Geometric dynamics of optimization

Transcription

1 Geometric ynamics of optimization François Gay-Balmaz 1, Darryl D. Holm 2, an Tuor S. Ratiu 3 Abstract arxiv: v3 [nlin.cd] 19 Jun 211 This paper investigates a family of ynamical systems arising from an evolutionary re-interpretation of certain optimal control an optimization problems. We focus particularly on the application in image registration of the theory of metamorphosis. Metamorphosis is a means of tracking the optimal changes of shape that are necessary for registration of images with various types of ata structures, without requiring that the transformations of shape be iffeomorphisms, but penalizing them if they are not. This is a rich fiel whose possibilities are just beginning to be evelope. In particular, metamorphosis an its relate variants in the geometric approach to control an optimization can be expecte to prouce many exciting opportunities for new applications an analysis in geometric ynamics. Contents 1 Introuction LDM approach, EPDiff, an momentum maps Distribute optimization ynamics, or evolutionary metamorphosis Plan an main contributions of the paper Review of optimal control problems Definitions Examples: Lie group controls acting on state manifols Distribute optimization problems Overview Distribute optimization Review of Clebsch optimal control Optimization using penalties Lagrange-Poincaré an metamorphosis reuction 32 1 Control an Dynamical Systems, California Institute of Technology 17-81, Pasaena, CA 91125, USA an Laboratoire e Météorologie Dynamique, École Normale Supérieure/CNRS, Paris, France. fgbalmaz@cs.caltech.eu 2 Department of Mathematics, Imperial College Lonon. Lonon SW7 2AZ, UK..holm@imperial.ac.uk 3 Section e Mathématiques an Bernoulli Center, École Polytechnique Féérale e Lausanne. CH 115 Lausanne. Switzerlan. Tuor.Ratiu@epfl.ch 1

2 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 2 5 Optimization, the Lagrangian approach 36 6 Hamilton-Poincaré an metamorphosis reuction 38 7 Optimization, the Hamiltonian approach 42 8 Examples Action by representation an avecte quantities Heavy top Ajoint representations Action by affine representation Actions by multiplication on Lie groups The N-imensional rigi boy Euler flui equations Optimization ynamics of a compressible flui N-imensional Camassa-Holm equation Left action of iffeomorphisms on embee subspaces Back-to-labels map for fluis Metamorphosis ynamics Subgroup actions Example: Metamorphosis equations on SE( Lie-Poisson Hamiltonian formulation of metamorphosis for right action 7 9 Conclusions an outlook 72 References 73 1 Introuction With the avent of new evices capable of seeing objects an structures not previously imagine, the realm of science an meicine has been extene in a multitue of ifferent ways. The impact of this technology has been to generate new challenges associate with the problems of formation, acquisition, compression, transmission an analysis of images. These challenges cut across the isciplines of mathematics, physics, computational science, engineering, biology, meicine, an statistics. For example, in computational anatomy (CA biomeical images are compare quantitatively by calculating the istance between them, along a path that is optimal in transforming one such image to another. The optimal path is traverse along a curve of eformations in the group of smooth invertible maps with smooth inverses (i.e., the iffeomorphisms an it is governe by a partial ifferential equation (PDE calle the EPDiff equation. In particular, EPDiff governs the geoesic flow on the group of iffeomorphisms, with respect to any prescribe metric. This flow from one shape to another also has an evolutionary interpretation that invites ieas from the analysis of evolutionary equations. In particular, the momentum map for EPDiff ientifie first in [16] an explaine more completely in [39] yiels the canonical Hamiltonian formulation of the ynamics of the singular evolutionary solutions of EPDiff. Moreover, in an optimization

3 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 3 sense, this momentum map also provies a complete representation of the lanmarks an contours (outlines of images to be matche, in terms of the canonical positions an momenta associate with the evolutionary interpretation [43]. In aition, it provies a natural strategy for fining the optimal path between two configurations of either lanmarks or contours [7]. Thus, the momentum map (a concept from Hamiltonian systems is crucial in the construction of an isomorphism between the ata structures use in the optimal matching of images an the evolutionary singular solutions of the EPDiff equation. This isomorphism has alreay suggeste new ynamical paraigms for CA, as well as new strategies for assimilation of ata in other image representations, for example, as gray-scale ensities [46, 7]. The converse benefit may also evelop, in which methos of optimal control an optimization of ata assimilation use in image matching for CA may suggest new strategies for investigating ynamical systems of evolutionary PDE. In short, the variational formulations, Lie symmetries an associate momentum maps encountere in applications of EPDiff have le to a convergence in the analysis of both its evolutionary properties an its optimization equations. This paper focuses on the evolutionary aspects of the PDE that are summone by aopting a ynamical interpretation of the optimal control an optimization methos use in the registration of various types of images. The paper oes not perform any applications of optimization methos to image registration, nor oes it evelop any numerical algorithms for making such applications. Instea, the paper re-interprets the eneavor of image registration from a ynamical systems viewpoint. In particular, as we shall explain, a recent evelopment in the large eformation iffeomorphic matching methos (LDM, in an approach for image registration calle metamorphosis 1 [6, 66, 46] introuces a new type of evolutionary equation that may be calle optimization ynamics. In following this line of reasoning, the geometric mechanics approach for evolutionary PDE provies a framework that we hope will inform both optimization an ynamics. The primary example in the line of reasoning leaing to optimization ynamics is the EPDiff equation [41, 42, 7]. A brief history of the EPDiff equation EPDiff stems from the recognition by Arnol in [1] that incompressible flui ynamics coul be characterize as geoesic flow in the group of volume preserving iffeomorphisms, with respect to the kinetic energy metric (L 2 norm of the flui velocity. A few years later, the one-imensional compressible version of EPDiff reappeare as the ispersionless limit of the Camassa-Holm (CH equation [16]. The CH equation is a completely integrable evolution equation for shallow water waves, whose soliton solutions evelop sharp peaks in the ispersionless limit. Its peake soliton solutions (peakons correspon to concentrations of momentum into elta-function singularities an are solutions of EPDiff in one imension with the H 1 kinetic energy metric. Slightly later, the incompressible version of EPDiff with the H 1 kinetic energy metric was generalize to higher imensions in [41, 42] by using its symmetry-reuce variational principle, an was interprete as Euler s flui equations, average following Lagrangian particle trajectories. This interpretation soon le to the 1 Although the term metamorphosis has a precise mathematical efinition that will be given below, it also satisfies its proper ictionary efinition, as a change of physical form, structure or substance. This paper interprets the change as a type of evolution.

