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

Authors: Lígia Abrunheiro; Margarida Camarinha; Jesús Clemente-Gallardo
Journal: Conference Papers in Mathematics
 (2013)

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## Abstract

We consider a second-order variational problem depending
on the covariant acceleration, which is related to the notion of Riemannian cubic polynomials. This problem and the corresponding optimal control problem
are described in the context of higher order tangent bundles using geometric
tools. The main tool, a presymplectic variant of Pontryagin’s maximum
principle, allows us to study the dynamics of the control problem.


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## Body

## 1. 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., [6–9] 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 [12–14].The organization of the paper is as follows. In Section2, 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.


## 2. Preliminary Results

Consider a differentiable manifoldM of finite dimension n. Let (x1,…,xn) be a local coordinate system on M, simply denoted by (xi). In the paper, we assume similar simplifications to coordinate notations.


### 2.1. Higher Order Tangent Bundles

The tangent bundle ofM 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 inM. We say that two smooth curves inM, γ1, and γ2, defined on an interval (-a,a) with a∈ℝ, have contact of order k at 0 if γ1(0)=γ2(0)=x and for a local coordinate system (U,φ) on M around x, the derivatives of φ∘γ1 and φ∘γ2 up to order k, included, coincide at 0.The equivalence class determined by a curveγ is denoted by [γ]0k and is called a k-jet or k-velocity. The tangent bundle of orderkofM, represented by TkM, is defined as being the set of all equivalence classes. The tangent bundle TkM is a (k+1)  n-dimensional manifold, and it is also a fibered manifold over M with projection
(1)πk:TkM⟶M,[γ]0k⟼γ(0)=x.
A system (U,xi) of local coordinates on M induces natural local coordinates on TkM given by (πk-1(U),x0i;x1i;x2i;…;xki), where
(2)xli:πk-1(U)⊂TkM⟶ℝ,[γ]0k⟼dldtl(xi∘γ)(t)|t=0,
for l=0,…,k and i=1,…,n. If k=0, the tangent bundle T0M is identified with the manifold M and for k=1, T1M is just the tangent bundle of M, TM.We have the canonical projections(3)τkl:TkM⟶TlM,[γ]0k⟼[γ]0l,l=0,…,k,
which define several different fibered structures on TkM. Note that τk0=πk. Locally, τkl(x0i;x1i;x2i;…;xki)=(x0i;x1i;x2i;…;xli).Given a smooth curveγ in M, the lift to TkM of γ is a smooth curve in TkM defined by γk(t)=[γt]0k, where γt(s)=γ(t+s). If γ is given locally by (xi), then γk is locally represented by (xi;dxi/dt;…;dkxi/dtk).We can also consider natural injectionsik,l:Tk+lM→Tk(TlM),[γ]0k+l↦[γl]0k, for l=1,…,k.Here, we are particularly interested in the second-order tangent bundleT2M. We denote the canonical local coordinates on TM and T2M by (x1,…,xn;y1,…,yn)=(xi;yi) and (x1,…,xn;y1,…,yn;u1,…,un)=(xi;yi;ui), respectively. The natural bases of the tangent spaces TxM, T(x,y)TM, and T(x,y,u)T2M are denoted, respectively, by {∂/∂xi}, {∂/∂xi,∂/∂yi}, and {∂/∂xi,∂/∂yi,∂/∂ui}, for x∈M, (x,y)∈TM, and (x,y,u)∈T2M. For clarity, to distinguish the projections π1 of the tangent bundles TM and TTM, we use the notations πM:TM→M and πTM:TTM→TM. Moreover, the natural injection i1,1:T2M→TTM is locally defined by
(4)i1,1(xi;yi;ui)=∑i=1n‍(yi∂∂xi+ui∂∂yi).


