedness of the ontrol systems follows. Rewriting (4.9) as: (LT XhN Æ g)(x) = (LT XhM g)(x) 8g 2 C1(N) , (d(f Æ ) XhNÆ )(x) = (d(f Æ ) XhM )(x) , fhN Æ ; f Æ gM (x) = fhM ; f Æ gM (x) , ff Æ ; hN Æ gM (x) = ff Æ ; hMgM (x) , (d(hN Æ hM ) XfÆ )(x) = 0 Sin e f Æ is onstant on the leaves of K it follows that XfÆ 2 B# M (Ko) T i(TN). Therefore (d(hN Æ hM ) XfÆ )(x) = 0 sin e hN equals hM on the integral manifold (whi h is just N) of T i(TN) passing through the point x. Sin e the argument does not depend on the point x the on lusion holds for all x 2M . To show that the Hamiltonian ontrol system indu ed by HN is the smallest one -related to SHM we will onsider any other family of Hamiltonians GN indu ing a ontrol system -related to SHM . This family de nes a odistribution on M verifying properties 1 and 2 by d(GN Æ ). However sin e dHM is ontained in any other su h odistribution it follows that dHM d(GN Æ ) whi h is equivalent to B(dHM ) B(dGN Æ ) and lead to the desired in lusion dHN dGN . As asserted by Proposition 4.9 the abstra tion obtained by the anoni al onstru tion is the smallest Hamiltonian ontrol system -related to SHM , therefore we are always able to ompute the minimal -abstra tion of any Hamiltonian ontrol system given an abstra ting Poisson map . 5. Lo al A essibility Equivalen e In addition to propagating traje tories and Hamiltonians from the original Hamiltonian ontrol system to the abstra ted Hamiltonian system, we will investigate how a essibility properties an be preserved in the abstra tion pro ess. We rst review several (lo al) a essibility properties for ontrol systems [4, 6, 11℄. De nition 5.1 (Rea hable sets [6℄). Let SM be a ontrol system on a smooth manifold M . For ea h T > 0 and ea h x 2 M , the set of points rea hable from x at time T , denoted by Rea h(x; T ), is equal to the set of terminal points M (T ) of SM traje tories that originate at x. The set of points rea hable from x in T or fewer units of time, denoted by Rea h(x; T ) is given by Rea h(x; T ) = [t TRea h(x; T ). De nition 5.2. A ontrol system SM is said to be Lo ally a essible from x there is a neighborhood V of x su h that Rea h(x; T ) ontains a non-empty open set of M for all T > 0 and Rea h(x; T ) V . Lo ally a essible if it is lo ally a essible from all x 2M . Controllable if for all x 2M , Rea h(x; T ) =M for some T . 10 PAULO TABUADA AND GEORGE J. PAPPAS Re all rst that lo al a essibility properties of Hamiltonian ontrol systems an be hara terized by simple rank onditions of the Poisson algebra generated by the ontrolled Hamiltonian. Proposition 5.3 (A essibility Rank Conditions [11℄). Let SHM be a Hamiltonian ontrol system on a Poisson manifold (M; f; gM ) of dimension m and denote by P(HM ) the Poisson algebra freely generated by the olle tion of Hamiltonians HM . Then: If dim(d(P(HM (x)))) = m, then the ontrol system SHM is lo ally a essible at x 2M . If dim(d(P(HM (x)))) = m for all x 2M , then ontrol system SHM is lo ally a essible. If dim(d(P(HM (x)))) = m for all x 2 M , HM is symmetri , that is h 2 HM ) h 2 HM , and M is onne ted, then ontrol system SHM is ontrollable. Theorem 4.3 immediately propagates lo al a essibility from the original Hamiltonian system to its abstra tion. Proposition 5.4 (Lo al A essibility Propagation). Let