4 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 4 introuction of viscosity an some interesting applications of the resulting viscous equations as a turbulence moel by Chen et al. [19, 2]. Aroun the same time, EPDiff arose inepenently in a completely ifferent context. Namely, it arose as the governing equation in the optimization problem for large eformation iffeomorphic matching (LDM in image registration [64, 65, 68]. The recognition that EPDiff was arising in these two ifferent contexts provie a fruitful opportunity for ual interpretations of the solutions of the same equation. In particular, the peakons of the CH equation in the water wave context were soon recognize to be the lanmarks in images in the LDM context. Since then, the two types of problems have continue their optimizationynamics interplay an have been foun to inform each other, while also showing intriguing ifferences an similarities that arise in their ual formulations as initial value problems on one han an bounary value problems on the other. In particular, the concept of symmetry reuction an momentum maps from geometric mechanics that ha previously been applie so effectively in flui ynamics [1] an shallow water soliton theory [16], has recently been recognize as a unifying approach for eveloping multi-moe LDM methos for images whose ata structure may comprise arbitrary tensors, or tensor ensities [15]. This is a rich an rapily eveloping area of science, for which a complete literature review woul be beyon our scope here. A convergence of these two inepenent eneavors has le to ual interpretations of the same equation an the same key ieas in such ifferent but complementary contexts. This convergence is fascinating, an we continue our investigation of it here. In the present paper, we emphasize the ynamical interpretations of the equations an approaches that are applie in optimal image matching. This is not to say that we solve optimal matching problems for images at all in this paper. Rather, being cognizant of the ieas an variational formulations unerlying the optimal matching approach, we shall apply these formulations to stuy certain classes of equations that arise in the problem of image registration, not from the viewpoint of optimization, but rather from the evolutionary viewpoint of geometric mechanics [44, 53]. The geometric mechanics approach emphasizes Lie group actions on manifols, momentum maps, an reuction by symmetry. This approach leas to an unerstaning of certain classes of control an optimization problems as systems of evolutionary equations. In particular, the Lie symmetry ieas unerlying the process of optimal image assimilation known as metamorphosis [6, 66, 46] in combination with the evolutionary geometric mechanics viewpoint leas the family of EPDiff equations into the realm of optimization ynamics. Optimization ynamics extens the previous association of image matching ieas with soliton theory [43] to prouce new results, such as the erivation an re-interpretation of the twocomponent CH system (CH2 as a equation for the ynamics of metamorphosis of gray-scale images [46]. The CH2 system is a completely integrable evolutionary system of equations that was recently iscovere using isospectral methos for solitons [21]. Its inverse scattering transform is iscusse in [37]. Recognizing that some systems of equations arising in optimization ynamics for image analysis may be associate with soliton theory raises many questions about the mathematical properties of these systems an their solutions, particularly when the equations are nonlocal. For example, the initial value problems for some of the nonlocal equations obtaine in optimization ynamics investigate here allow emergent singular solutions, in which the evolution of a smooth, spatially confine, initial

5 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 5 conition becomes singular by concentrating itself into elta function istributions. In particular, EPDiff has that property an so oes the corresponing system of equations for the optimization ynamics of metamorphosis. See [41, 42, 44] an [69, 7], respectively, for further iscussions of EPDiff from the ifferent but complementary viewpoints of geometric mechanics an image matching. 1.1 LDM approach, EPDiff, an momentum maps The LDM approach is base on minimizing the sum of a time-integrate kinetic energy metric whose value efines the length of an optimal eformation path, plus a penalty norm that ensures an acceptable tolerance in image mismatch. (The matching cannot be exact because of the unavoiable errors that arise in real applications. LDM approaches were introuce an systematically evelope in Trouvé [64, 65], Dupuis et al. [24], Joshi an Miller [47], Miller et al. [6, 59], Beg [4], an Beg et al. [5]. The LDM approaches of those papers are base on Grenaner s eformable template paraigm for image registration [31]. Grenaner s paraigm, in turn, is a evelopment of a biometric strategy introuce by D Arcy Thompson [63] of comparing a template image I to a target image I 1 by fining a smooth invertible transformation of coorinates th at maps one image to the other. This transformation is assume to belong to a Lie group G of iffeomorphisms that acts on the set of templates containing I an I 1. The effect of the transformation on the ata structure that is encoe in the set of templates is calle the action of the Lie group G on the set of images. The optimal path in the transformation group is the one that costs the least in time-integrate kinetic energy for a given tolerance. This concept of optimization summons a control theory approach into the analysis an registration of images. In applications of the LDM approach, the optimal transformation path is often sought by using a variational optimization metho such as the one evelope in [24, 64, 65]. Using this metho, the optimal path for the matching transformation in this problem is obtaine from a graient-escent algorithm base on the Euler-Lagrange equation arising from stationary balance between kinetic energy an tolerance. This graient-escent approach oes inee etermine an optimal matching path. However, from the viewpoint of ynamical systems theory, it misses the following potentially interesting question: What information an perspective might be obtaine by interpreting the Euler-Lagrange equations associate to the LDM approach from a ynamical systems viewpoint? The answer to this question may be sought by interpreting the variational optimization metho in the LDM approach as a form of Hamilton s principle. Hamilton s principle for the variational construction of optimal paths with minimal kinetic energy for a given tolerance in image mismatch yiels an associate set of Euler-Lagrange equations that may then be given an evolutionary interpretation. The optimal solutions of these equations have been investigate as evolutionary motion on the Lie group of iffeomorphisms in the absence of aitional penalty terms by Arnol [1, 2], Holm et al. [41, 42], Marsen an Ratiu [53], an for the particular application to template matching in Miller et al. [59]. As mentione earlier, the optimal paths in these cases are geoesics with respect to the metric provie by

