Geometric Hamiltonian Formulation of a Variational Problem Depending on the Covariant Acceleration

1 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal 2 ISCA, University of Aveiro, 3810-500 Aveiro, Portugal 3 CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal 4 BIFI, Department of Theoretical Physics and Unidad Asociada IQFR-BIFI, University of Zaragoza, Edificio I+D, Campus Rı́o Ebro, C/ Mariano Esquillor s/n, 50018 Zaragoza, Spain


Introduction
In [1], Skinner and Rusk obtained a unified formalism for Lagrangian and Hamiltonian dynamics of autonomous mechanical systems, and this issue has been extended in many directions. In particular, there is an increasing interest in the study of optimal control problems from that geometric viewpoint, which involves the presymplectic algorithm of Gotay-Nester [2].
Riemannian cubic polynomials can be seen as a generalization of cubic polynomials to non-Euclidean spaces [3,4]. These objects are stationary curves in a Riemannian manifold for a second-order variational problem with Lagrangian given by the norm squared covariant acceleration. There are many applications that inspire the study of those curves, namely, problems of interpolation in computer graphics and problems in the context of robotics and aeronautics as the trajectory planning of a rigid body.
As far as we know, the first Hamiltonian description of the optimal control problem whose control system is associated with the variational problem mentioned in the previous paragraph was considered in [5] but from a non-geometric perspective. The aim of this paper is to give a precise and geometric description of that optimal control problem. For this purpose, we adapt the presymplectic geometric version of the Pontryagin maximum principle based on the Skinner-Rusk methodology, which was proposed for the control theory by several authors (see, e.g., [6][7][8][9] and the references mentioned in these papers). Here, we develop the work started in [10,11] where that intrinsic version of the problem was first presented. We specify to our problem all the details of the presymplectic approach and reduce the study of the problem to the study of an interesting symplectic Hamiltonian system.
The approach used in this work has important implications from the point of view of the integrability of the dynamical system on compact and connected Lie groups. For a detailed description of the optimal control problem for compact and connected Lie groups, we refer the interested reader to [12][13][14].

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The organization of the paper is as follows. In Section 2, we review the concept and some properties of higher order tangent bundles. We also recall the geometric formulation of optimal control problems and its adaptation to Skinner-Rusk methodology. Section 3 contains the geometric formulation of the second-order variational problem whose Euler-Lagrange equation is the fourth-order differential equation that defines the cubic polynomial curves on Riemannian manifolds. Section 4 is devoted to the main results of this paper; we consider the second-order optimal control problem corresponding to the variational problem presented in Section 3 and use the presymplectic constraint algorithm to describe its dynamics. In the last section, some examples are provided in order to illustrate these geometric tools.

Preliminary Results
Consider a differentiable manifold of finite dimension . Let ( 1 , . . . , ) be a local coordinate system on , simply denoted by ( ). In the paper, we assume similar simplifications to coordinate notations.

Higher Order Tangent Bundles. The tangent bundle of
can be seen as a trivial example of higher order tangent bundles. We recall very briefly some basic tools from the geometry of those bundles. For further details, see [15].
Consider the following well-defined equivalence relation on the set of smooth curves in .
We say that two smooth curves in , 1 , and 2 , defined on an interval (− , ) with ∈ R, have contact of order at 0 if 1 (0) = 2 (0) = and for a local coordinate system ( , ) on around , the derivatives of ∘ 1 and ∘ 2 up to order , included, coincide at 0. The equivalence class determined by a curve is denoted by [ ] 0 and is called a -jet or -velocity. The tangent bundle of order of , represented by , is defined as being the set of all equivalence classes. The tangent bundle is a ( + 1) -dimensional manifold, and it is also a fibered manifold over with projection A system ( , ) of local coordinates on induces natural local coordinates on given by ( −1 ( ), 0 ; 1 ; 2 ; . . . ; ), where for = 0, . . . , and = 1, . . . , . If = 0, the tangent bundle 0 is identified with the manifold and for = 1, 1 is just the tangent bundle of , . We have the canonical projections which define several different fibered structures on . Note that 0 = . Locally, ( 0 ; 1 ; 2 ; . . . ; ) = ( 0 ; 1 ; 2 ; . . . ; ).
Given a smooth curve in , the lift to of is a smooth curve in defined by ( ) = [ ] 0 , where ( ) = ( + ). If is given locally by ( ), then is locally represented by ( ; / ; . . . ; / ). We can also consider natural injections , : Here, we are particularly interested in the secondorder tangent bundle 2 . We denote the canonical local coordinates on and 2 by