Hamiltonian ontrol systems SHM and SHN be related with respe t to a Poisson map : M ! N . Then, if SHM is (symmetri ally) lo ally a essible (at x 2 M) then SHN is also (symmetri ally) lo ally a essible (at (x) 2 N). Also, if SHM is ontrollable then SHN is ontrollable. We now determine under what onditions on the abstra ting maps, lo al a essibility of the original system SHM is equivalent to lo al a essibility of its anoni al abstra tion SHN . In parti ular, we need to address the problem of propagating a essibility from the abstra ted system SHN to the original system SHM . We start by exploring the relationship between the Poisson algebras of anoni ally -related Hamiltonian systems. Lemma 5.5. Let SHN be anoni ally -related to SHM , then for all x 2M we have B (dP(HM (x))) = dP(HN )( (x)) Proof. We start by showing that for any two fun tions hM ; h0M 2 C1(M) and any point x 2M we have: Tx XfhM ;h0MgM (x) = XfhN ;h0NgN ( (x)) (5.1) with hN = hM Æ i, h0N = h0M Æ i and any i embedding N into M through the point x. Sin e is Poisson we have that: Tx XfhN ;h0NgNÆ (x) = XfhN ;h0NgN ( (x)) (5.2) so that we only have to show that: Tx XfhM ;h0MgM (x) = Tx XfhN ;h0NgNÆ (x) (5.3) This argument parallels the one in the proof of Proposition 4.9. The previous expression is equivalent to: (LT XfhM;h0M gM f)(x) = (LT XfhNÆ ;h0NÆ gM f)(x) 8f 2 C1(N) , ff Æ ; fhM ; h0MgMgM (x) = ff Æ ; fhN Æ ; h0M Æ gMgM (x) , ff Æ ; fhM hN Æ ; h0M h0N Æ gMgM (x) = 0 , fhM hN Æ ; fh0M ; h0N Æ ; f Æ gMgM (x) fh0M h0N Æ ; ff Æ ; hM hN Æ gMgM (x) = 0 (5.4) ABSTRACTIONS OF HAMILTONIAN CONTROL SYSTEMS 11 But sin e fh0M h0N Æ ; f Æ gM (x) = (LXfÆ h0M h0N Æ )(x), XfÆ 2 B# M (Ko) T i(TN) and h0M = h0N on T i(TN) we on lude that fh0M h0N Æ ; f Æ gM (x) = 0 and similarly for fhM hN Æ ; f Æ gM (x). This shows that equality (5.4) holds, and a indu tion argument extends (5.1) to: Tx XP(HM )(x) = XP(HN)( (x)) (5.5) By making use of Proposition 4.4 the above expression is equivalent to: B (dP(HM (x))) = dP(HN )( (x)) (5.6) Using the above lemma, a essibility equivalen e between the two ontrol systems an be now asserted. Theorem 5.6 (Lo al A essibility Equivalen e). Let SHN be anoni ally -related to SHM . If every ve tor eld Ki 2 Ker(T ) is Hamiltonian with Hamiltonian fun tion hi 2 C1(M) and hi 2 P(HM ), then SHM is lo ally a essible if and only if SHN is lo ally a essible. Proof. We begin by showing how a essibility properties of SHM are propagated to SHN . Suppose that SHM is lo ally a essible, that is dP(HM )(x) = T xM for all x 2M , then by Lemma 5.5 dP(HN)( (x)) = B(x)T xM . Sin e B = (B# N ) 1 Æ T Æ B# M and both B# N and B# M are isomorphisms, and T is surje tive, B is also surje tive. We on lude therefore that dP(HN )(y) = T yN , for all y = (x). But is surje tive so SHN is lo ally a essible. Let us now show how a essibility properties of SHN an be pulled ba k to SHM . We pro eed by ontradi tion. Assume that every Ki 2 Ker(T ) is Hamiltonian with Hamiltonian fun tion hi 2 P(HM ) and that SHN is lo ally a essible while SHM is not. Then dP(HN )(y) = T yN and by Lemma 5.5 BdP(HM )(x) = T yN for all x su h that (x) = y. Sin e SHM is not lo ally a essible there exists some g 2 C1(M) su h that dg(x) = 2 dP(HM )(x), but B is surje tive so dg(x) must belong to Ker( B(x)). Taking into