6 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 6 the kinetic energy. The kinetic energy for LDM is invariant uner right translations on the iffeomorphism group. Reucing Hamilton s principle with respect to this symmetry an then invoking the Euler-Poincaré theory applie to iffeomorphisms prouces an evolution equation known as the EPDiff equation [41, 42], whose erivation in the present context is explaine in Section 8.4. The solution of the EPDiff equation yiels the spatial representation of the geoesic velocity, i.e., the tangent vector to the optimal path of eformations along which the minimal istance from one image to another is measure. The geoesics themselves may be obtaine from the solutions of EPDiff for the velocity by a reconstruction process that inverts the previous reuction by symmetry after the solution to the EPDiff equation for velocity has been obtaine. This is analogous to the reconstruction process in classical mechanics that recovers the symmetry coorinate conjugate to a conserve momentum as the final step in the solution, after the other egrees of freeom have been etermine in the reuce space. Composing the evolutionary solutions of EPDiff with the reconstruction process provies an important representation of iffeomorphisms that relates the enpoint of a geoesic to the initial value for momentum in the EPDiff equation. This relation is the momentum representation of the eformation. The long-time existence of this representation is base on conservation by EPDiff of the kinetic energy norm, which may be chosen so that its bouneness affors enough smoothness on the velocities to ensure the long-time existence of solutions of EPDiff. In this case, EPDiff amits emergent weak momentum solutions; for example, elta-function istributions of momentum that emerge from smooth, spatially confine initial conitions [16, 39]. This singular behavior is well unerstoo analytically only in certain one-imensional cases. In particular, it is unerstoo for the completely integrable case of the Camassa-Holm equation, see, e.g., [51, 61] an references therein. The EPDiff equation is of central importance in computational anatomy [7]. This is because the optimal paths sought by LDM on the image template space efine on a manifol M are inherite from the geoesics on Diff(M, the Lie group of iffeomorphisms acting on the manifol M. These, in turn, are governe by EPDiff. Consequently, any solution of the LDM problem for optimal geoesics must involve EPDiff [7]. Conversely, solving the LDM problem irectly prouces the momentum representation of the optimal iffeomorphism. The momentum representation arising from this evolutionary interpretation is then available for analyzing anatomical ata sets. In any case, espite the isparate forms that the geoesic equations may take for the various ata structures in the various types of images, all of them are instances of EPDiff with the corresponing representation for momentum. The specific representation for momentum in terms of the image ata structure in a given case is calle the momentum map. The momentum map for images is another ynamical systems concept that emerges as a central feature in this paper. The EPDiff equation an its associate momentum map for various image ata structures are iscusse in Section 8.4. An interesting example of the momentum map relating solutions of LDM to solutions of EPDiff arises for the case of lanmark ata structure, in which the momentum is singularly concentrate at points. The relation between these singular geoesic solutions an evolutionary soliton solutions, calle peakons for a shallow water wave equation introuce in Camassa an Holm [16], has been examine in the context of computational anatomy in Holm et al. [43]. A numerical analysis of the stability of these equations is also given in

7 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 7 McLachlan an Marslan [56]. See also Micheli [57] for other recent evelopments involving the curvature of the space of lanmark shapes. Holm an Marsen [39] explain that two inepenent momentum maps for EPDiff are available in the case that the image ata structure comprises the manifol Emb(S 1,R 2 of embee close curves (embee images of S 1 in the plane R 2. The left action of the group of iffeomorphisms Diff(R 2 of the plane eforms the curve by a smooth invertible transformation of the coorinate system in which it is embee, while leaving the parameterization of the curve invariant. The right action of the group of iffeomorphisms Diff(S 1 of the circle correspons to smooth invertible reparameterizations of the omain S 1 of the coorinates of the curve. In this case, one momentum map correspons to action from the left by the iffeomorphisms on R 2, the other to their action from the right on the embee curves. Optimal control an reparameterization methos for matching close curves in the plane using these two momentum maps for the space of close curves in the plane have recently been evelope in Cotter an Holm [23]. In summary, LDM image analysis is base on optimization methos that are formulate as bounary value problems. However, the re-interpretation of their governing equations as evolutionary systems by using symmetry reuction of the corresponing Hamilton s principle allows various concepts from ynamical systems theory to be profitably applie in the solution an interpretation of image analysis problems. Thus, the transfer of concepts an ieas between these two fiels in the context of image registration has the potential to enrich them both. 1.2 Distribute optimization ynamics, or evolutionary metamorphosis As we have been iscussing, the paper focuses on the geometric ynamics interpretation of the optimization problems esigne for image registration. However, rather than concentrating on the evelopment of solutions of optimization problems, the treatment here focuses on the ynamics that are prouce in applying the metho of reuction by Lie group symmetry to families of optimization problems pose in a geometric setting. This is a new arena for geometric ynamics an several new epartures are being taken. Among these new epartures is the investigation of the evolutionary ynamics that arises when istribute or nonlocal penalties are impose in Hamilton s principle, rather than local constraints. For lack of a better name, we call this sort of problem istribute optimization ynamics. It is the evolutionary counterpart of the metamorphosis approach in imaging science [6, 66, 46], which, in turn, is a moification an evelopment of LDM that allows the evolution n(t of the image template to eviate from pure eformation. That is, metamorphosis only penalizes the spatial average of the eviation away from the infinitesimal action of the vector fiels on an image manifol, rather than enforcing it as a local pointwise constraint. This approach, in turn, moifies the EPDiff equation an thereby introuces a wealth of new structure an new examples that we shall investigate in this paper. An explicit comparison for the case that the image templates are gray-scale ensity istributions may help to unerstan the ifference between the LDM approach an the metamorphosis approach. LDM approach: Given the source an target templates for the images characterize as scalar ensities n an n T at the initial time t = an the final time t = T, respectively,