Geometric Description of an Optimal Control Problem.
Let be a fiber bundle over with projection : → . Consider also a vector field Π along the projection ; that is, a smooth map Π : → such that the diagram is commutative, where represents the natural canonical projection from to . An optimal control problem with state space and control bundle consists in finding the curves : [0, ] → of class 2 , piecewise smooth and with ∈ R + , with fixed initial and final conditions in the state space, satisfying and minimizing an integral functional ∫ 0 ( ( )) , where is a smooth function : → R called cost function. Equation (6) is known as the control system, while an integral curve of Π, that is, a curve in satisfying the control system, is called a trajectory of the control system. Note that if ( ) is a local coordinate system on and ( ; ) are natural coordinates on ( = 1, . . . , , = 1, . . . , , with + = dim ), then the control system is characterized by the system of differential equationṡ Conference Papers in Mathematics 3 for ( , ) ∈ and where Π represent the canonical local coordinates of Π on , that is, The costate space of the system is the cotangent bundle * with natural canonical projection : * → . The dynamics of the control system can be described by a symplectic or a presymplectic Hamiltonian system (see, e.g., [6][7][8][9]). Here, we are interested in the presymplectic description, and hence we should consider the presymplectic Hamiltonian system (T, Ω, ) with the following.
(ii) The canonical presymplectic form Ω on T (i.e., a closed 2-form which may be degenerate) given by the pullback of the canonical symplectic form 0 on * (i.e., a closed and nondegenerate 2-form) by the where ( ; ) are the natural local coordinates on * induced by ( ). Note that the kernel of Ω is locally given by / = 0.
The dynamics of the presymplectic Hamiltonian system (T, Ω, ) is determined by the vector field solution of the equation Equation (10) is interpreted as an intrinsic version of the Hamiltonian equations that come from the maximum principle of Pontryagin in the sense that a curve in is a trajectory of the optimal control problem if there exists a lifting of to the total space T which is an integral curve of the vector field . Notice the following.
(i) Locally, = ∑ =1 ( ( / ) + ( / )) + ∑ =1 ( / ), where , , and are smooth functions on T. Hence, (ii) On the other hand, . (iv) Therefore, an integral curve of , locally given by ( ; ; ), is such thaṫ That is, is the vector field defined in the subset 1 = {( , , ) ∈ T : In the geometric framework, we have where Ker Ω( ) = {V ∈ T : Ω( )(V , ⋅) = 0}. Indeed, since Ω is presymplectic, we have to consider the points of T where (10) has solution. We assume that 1 is a submanifold of T. The dynamical vector field is determined by However, the solution of that equation is not necessarily unique, and it is possible that there exist points on 1 where the solution vector field is not tangent to 1 , and thus it does not necessarily induce a dynamics on 1 . If it is the case, we construct a second constraint submanifold 2 , that we assume to be a submanifold of 1 , defined by the points on 1 , where such a solution exists. But, again, it may happen, that we cannot guarantee the existence of a dynamics on 2 , and so the process may have to continue. This procedure is called the presymplectic constraint algorithm of Gotay-Nester [2]. The idea of the algorithm is to construct a chain of constraint manifolds until we find (if it exists) a final submanifold , where exists at least one vector field tangent to that submanifold and satisfying the dynamical equation. If the optimal control problem is regular, then = 1 .