onsideration that dg(x) 2 Ker( B(x)) , Xg(x) 2 Ker(Tx ) we have a ontradi tion sin e we were assuming that all Hamiltonian fun tions of the ve tors belonging to Ker(Tx ) were also in P(HM )(x) and g(x) = 2 P(HM )(x). This shows that SHM is in fa t lo ally a essible from x. Sin e the argument does not depend on the parti ular point x, SHM is lo ally a essible. Corollary 5.7. Let SHN be anoni ally -related to SHM . If every Ki 2 Ker(T ) is Hamiltonian with Hamiltonian fun tion hi 2 C1(M), hi 2 P(HM ) and both HM and HN are symmetri and furthermore both M and N are onne ted then SHM is ontrollable i SHN is ontrollable. Theorem 5.6 provides moderate onditions to propagate a essibility properties in a hierar hy of abstra tions. In fa t when dealing with systems aÆne in ontrols, that is, of the form H = H0 +PiHiui we an always build a map satisfying the onditions of Theorem 5.6 by de ning its kernel to be XHi for some i provided that the onjugate of Hi belongs to the Poisson algebra generated by the ontrol system. A example of this onstru tion is presented in the next se tion. 6. A spheri al pendulum example As an illustrative example, onsider the spheri al pendulum as a fully a tuated me hani al ontrol system. This system an be used to model, for example, the stabilization of the spinning axis of a satellite or a panIONS OF HAMILTONIAN CONTROL SYSTEMS 11 But sin e fh0M h0N Æ ; f Æ gM (x) = (LXfÆ h0M h0N Æ )(x), XfÆ 2 B# M (Ko) T i(TN) and h0M = h0N on T i(TN) we on lude that fh0M h0N Æ ; f Æ gM (x) = 0 and similarly for fhM hN Æ ; f Æ gM (x). This shows that equality (5.4) holds, and a indu tion argument extends (5.1) to: Tx XP(HM )(x) = XP(HN)( (x)) (5.5) By making use of Proposition 4.4 the above expression is equivalent to: B (dP(HM (x))) = dP(HN )( (x)) (5.6) Using the above lemma, a essibility equivalen e between the two ontrol systems an be now asserted. Theorem 5.6 (Lo al A essibility Equivalen e). Let SHN be anoni ally -related to SHM . If every ve tor eld Ki 2 Ker(T ) is Hamiltonian with Hamiltonian fun tion hi 2 C1(M) and hi 2 P(HM ), then SHM is lo ally a essible if and only if SHN is lo ally a essible. Proof. We begin by showing how a essibility properties of SHM are propagated to SHN . Suppose that SHM is lo ally a essible, that is dP(HM )(x) = T xM for all x 2M , then by Lemma 5.5 dP(HN)( (x)) = B(x)T xM . Sin e B = (B# N ) 1 Æ T Æ B# M and both B# N and B# M are isomorphisms, and T is surje tive, B is also surje tive. We on lude therefore that dP(HN )(y) = T yN , for all y = (x). But is surje tive so SHN is lo ally a essible. Let us now show how a essibility properties of SHN an be pulled ba k to SHM . We pro eed by ontradi tion. Assume that every Ki 2 Ker(T ) is Hamiltonian with Hamiltonian fun tion hi 2 P(HM ) and that SHN is lo ally a essible while SHM is not. Then dP(HN )(y) = T yN and by Lemma 5.5 BdP(HM )(x) = T yN for all x su h that (x) = y. Sin e SHM is not lo ally a essible there exists some g 2 C1(M) su h that dg(x) = 2 dP(HM )(x), but B is surje tive so dg(x) must belong to Ker( B(x)). Taking into onsideration that dg(x) 2 Ker( B(x)) , Xg(x) 2 Ker(Tx ) we have a ontradi tion sin e we were assuming that all Hamiltonian fun tions of the ve tors belonging to Ker(Tx ) were also in P(HM )(x) and g(x) = 2 P(HM )(x). This