8 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 8 minimize the quantity l(u(tt+ 1 2σ n 2 η 1 T n T 2 L2, (1.1 over the time epenent vector fiel u(t, where η T is the flow of u(t evaluate at time t = T, an the formula ṅ(t+iv ( n(tu(t = is its infinitesimal action on a smooth ensity n(t = n ηt 1 the omain of flow. efine over time t T on Metamorphosis approach: Given n an n T, minimize ( l(u(t+ 1 2σ ṅ(t+iv( n(tu(t 2 2 L t (1.2 2 over time epenent vector fiel u(t an scalar ensities n(t. As one sees in Figure 1.1 for the metamorphosis of shapes characterize as ensities, the term metamorphosis introuce in [66] for this process can be unerstoo in practice by its orinary meaning, as change of shape, such as the graual an continuous metamorphosis of a tapole into a frog. Figure 1.1: These gray-scale images show optimal metamorphoses between two ensity istributions with equal total mass from [46]. The optimization approach woul compute the istance along the optimal path between between the first an last ensity in each row. In the evolutionary approach, the optimal trajectories for n(t are compute. The images between the enpoints show snapshots along the optimal path n(t in each row at intermeiate points in time. In particular, the secon row shows that metamorphosis allows a change in topology along its optimal path. Our interest focuses on the evolutionary equations for the process of metamorphosis. The ynamical system of metamorphosis equations obtaine in registering such gray-scale image ensities is given in Section 8 as one of the examples of the general approach. In one imension, the metamorphosis equations for this class of images comprises a completely integrable Hamiltonian system.

9 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 9 The paper begins by contrasting optimal control problems with istribute optimization problems in a geometric setting. In particular, we iscuss the geometric properties of Lie algebra controls acting on state space manifols. The latter optimal control approach parallels the familiar Clebsch variational formulation of ynamical equations continuum mechanics (e.g., [38]. In fact, continuum mechanics was one of the early paraigms for image registration [68]. The Clebsch variational formulation of continuum mechanics has recently been evelope an applie in the stuy of the ynamical aspects of optimal control problems in a geometric setting (see [28, 36]. Conversely, our concern here is to continue this parallel evelopment by stuying the implications for ynamics of the geometric approach to istribute optimization problems. 1.3 Plan an main contributions of the paper In the remainer of the paper, we compare the ynamical equations that arise from optimal control problems with those arising from istribute optimization. This comparison provies several examples of how the two approaches iffer an, in particular, how their ynamical equations iffer when their variational problem is regare as Hamilton s principle for the ynamics. Their comparison also ientifies the aspects of these approaches that are funamentally the same. Section 2 begins by explaining the ynamical set up for stanar optimal control problems treate by the Pontryagin Maximum Principle. Section 2.2 provies several examples illustrating the consequences of applying Lie group controls acting on state manifols by using the Clebsch framework for optimal control. These examples introuce the momentum map for the cotangent-lifte action of the Lie group controls on the state manifol. The cotangent-lift momentum map is a funamental concept in the application of geometric mechanics methos in the Clebsch framework for optimal control. It turns out that the same momentum map is also the organizing principle for the istribute optimization ynamics introuce in Section 2.3. After establishing this backgroun for our comparison of optimization an ynamical systems methos, Section 2.4 provies an overview of the rest of the paper. Section 3 begins by reviewing the Clebsch framework for optimal control problems introuce an stuie in [28]. A new class of optimization problems is then introuce which is the subject of stuy of this paper. The stationarity conitions are obtaine an the associate equations of motion are etermine. Inspire by the extremum problems presente earlier, Section 4 presents two Lagrangian reuction proceures for Lagrangian functions efine on T (G Q, where G is a Lie group acting on the manifol Q. These reuction methos are use in Section 5 to reerive the equations of motion that were foun in Section 3. Hamiltonian reuction is carrie out in Section 6. As before, there are two reuction methos an, in the case of a representation, one of them leas to Lie-Poison equations with a symplectic cocycle on the ual of a larger semiirect prouct Lie algebra. In Section 7 we apply these Hamiltonian reuction methos to the optimization problems introuce earlier. Section 8, by far the longest of the paper, presents a number of examples. We begin by stuying examples where G is represente on a vector space. The concrete examples treate are the heavy top an a class of problems using the ajoint representation. For example, we fin a moification of the pair of ouble bracket equations stuie in [7], [8]. Next, we stuy optimization problems associate to affine actions. Actions by group multiplication is the