Second-Order Variational Problem
From this section onwards, is a Riemannian manifold with Riemannian metric ⟨⋅, ⋅⟩. We denote the symmetric connection on compatible with this metric by ∇ and the corresponding covariant derivative along a curve in by / , where is a vector field along the curve. Moreover, we denote the curvature tensor field by .
We are interested in the following second-order variational problem: find the curves that minimize over the class C of smooth curves : [0, ] → satisfying the boundary conditions   [4] and later, in 1995, by Crouch and Silva Leite in the context of dynamic interpolation [3]. The Euler-Lagrange equation of the problem is the fourth-order differential equation The solutions of this equation are called cubic polynomials on .
Recall that if a curve in is locally represented by ( ), then the velocity vector field along is / = ∑ =1̇( / )| ( ) and the covariant acceleration of is given by Here, Γ are the Christoffel symbols defining the Riemannian connection, which can be obtained using the identity where are the components of the Riemannian metric and [ ] 1≤ , ≤ is the inverse matrix of the matrix [ ] 1≤ , ≤ . Moreover, the lift 2 to 2 of the curve is locally represented by ( ;̇;̈). Therefore, the action functional : C → R of our problem can be written as ( ) = ∫ 0 ( 2 ( )) , where : 2 → R is the Lagrangian of the problem. Locally, we have with = + ∑ , =1 Γ . Observe that the Lagrangian : 2 → R of the problem is defined, for each [ ] 2 0 ∈ 2 , by for 1,1 : 2 → the natural injection defined by (4) and : → the connection application locally given by The Lagrangian defines a dynamics on the third-order tangent space 3 since the Euler-Lagrange equation (16) can be interpreted as a vector field on 3 , whose integral curves are lifts to 3 of curves in solutions of the Euler-Lagrange equation.

Second-Order Optimal Control Problem
The control system associated with the variational problem of the previous section is a control system of second-order on . We now adapt for that situation the geometric description of Section 2.2.

Geometric Formulation of the Optimal Control Problem.
The second-order control system on that we are interested in is where : [0, ] → is a curve in and are real smooth functions called control functions. If is locally represented by ( ), then the control system is written as the set of differential equations dependent of the parameters since 2 / 2 is given by (17). Note that the system is affine on the controls. From a geometric point of view, the control system can be described by a vector field Π along the natural projection , that is, a smooth map Π : 2 → such that the following diagram is commutative: where is the canonical projection. The state space is and 2 is the control bundle. If [ ] 2 0 ∈ 2 , we know that Conference Now, if is locally represented by ( ; ; ), in order to describe the differential equations (23), we should consider (25) with The variables ( ; ) are called the state variables and are the control variables of our control problem. The optimal control problem consists in finding the curves : [0, ] → 2 of class 2 , piecewise smooth, with fixed endpoints in the state space satisfying the control system ( / )( 1 2 ∘ ) = Π ∘ and minimizing the functional ∫ 0 ( ( )) , for : 2 → R the cost function defined by for each [ ] 2 0 ∈ 2 , where is the connection application defined by (21). Notice that in local coordinates, the cost function is given by The relation between (19) and (30)