shows that SHM is in fa t lo ally a essible from x. Sin e the argument does not depend on the parti ular point x, SHM is lo ally a essible. Corollary 5.7. Let SHN be anoni ally -related to SHM . If every Ki 2 Ker(T ) is Hamiltonian with Hamiltonian fun tion hi 2 C1(M), hi 2 P(HM ) and both HM and HN are symmetri and furthermore both M and N are onne ted then SHM is ontrollable i SHN is ontrollable. Theorem 5.6 provides moderate onditions to propagate a essibility properties in a hierar hy of abstra tions. In fa t when dealing with systems aÆne in ontrols, that is, of the form H = H0 +PiHiui we an always build a map satisfying the onditions of Theorem 5.6 by de ning its kernel to be XHi for some i provided that the onjugate of Hi belongs to the Poisson algebra generated by the ontrol system. A example of this onstru tion is presented in the next se tion. 6. A spheri al pendulum example As an illustrative example, onsider the spheri al pendulum as a fully a tuated me hani al ontrol system. This system an be used to model, for example, the stabilization of the spinning axis of a satellite or a pan 12 PAULO TABUADA AND GEORGE J. PAPPAS and tilt amera. Consider a massless rigid rod of length l xed in one end by a spheri al joint and having a bulb of mass m on the other end. The on guration spa e for this ontrol system is S2 parameterized by 2 [0; [ and 2 [0; 2 [. The kineti energy of the system is given by T = 1 2ml2( _ 2 + sin2 _ 2) (6.1) and the potential energy of the system is V = mgl os (6.2) Trough the Legendre transform of the Lagrangian L = T V one arrives at the Hamiltonian H0 = 1 2ml2 p2 + 1 2ml2 sin2 p2 mgl os (6.3) where p is given by p = ml2 _ and p = ml2 sin2 _ . Sin e the system is fully a tuated the Hamiltonian ontrol system SHM de ned over M = T S2 with the anoni al Poisson bra ket is given by: HM = H0 +H1u1 +H2u2 (6.4) with H1 = and H2 = and where u1 and u2 are the ontrol inputs. The drift ve tor eld asso iated with H0 is invariant under rotations around the verti al axis and ould be redu ed using this symmetry. However to emphasize the advantages to the abstra tion method we will abstra t away pre isely the dire tions were there are no symmetries. Consider the abstra ting map: : T S2 ! T S1 (6.5) ( ; ; p ; p ) 7! ( ; p ) (6.6) It is lear that 2 P(HM ) and the onjugate variable to , p , also belongs to P(HM ) sin e fH0; gM = 1 ml2 p , so the onditions of Proposition 4.6 are ful lled meaning that is a Poisson map indu ing a nondegenerate bra ket in T S1. Following the steps of the anoni al onstru tion one omputes the family of maps: HM = HM [ f ;HMgM [ fp ;HMgM [ : : : (6.7) However it is enough to ompute f ;H0gM = 1 ml2 p sin e dim(d(P(HM [ f 1 ml2 p g))) = 4 and all remaining bra kets an be generated by HM [ f ;H0gM . The olle tion of Hamiltonians anoni ally -related to HM is given by HN = Spanf 1 2ml2 p2 + 1 2ml2 sin2 p2 mgl os ; ; ; 1 ml2 p g (6.8) but where and p are now regarded as ontrol in puts sin e they are ranging in K. Introdu ing the new ontrol inputs v1 = and v2 = u2 the abstra ting ontrolled Hamiltonian an be written as: HN = 1 2ml2 sin2 v1 p2 + v2 (6.9) Note that the terms depending only on or p have disappeared sin e regarding and p as ontrol inputs redu ed those terms as onstants multiplying the ontrol inputs and onstants are asso iated with the null ABSTRACTIONS OF HAMILTONIAN CONTROL SYSTEMS 13 ve tor eld. The equations of the new ontrol system on N = T S1 are obtained through the indu ed Poisson bra ket (whi h is just the anoni al one on N) and are given by: _ = 1 ml2 sin2 v1 p (6.10) _ p = v2 (6.11) whi h de ne a ontrollable Hamiltonian ontrol system on N . 