10 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 1 next topic. The concrete examples inclue the N-imensional free rigi boy, Euler s equations for an ieal incompressible homogeneous an for a barotropic flui. The N-imensional Camassa-Holm equation is presente from this optimization point of view, inspire by the construction of singular solutions. Finally, the optimization problem is use to obtain the equations of metamorphosis ynamics for use in computational anatomy. Section 9 briefly summarizes the paper an gives an outlook for future work. 2 Review of optimal control problems 2.1 Definitions We begin by recalling the efinition of optimal control problems. Definition 2.1. (Optimal control problems A stanar optimal control problem comprises: a ifferentiable manifol Q on which state variables n Q evolve in time t uring an interval I = [,T ] along a curve n : I Q from n( = n to n(t = n T, with specifie values n,n T Q; a vector space U of control variables u U whose time epenence u : I U is at our isposal to affect the evolution n(t of the state variables; a smooth map F : Q U T Q such that F (,u : Q T Q is a vector fiel on Q for any u U whose associate evolution equation 2 ṅ = F (n,u (2.1 relates the unknown state an control variables (n(t,u(t : I Q U; a cost functional epening on the state an control variables S := l(u(t,n(tt, (2.2 subject to the prescribe initial an final conitions, at n( = n an n(t = n T. The integran l : Q U R, calle the Lagrangian, is assume to be C 1 on Q U. The goal of the optimal control problem is to fin the evolution (n(t,u(t of the state an control variables such that S is minimal subject to the prescribe ynamics (2.1 an the prescribe initial an final conitions n( = n, n(t = n T. 2 The over-ot notation in ṅ means time erivative. Several forms of time erivative appear in applications an the meaning shoul be clear from the usage. Besies the over-ot notation, we shall use the equivalent notation /t to mean either partial or orinary time erivative in the abstract formulas, as neee in the context. For fluis, we shall also use t for the Eulerian time erivative at fixe spatial location. Finally, the covariant time erivation on a Riemannian manifol will be enote as D/Dt.

11 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 11 The coupling between the control an state variables may be mae explicit by using the pairing, Q : T Q T Q R an a Lagrange multiplier α T Q that imposes the state system as a constraint on the cost functional, ] S c := [l(u,n+ α,ṅ F (n,u Q t. (2.3 This is a consequence of the well-known Pontryagin maximum principle [6, 48]. The variable α T Q is calle a costate variable. We now compute the equations associate to the variational principle δs c =. For simplicity, we suppose here that the state manifol Q is a vector space, say W. In this case the cotangent space is T W = W W an the costate variable is of the form α = (n,p W W. The stationary variations of the constraine cost function S c in (2.3 yiel = δs c = [ ( T δl δf p ṗ,δn δn + δn W ] + δp,ṅ F (n,u W ( T δl δf p,δu δu δu U T t+ p,δn Q, where, U : U U R enotes the uality pairing for the control vector space U. Stationarity in the variations δu gives a relation that etermines the controls u in terms of the state an costate variables, n an α, respectively, while stationarity in the variations (δn,δα etermines the evolution equations for the state an costate variables that minimize the cost function S. Since the values of n at the enpoints in time are fixe, δn vanishes at the enpoints. We thus get the stationarity conitions ( T δl δf δu = p, ṅ = F (n,u, ṗ = δl ( T δf δu δn p. δn Remark 2.2. Although we shall confine our consierations to the Lagrangian escription, we point out that the relation to the Pontryagin Maximum Principle in the Hamiltonian escription is obtaine via the Legenre transformation of the integran in the cost functional given by (2.3 which, for each point u in the control space U, efines the corresponing Hamiltonian H u : T Q R by H u (α n = α n,f (n,u Q l(n,u. (2.4 The notation α n for a covector in T Q means that it belongs to the fiber TnQ of the cotangent bunle. For more information about the Hamiltonian approach to geometric optimal control theory an the Pontryagin Maximum Principle, see [6, 48]. 2.2 Examples: Lie group controls acting on state manifols As an example that illustrates the theory evelope in this paper, we consier the case of continuum mechanical systems with avecte quantities; see Section 6 in [41]. In this case, the state manifol M is some vector subspace V of T(D Den(D, the tensor fiel ensities

12 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 12 on a manifol D. We will enote by a V these tensor fiel ensities. The group Diff(D of all iffeomorphisms of the manifol D acts on V by pull back, that is, a η a = a η, for all η Diff(D. It is thus a right representation of Diff(D on T(D Den(D. We consier here the group Diff(D of iffeomorphism as an infinite imensional Lie group (either formally or in some Fréchet sense whose Lie algebra is given by vector fiels v X(D. The right action of the Lie algebra X(D on V is given by the Lie erivative t exp(tv a := v a, t= where t exp(tv enotes the flow of v. Example 1 We present a simple example of optimal control problem base on the geometric formulation of continuum mechanics escribe above. In this example, the control space U is the Lie algebra X(D an thus the control variable is a vector fiel u := v X(D. The state manifol Q is the vector space V of tensor fiel ensities. The state variable n := a V is constraine to evolve accoring to the ODE ȧ = F (a,v := v a an one wants to minimize S := 1 2 v 2 gt, where g is an inner prouct norm on the Lie algebra g = X(D. Note that we are in the setting of Definition 2.1 with M = V an U = X(D. This is an example of a Clebsch optimal control problem, as stuie from a geometric point of view in [28]. For this class of problems, the vector fiel F is given by the infinitesimal generator associate to a group action on the state manifol. In the present example, this infinitesimal generator turns out to be the Lie erivative. Accoring to (2.3, the constraine cost function in this case is ( 1 S c = 2 v 2 g + p,ȧ v a V t, where p V is the costate variable. This is nothing else than the Clebsch approach to continuum mechanics; see, e.g., [38]. The variational principle δs c = gives the control v = (p a g, where : g g is the sharp operator associate to the inner prouct on g an the bilinear operator : V V g is efine by p a,v := v a,p, for all p V, a V, v g. (2.5