Presymplectic Hamiltonian
System. The Hamiltonian description of our problem has the cotangent bundle * as costate space. We consider the presymplectic system (T, Ω, ) characterized as follows.
(i) The total space is the bundle over given by where the fiber of T over a point ( , ) ∈ is * (ii) The presymplectic 2-form Ω on T is defined by the pullback where 1 denotes the canonical symplectic 2-form on * .
(iii) The Hamiltonian function : T → R is defined by where Π and are defined, respectively, by (27) and (29), and ⟨⟨⋅, ⋅⟩⟩ represents the pairing duality of vectors and covectors on .
(iv) The dynamical vector field : T → (T) is the solution of the dynamical system Ω = . (34) We now apply the geometric algorithm of presymplectic systems to (T, Ω, ). We first consider the submanifold 1 defined by (13), but adapted to our second-order problem. In this stage, it is important to do a local analysis of the presymplectic structure Ω defined by (32). If ( ) is a local coordinate system on and ( ; ; ; ) and ( ; ; ) represent, respectively, the natural local coordinates on * and 2 , then ( ; ; ; ; ) is a coordinate system on the total space T = * × 2 . In this context, it is obvious that Ω is expressed by and so Ker Ω = span{ / 1 , . . . , / }. It follows that 1 is locally defined by the constraints Note also that since our control system is affine on the controls, from (33) and (30), we get ( 2 )/( ) = −( 2 )/( ) = − , 1 ≤ , ≤ . As a consequence, the matrix is invertible, and this means that the system is regular at any point.
Consider Ω 1 the restriction to 1 of the presymplectic form Ω defined by (32).

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To conclude the proof, we have just to observe that Ker Ω( ) = ( 1 ) ⊥ . It is sufficient to verify that ( 1 ) ⊥ ⊂ Ker Ω( ) since, by definition, the opposite inclusion always happens. If V ∈ ( 1 ) ⊥ , then Ω( )(V , ) = 0, for all ∈ 1 . Furthermore, from / ∈ Ker Ω, we get Ω( )(V , / ) = 0, for all = 1, . . . , . In this way, according to (39), we conclude that V ∈ Ker Ω( ), and hence Ker Ω( ) = ( 1 ) ⊥ . Consequently, as we have the direct sum (39), we get The previous proposition assures us that ( 1 , Ω 1 ) is a symplectic manifold. As a result, the algorithm stops after the first step because we can state that there exists a unique vector field 1 on 1 solution of the dynamical system (34) when restricted to 1 , that is, such that where 1 is the restriction of to 1 . We proceed with the analysis of the obtained system and an important simplification of our study. Using (27) and (30), we obtain the following local expression for the Hamiltonian : T → R defined by (33): Then, the submanifold 1 is defined by and this implies that the optimal controls are = ∑ =1 , = 1, . . . , .
So, we can consider the diffeomorphism Observe that the inverse function of is the restriction to 1 of the projection 1 . It is easy to show that * Ω 1 = 1 , which means that defines a symplectomorphism between the symplectic manifolds ( * , 1 ) and ( 1 , Ω 1 ). This allows us to reduce the study of the dynamical system (40) on 1 , to the study of the following system on * : where 1 : * → R is defined by 1 = 1 ∘ . Locally, and the vector field 1 on * is the pushforward of 1 by −1 ; that is, The solution vector field is determined according to that is, As a consequence, the Hamiltonian equations arė for = 1, . . . , .

Optimal Control Problem on the Euclidean Space R .
A trivial example of the optimal control problem discussed in the previous section is the case = R . The tangent space of R at an arbitrary point can be identified with R , and the Riemannian metric on R is the Euclidean one. By means of Conference Papers in Mathematics 7 the canonical basis { } of R , the components of the metric are = ⋅ = and the Christoffel symbols are all null. The cost functional of the control problem is and the control system, described by a vector field Π : R 3 → R 4 along the projection 1 2 : R 3 → R 2 , is locally given bẏ = ,̇= , = 1, . . . , .
The Hamiltonian : R 5 → R of the presymplectic system that describes the dynamics of the problem is for = 1, . . . , . Note that these equations give the equations 4 / 4 = 0, = 1, . . . , , and so the curve in R locally represented by ( ) is such that 4 / 4 = 0. This corresponds to (16), since, on the Riemannian manifold R , the covariant derivative along a curve is the usual derivative along a curve in R and the curvature tensor is null. We have obtained the equation of cubic polynomials on the Euclidean space as we would expect.
As a consequence, the Hamiltonian 1 is defined on * H 2 by