7. Con lusions In this paper, we have presented a hierar hi al abstra tion methodology for Hamiltonian nonlinear ontrol systems. The extra stru ture of me hani al systems was utilized in to provide onstru tive methods for generating abstra tions while maintaining the Hamiltonian stru ture. Furthermore we have hara terized a essibility equivalen e through easily he kable onditions. These results are very en ouraging for hierar hi al ontrolling me hani al systems. Re ning ontroller design from the abstra ted to the original system is learly important. Other resear h topi s under urrent resear h in lude the propagations of nonholonomi onstraints among the di erent levels of the hierar hy, and better understanding the relationship between Hamiltonian abstra tions and more established notions of redu tion based on symmetries. Referen es [1℄ R. Abraham, J. Marsden, and T. Ratiu. Manifolds, Tensor Analysis and Appli ations. Applied Mathemati al S ien es. Springer-Verlag, 1988. [2℄ W. M. Boothby. An Introdu tion to Di erentiable Manifolds and Riemannian Geometry. A ademi Press, 1975. [3℄ Ana Cannas da Silva. Le tures on Symple ti Geometry. Preprint, 2000. [4℄ Alberto Isidori. Nonlinear Control Systems. Springer-Verlag, third edition, 1996. [5℄ A. Weinstein J. E. Marsden. Redu tion of symple ti manifolds with symmetry. Reports on Mathemati al Physi s, 5:121{120, 1974. [6℄ Velimir Jurdjevi . Geometri Control Theory. Number 51 in Cambridge Studies in Advan ed Mathemati s. Cambridge University Press, 1997. [7℄ W. S. Koon and J. E. Marsden. The poisson redu tion of nonholonomi me hani al systems. Reports on Mathemati al Physi s, 42:101{134, 1998. [8℄ Gerardo La erriere, George J. Pappas, and Shankar Sastry. O-minimal hybrid systems. Mathemati s of Control, Signals, and Systems, 13(1):1{21, 2000. [9℄ Jerold Marsden and Tudor Ratiu. Redu tion of poisson manifolds. Letters in Mathemati al Physi s, 11:161{170, 1986. [10℄ Jerrold E. Marsden and Tudor S. Ratiu. Introdu tion to Me hani s and Symmetry. Texts in Applied Mathemati s. SpringerVerlag, 1999. [11℄ H. Nijmaijer and A.J. van der S haft. Nonlinear Dynami al Control Systems. Springer-Verlag, 1995. [12℄ George J. Pappas, Gerardo La erriere, and Shankar Sastry. Hierar hi ally onsistent ontrol systems. IEEE Transa tions on Automati Control, 45(6):1144{1160, June 2000. [13℄ George J. Pappas and Slobodan Simi . Consistent hierar hies of nonlinear abstra tions. In Pro eedings of the 39th IEEE Conferen e in De ision and Control. Sydney, Australia, De ember 2000. [14℄ A. J. van der S haft. Hamiltonian dynami s with external for es and observations.Mathemati al Systems Theory, 15:145{168, 1982. [15℄ Alan Weinstein. The lo al stru ture of poisson manifolds. Journal of Di erential Geometry, 18:523{557, 1983.IONS OF HAMILTONIAN CONTROL SYSTEMS 13 ve tor eld. The equations of the new ontrol system on N = T S1 are obtained through the indu ed Poisson bra ket (whi h is just the anoni al one on N) and are given by: _ = 1 ml2 sin2 v1 p (6.10) _ p = v2 (6.11) whi h de ne a ontrollable Hamiltonian ontrol system on N . 