13 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 13 The other stationarity conitions are {ȧ+ (p a a =, ṗ T (p a p =, (2.6 where T vp V is efine by a, T vp = v a,p, for all p V, a V, v g. (2.7 The Clebsch state-costate equations (2.6 are canonically Hamiltonian with H(a,p = 1 2 (p a 2 g = 1 p a, (p a. 2 g As is well known, [38], using the cotangent-lift momentum map given by Π = p a to project the equations (2.6 on T M to g, yiels the (left Lie-Poisson bracket on the ual Lie algebra g. Explicitly, this Lie-Poisson bracket is given by where the Hamiltonian has the expression Example 2 Π = a δh/δππ = a Π Π (2.8 h(π = 1 2 Π, Π g. (2.9 This example will use the geometric setting of continuum mechanics as escribe before. However, the control vector space will now be given by U := g V (v,ν. We choose the quaratic Lagrangian l(v,ν := 1 2 v 2 g + 1 2σ 2 ν 2 L 2, where L 2 enotes an L 2 norm on V T(D Den(D. As before, the state manifol Q is V an the state variable a V is constraine to evolve as ȧ = F (a,v,ν := v a+ν. Note that the avection law ȧ = v a is not impose. Instea, the penalty term in the Lagrangian introuces the aitional term ν into the avection law. Thus, the constraine action (2.3 becomes in this case ( 1 S c = 2 v 2 g + 1 2σ 2 ν 2 L + p,ȧ va ν 2 V t, (2.1 whose stationary variation results in = δs c = [ T vp ṗ,δa + v +p a,δv V g 1 ] T + σ 2 ν p,δν + δp,ȧ v a ν V t+ p,δa V, V

14 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 14 where the flat operators : g g an : V V are associate to the inner proucts on g an V, respectively. Here the enpoint terms vanish because the values of a at the enpoints in time are fixe. Accoring to the variational formula for δs c, the cost functional in (2.1 is optimize when the controls satisfy v = (p a g an ν = σ 2 p V, (2.11 in which the sharp maps are the inverses of the flat maps efine above. For the controls (v,ν g V, the state an costate variables (a,p V V evolve accoring to the following close system {ȧ+ (p a a = σ 2 p, ṗ T (p a p =. (2.12 These are Hamilton s canonical equations for the Hamiltonian H(p,a = 1 2 (p a, (p a + σ2 g p, p 2. (2.13 V Remark 2.3. Thus, the evolution of the state a an costate p variables occurs by the corresponing Lie erivative actions of the vector fiel (p a g = X(D calculate by applying the sharp map to raise inices on the cotangent momentum map (a,p V V = T V J(a,p = p a g of the cotangent-lifte action. The evolution of the momentum V V g itself is the last formula to be foun, just as in the Clebsch approach, [38]. Proposition 2.4. Denote the momentum map of the cotangent-lifte action by an its ual vector fiel by Π := p a v := (p a = Π. Then the state an costate equations (2.12 imply the following Euler-Poincaré equation for the evolution for the momentum map: Π = vπ σ 2 p p, (2.14 where the operator v : g g is efine by vπ,u := Π,[v,u] JL for any u,v g = X(D, Π g = Ω 1 (D Den(D an [v,u] JL = v u enotes the stanar Lie bracket of vector fiels. Proof. The proof procees by a irect calculation. In the computation below we use the stanar Jacobi-Lie bracket of vector fiels [X,Y ] JL (f = X(Y (f Y (X(f for any

15 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 15 f C (D. For a fixe Lie algebra element Z g = X(D we compute, Π,Z = ṗ a+p ȧ,z = ṗ, Z a + p, Z ȧ = T vp, Z a + p, Z v a +σ 2 p, Z p = p, [Z,v] a +σ 2 p, Z p = p a,[z,v] σ 2 p p,z = Π, v Z σ 2 p p,z = vπ,z σ 2 p p,z, which proves the Proposition 2.4. Remark 2.5. (Lie algebra formulation of the equations Recall the the Lie algebra bracket [u,v] = a u v on g is minus the Lie bracket of vector fiels, that is, [u,v] = [u,v] JL := (u v v u. We may thus ientify v = a v an the previous equations can be rewritten as Π = a vπ σ 2 p p, ȧ = v a+σ 2 p, ṗ = T vp. (2.15 These are Lie-Poisson equations with a cocycle for the Hamiltonian h(π,a,p = 1 2 Π, Π + σ2 g p, p 2, (2.16 V with respect to the Lie-Poisson bracket given by, Π a Π a p h/ Π = Π = v ȧ = a 1 h/ a = (2.17 ṗ T p 1 h/ p = σ 2 p in which the variational erivatives of the Hamiltonian are to be substitute into the corresponing places inicate by a box (. This matrix is ientifie as the Hamiltonian operator for the Lie-Poisson bracket ual to the semiirect prouct Lie algebra gs(v V plus a symplectic 2-cocycle on (a,p V V. Remark 2.6. This Hamiltonian matrix will block-iagonalize in the Lagrange-Poincaré formulation iscusse in Section 4. Roughly speaking, this amounts to transforming variables Π Π := (Π+p a an (a,ν (a,ȧ.