7. Con lusions In this paper, we have presented a hierar hi al abstra tion methodology for Hamiltonian nonlinear ontrol systems. The extra stru ture of me hani al systems was utilized in to provide onstru tive methods for generating abstra tions while maintaining the Hamiltonian stru ture. Furthermore we have hara terized a essibility equivalen e through easily he kable onditions. These results are very en ouraging for hierar hi al ontrolling me hani al systems. Re ning ontroller design from the abstra ted to the original system is learly important. Other resear h topi s under urrent resear h in lude the propagations of nonholonomi onstraints among the di erent levels of the hierar hy, and better understanding the relationship between Hamiltonian abstra tions and more established notions of redu tion based on symmetries. Referen es [1℄ R. Abraham, J. Marsden, and T. Ratiu. Manifolds, Tensor Analysis and Appli ations. Applied Mathemati al S ien es. Springer-Verlag, 1988. [2℄ W. M. Boothby. An Introdu tion to Di erentiable Manifolds and Riemannian Geometry. A ademi Press, 1975. [3℄ Ana Cannas da Silva. Le tures on Symple ti Geometry. Preprint, 2000. [4℄ Alberto Isidori. Nonlinear Control Systems. Springer-Verlag, third edition, 1996. [5℄ A. Weinstein J. E. Marsden. Redu tion of symple ti manifolds with symmetry. Reports on Mathemati al Physi s, 5:121{120, 1974. [6℄ Velimir Jurdjevi . Geometri Control Theory. Number 51 in Cambridge Studies in Advan ed Mathemati s. Cambridge University Press, 1997. [7℄ W. S. Koon and J. E. Marsden. The poisson redu tion of nonholonomi me hani al systems. Reports on Mathemati al Physi s, 42:101{134, 1998. [8℄ Gerardo La erriere, George J. Pappas, and Shankar Sastry. O-minimal hybrid systems. Mathemati s of Control, Signals, and Systems, 13(1):1{21, 2000. [9℄ Jerold Marsden and Tudor Ratiu. Redu tion of poisson manifolds. Letters in Mathemati al Physi s, 11:161{170, 1986. [10℄ Jerrold E. Marsden and Tudor S. Ratiu. Introdu tion to Me hani s and Symmetry. Texts in Applied Mathemati s. SpringerVerlag, 1999. [11℄ H. Nijmaijer and A.J. van der S haft. Nonlinear Dynami al Control Systems. Springer-Verlag, 1995. [12℄ George J. Pappas, Gerardo La erriere, and Shankar Sastry. Hierar hi ally onsistent ontrol systems. IEEE Transa tions on Automati Control, 45(6):1144{1160, June 2000. [13℄ George J. Pappas and Slobodan Simi . Consistent hierar hies of nonlinear abstra tions. In Pro eedings of the 39th IEEE Conferen e in De ision and Control. Sydney, Australia, De ember 2000. [14℄ A. J. van der S haft. Hamiltonian dynami s with external for es and observations.Mathemati al Systems Theory, 15:145{168, 1982. [15℄ Alan Weinstein. The lo al stru ture of poisson manifolds. Journal of Di erential Geometry, 18:523{557, 1983. 14 PAULO TABUADA AND GEORGE J. PAPPAS Instituto de Sistemas e Rob oti a, Instituto Superior T e ni o, Torre Norte Piso 6, Av. Rovis o Pais, 1049-001 Lisboa Portugal E-mail address: tabuada isr.ist.utl.pt Department of Ele tri al Engineering, 200 South 33rd Street, University of Pennsylvania, Philadelphia, PA 19104 E-mail address: pappasg ee.upenn.edu