16 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 16 Example 3 We now consier an example analogous to the preceing one but in finite imensions. We let the orthogonal group G = SO(3 act on R 3 by matrix multiplication on the left an we choose U := so(3 R 3 (Ω,ν as control space. As usual, we ientify the Lie algebra so(3 with R 3. We choose the quaratic Lagrangian l : so(3 R 3 R given by l(ω,ν := 1 2 IΩ Ω+ 1 Kν ν, 2σ2 for symmetric positive efinite matrices I an K. We impose the evolution equation Ẋ = Ω X+ν (2.18 for the state variable X R 3 =: Q. As before, the variational principle δs c = with ( 1 S c = 2 IΩ Ω+ 1 Kν ν +P (Ẋ+Ω X ν t 2σ2 yiels the controls IΩ = P X an Kν = σ 2 P, as in (2.11. Note that Ω = I 1 (P X = (P X an K 1 P = P, by the efinition of the sharp maps. Then the state an costate evolution equations (2.12 take canonical Hamiltonian form with Hamiltonian function H(X,P = 1 2 (P X (P X + σ2 2 P P. (2.19 Intriguingly, the resulting canonical Hamiltonian equations, Ẋ = H P = (P X X+σ 2 P, Ṗ = H X = (P X P, (2.2 involve the ouble cross prouct of the state an costate vectors (X,P R 3 R 3. The ouble cross proucts correspon to the Lie erivatives in equations (2.12 which for this case become cross proucts. For more information about the roots of the Hamiltonian approach in geometric control theory, see [3]. Upon efining the vector Π := IΩ = P X, equations (2.2 imply Π = Ω Π σ 2 (K 1 P P, Ẋ = Ω X+σ 2 P, Ṗ = Ω P, (2.21

17 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 17 which recovers the momentum map system (2.15 for this case. Inee, one may compute irectly that from which the result follows. Π = Ṗ X+P Ẋ = ( Ω P X+P ( Ω X+σ 2 P = (P Ω X+(Ω X P+σ 2 P P = (X P Ω+σ 2 P (K 1 P = Π Ω+σ 2 P (K 1 P, Remark 2.7. (Lie algebra formulation The Lie algebra bracket on se(3 so(3sr 3 may be written on R 3 R 3 as, a (Ω,α ( Ω, [ α = (Π,α, ( Ω, ] ( α = Ω Ω, Ω α Ω α Its ual operation is ( a (Ω,α(Π,P = Ω Π α P, Ω P. In terms of the a operation on se(3, the motion equations for (Π,P in (2.21 can be rewritten as ( Π, Ṗ ( = Ω Π σ 2 P P, Ω P ( = a ΩΠ+σ 2 P P, Ω P = a (Ω,σ 2 P ( Π, P. The result of the last calculation may be rewritten in Lie-Poisson bracket form as ( Π, Ṗ = a ( ( Π,P, (2.22 h/ Π, h/ P with Hamiltonian (2.19 rewritten in these variables as h(π,p = 1 2 Π Π + σ2 2 P P, (2.23 an using the (left Lie-Poisson bracket efine on the ual Lie algebra se(3. This is the Hamiltonian an Lie-Poisson bracket for the motion of an ellipsoial unerwater vehicle in the boy representation. See, e.g., [35] for more iscussion an references to the literature about the geometrical approach to the ynamics an control of unerwater vehicles. We have seen that equations (2.2 for the state-costate vectors (X,P are canonically Hamiltonian an that the system (2.22 for (Π,P is Lie-Poisson on the ual of a semiirect

18 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 18 prouct Lie algebra. Now, it remains to inclue the ynamics of the coorinate X into a single structure for the entire system (2.21 for (Π,X,P. We observe that equations (2.21 may be put into Lie-Poisson form, as Π Ẋ = Ṗ Π X P h/ Π Π X P X 1 h/ X = X 1 P 1 h/ P P 1 Ω σ 2 P. (2.24 This is the Lie-Poisson bracket ual to the semiirect prouct Lie algebra so(3s(r 3 R 3 plus a symplectic 2-cocycle on (X,P R 3 R 3. Remark 2.8. As mentione earlier, the Lagrange-Poincaré an Hamilton-Poincaré formulations in Sections 4 an 6 will block-iagonalize this Hamiltonian matrix. Remark 2.9. (Comparison of the examples The major ifference between Example 1 an Examples 2 an 3 is the following. In Example 1, we impose the avection equation ȧ = v a as a constraint on the minimization problem. This is one, as usual, by introucing a new variable p an aing the term p,ȧ v a in the action functional. In Examples 2 an 3, the avection law is not impose exactly, but only up to an error term ν := ȧ v a, whose norm is ae to the Lagrangian as a penalty, an nees to be minimize. Of course, in this case, the relation ν = ȧ v a is a constraint as seen in the term p,ȧ v a ν. As we have seen in Proposition 2.4, this error term implies a moification of the equations of motion. One of the aims of the present paper is to transform the control problem corresponing to the cost function in (2.1 into an optimization problem in which the penalty term ȧ v a 2 appears. This objective motivates the introuction of the istribute optimization problem in the next section. 2.3 Distribute optimization problems Definition 2.1. (istribute optimization problems A istribute optimization problem imposes the evolutionary state system in (2.1 as a penalty involving a chosen norm, rather than as a constraint. The resulting cost functional is thus taken to be of the form S := [ ] l(u,n+ 1 ṅ F (n,u 2 t, (2.25 2σ2 where the norm is associate to a Riemannian metric on Q. In this cost functional, the state system ynamics (2.1 is impose only in a istribute sense; namely, as a penalty enforce by the norm on Q, not pointwise on Q, as in (2.3. We assume that σ 2 >.

19 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 19 We may initially regar this secon approach as simply moifying the cost function in the optimal control problem (2.3 by introucing a penalty base on a norm of the state system. We will show later that the solutions of the two types of optimization problems coincie in the limit σ 2. In the case where Q is a vector space, enote by W, an the norm is associate to an inner prouct, the variations of the istribute cost function S in (2.25 now yiel δs = [ ( T δl δf p ṗ,δn δn + δn W ( T δl δf p,δu δu δu V ] T t+ p,δn W, (2.26 where the momentum variable p obtaine from the variation with respect to the vector fiel ṅ W is efine by ( σ 2 p := ṅ F (n,u W, (2.27 an in this case the map (inex lowering is applie with respect to the inner prouct on W. Let us return to Example 2 above an treat it as istribute optimization problem. Example As in 2.2, we consier the geometric setting of continuum mechanics. Contrary to Example 1 above, we o not impose the avection equation ȧ = v a as a constraint but as a penalty. The problem is now to minimize the expression [ 1 S := 2 v 2 g + 1 ] 2σ ȧ va 2 2 L t, 2 where L 2 is a L 2 norm on the space of tensor fiel ensities. This problem is clearly equivalent to that of Example 2 in 2.2. The variational principle δs = yiels the control an the same equations as before where we have efine the variable p by v = (p a g {ȧ+ (p a a = σ 2 p, ṗ T (p n p =, (2.28 p := 1 σ 2 (ȧ va V. (2.29 It is important to observe that in this approach the variable p is not really neee, since it is efine in terms of the other variables. This is not the case for the Clebsch approach escribe in the Examples of 2.2 for which p is an inepenent variable. For the Clebsch approach, the relation (2.29 is recovere as a consequence of the variational principle δs c =.

20 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics 2 Control problems versus optimization problems We now make some simple comments concerning the role of the variational principles in control problems an optimization problems. Let l = l(u,n : U Q R be a cost function an F a vector fiel as in the general Definition 2.1. As we have seen, one associates to these objects the following problems. (1 The optimal control problem consists of minimizing the integral S := l(u,nt subject to the conitions ṅ = F (n,u an the usual enpoint conitions. The resolution of this problem uses the Pontryagin maximum principle which, uner sufficient smoothness conition, implies that a solution of this problem is necessarily a solution of the variational principle ( δs c = δ l(ξ,n+ α,ṅ F (n,u t =. Example 1 in 2.2, for which the cost function is a kinetic energy an the vector fiel F is given by a Lie erivative, illustrates this metho. (2 The optimization problem with penalty escribe above consists of minimizing the integral ( S := l(u,n+ 1 ṅ F (n,u 2 t 2σ2 subject to the usual enpoint conitions. Of course, the solutions of this problem are necessarily solutions of the variational principle ( δs = δ l(ξ,n+ 1 2σ ṅ ξ Q(n 2 t =. 2 The examples in here illustrate this point. Remark Despite the analogy between the two variational principles δs c = an δs =, the origins of these principles are quite ifferent. In the first problem, the functional S is minimize uner a constraint, leaing to the construction of the functional S c by introucing the costate variable α. The well-known Pontryagin approach tells us that the solutions of the optimal control problem are necessarily critical points of S c. The variational principle of the secon problem is simply the stationarity conition implie by optimization of the functional S, without other constraints, except the enpoint conitions.

21 Gay-Balmaz, Holm an Ratiu Geometric optimization ynamics Overview In [28] a general formulation for a large class of optimal control problems was given. These problems, calle Clebsch optimal control problems, are associate to the action of a Lie group G on a manifol Q an to a cost function l : g Q R, where g enotes the Lie algebra of G. The Clebsch optimal control problem is, by efinition, subject to the following conitions: min ξ(t l(ξ(t,n(tt, (2.3 (A Either ṅ(t = ξ(t Q (n(t, or (A ṅ(t = ξ(t Q (n(t ; (B Both n( = n an n(t = n T, where ξ Q enotes the infinitesimal generator of the G-action, that is, ξ Q (n := t Φ exp(tξ (n. t= These optimal control problems comprise abstract formulations of many systems such as the symmetric representation of the rigi boy an Euler flui equations [8, 36], the ouble bracket equations on symmetric spaces [7], the singular solutions of the Camassa-Holm equation [16], control problems on Stiefel manifols [12], an others [6, 11]. Optimal control problems on Lie groups have a long history; see [6], [48] an references therein. Some of the earliest papers ealing with such problems are [13] an [32]. Goals of the paper The first goal of the present paper is to replace the constraints in the Clebsch optimal control problem with a penalty function ae to the cost function an to obtain in this way a classical (unconstraine optimization problem. The funamental iea is to use the constraints to form a quaratic penalty function in orer to get the Lagrangian ( l(u,n+ 1 2σ ṅ ξ Q(n 2 t. ( We first etermine necessary an sufficient conitions characterizing the critical points of this Lagrangian. Taking the time erivative of one of the conitions an using the others leas irectly to certain equations of motion. We then show that these equations are naturally obtaine by Lagrangian reuction an that they are the Lagrange-Poincaré equations of a Lagrangian function in the material representation that is the sum of the original Lagrangian plus the square of the norm on the velocity vector. This approach links irectly to the approach use in [46] in the stuy of the metamorphosis of shapes. From a variational point of view, one replaces the Hamilton-Pontryagin variational principle in the Clebsch framework δ (l(u,n+ α,ṅ u Q (n t =,