### 2.2. Geometric Description of an Optimal Control Problem

LetB be a fiber bundle over M with projection τ:B→M. Consider also a vector field Π along the projection τ; that is, a smooth map Π:B→TM such that the diagram
(5)
is commutative, where πM represents the natural canonical projection from TM to M.Anoptimal control problem with state space M and control bundle B consists in finding the curves α:[0,T]→B of class C2, piecewise smooth and with T∈ℝ+, with fixed initial and final conditions in the state space, satisfying
(6)ddt(τ∘α)=Π∘α
and minimizing an integral functional ∫0T‍L(α(t))dt, where L is a smooth function L:B→ℝ called cost function.Equation (6) is known as the control system, while an integral curve of Π, that is, a curve α in B satisfying the control system, is called a trajectory of the control system. Note that if (xi) is a local coordinate system on M and (xi;ua) are natural coordinates on B (i=1,…,n, a=1,…,m, with n+m=dimB), then the control system is characterized by the system of differential equations
(7)x˙i=Πi(x,u),i=1,…,n,
for (x,u)∈B and where Πi represent the canonical local coordinates of Π on TM, that is, Π(x,u)=∑i=1n‍Πi(x,u)(∂/∂xi)|τ(x,u).The costate space of the system is the cotangent bundleT*M with natural canonical projection τM:T*M→M. The dynamics of the control system can be described by a symplectic or a presymplectic Hamiltonian system (see, e.g., [6–9]). Here, we are interested in the presymplectic description, and hence we should consider the presymplectic Hamiltonian system (𝒯,Ω,H) with the following. (i)
The total space𝒯=T*M×MB a fiber bundle over the manifold M with canonical projections pr1:T*M×MB→T*M and pr2:T*M×MB→B.(ii)
The canonical presymplectic formΩ on 𝒯 (i.e., a closed 2-form which may be degenerate) given by the pullback of the canonical symplectic form ω0 on T*M (i.e., a closed and nondegenerate 2-form) by the projection pr1, Ω=(pr1)*ω0. Locally,
(8)Ω=∑i=1n‍dxi∧dpi,where(xi;pi) are the natural local coordinates on T*M induced by (xi). Note that the kernel of Ω is locally given by ∂/∂ua=0.(iii)
The HamiltonianH:T*M×MB→ℝ defined by H=〈〈pr1,Π∘pr2〉〉-L∘pr2, where 〈〈·,·〉〉 represents the pairing duality of vectors and covectors on M. We get
(9)H(x,p,u)=∑i=1n‍piΠi(x,u)-L(x,u),for each(x,p,u)∈𝒯 (i.e., x∈M, p∈Tx*M and (x,u)∈B, with τ(x,u)=x).The dynamics of the presymplectic Hamiltonian system(𝒯,Ω,H) is determined by the vector field XH solution of the equation
(10)iXHΩ=dH.
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 α in B is a trajectory of the optimal control problem if there exists a lifting of α to the total space 𝒯 which is an integral curve of the vector field XH. Notice the following.  (i)
Locally,XH=∑i=1n‍(Ai(∂/∂xi)+Bi(∂/∂pi))+∑a=1m‍Ca(∂/∂ua), where Ai, Bi, and Ca are smooth functions on 𝒯. Hence, iXHΩ=∑i=1n‍(Aidpi-Bidxi).(ii)
On the other hand,dH=∑i=1n‍((∂H/∂xi)dxi+(∂H/∂pi)dpi)+∑a=1m‍(∂H/∂ua)dua.(iii)
So, (10) is equivalent to Ai=∂H/∂pi, Bi=-∂H/∂xi, and ∂H/∂ua=0, i=1,…,n, a=1,…,m.(iv)
Therefore, an integral curve ofXH, locally given by (xi;pi;ui), is such that
(11)x˙i=∂H∂pi,p˙i=-∂H∂xi,∂H∂ua=0,i=1,…,n,a=1,…,mThat is,XH is the vector field
(12)XH=∑i=1n‍(∂H∂pi∂∂xi-∂H∂xi∂∂pi)defined in the subsetW1={(x,p,u)∈𝒯:∂H/∂ua=0,a=1,…,m}.In the geometric framework, we have(13)W1={z∈𝒯:dH(z)(vz)=0,∀vz∈KerΩ(z)},
where KerΩ(z)={vz∈Tz𝒯:Ω(z)(vz,·)=0}. Indeed, since Ω is presymplectic, we have to consider the points of 𝒯 where (10) has solution. We assume that W1 is a submanifold of 𝒯. The dynamical vector field XH is determined by (iXHΩ-dH)|W1=0. However, the solution of that equation is not necessarily unique, and it is possible that there exist points on W1 where the solution vector field is not tangent to W1, and thus it does not necessarily induce a dynamics on W1. If it is the case, we construct a second constraint submanifold W2, that we assume to be a submanifold of W1, defined by the points on W1, where such a solution exists. But, again, it may happen, that we cannot guarantee the existence of a dynamics on W2, 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 Wf, where exists at least one vector field tangent to that submanifold and satisfying the dynamical equation. If the optimal control problem is regular, then Wf=W1.


## 2.1. Higher Order Tangent Bundles

The tangent bundle ofM 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 inM. We say that two smooth curves inM, γ1, and γ2, defined on an interval (-a,a) with a∈ℝ, have contact of order k at 0 if γ1(0)=γ2(0)=x and for a local coordinate system (U,φ) on M around x, the derivatives of φ∘γ1 and φ∘γ2 up to order k, included, coincide at 0.The equivalence class determined by a curveγ is denoted by [γ]0k and is called a k-jet or k-velocity. The tangent bundle of orderkofM, represented by TkM, is defined as being the set of all equivalence classes. The tangent bundle TkM is a (k+1)  n-dimensional manifold, and it is also a fibered manifold over M with projection
(1)πk:TkM⟶M,[γ]0k⟼γ(0)=x.
A system (U,xi) of local coordinates on M induces natural local coordinates on TkM given by (πk-1(U),x0i;x1i;x2i;…;xki), where
(2)xli:πk-1(U)⊂TkM⟶ℝ,[γ]0k⟼dldtl(xi∘γ)(t)|t=0,
for l=0,…,k and i=1,…,n. If k=0, the tangent bundle T0M is identified with the manifold M and for k=1, T1M is just the tangent bundle of M, TM.We have the canonical projections(3)τkl:TkM⟶TlM,[γ]0k⟼[γ]0l,l=0,…,k,
which define several different fibered structures on TkM. Note that τk0=πk. Locally, τkl(x0i;x1i;x2i;…;xki)=(x0i;x1i;x2i;…;xli).Given a smooth curveγ in M, the lift to TkM of γ is a smooth curve in TkM defined by γk(t)=[γt]0k, where γt(s)=γ(t+s). If γ is given locally by (xi), then γk is locally represented by (xi;dxi/dt;…;dkxi/dtk).We can also consider natural injectionsik,l:Tk+lM→Tk(TlM),[γ]0k+l↦[γl]0k, for l=1,…,k.Here, we are particularly interested in the second-order tangent bundleT2M. We denote the canonical local coordinates on TM and T2M by (x1,…,xn;y1,…,yn)=(xi;yi) and (x1,…,xn;y1,…,yn;u1,…,un)=(xi;yi;ui), respectively. The natural bases of the tangent spaces TxM, T(x,y)TM, and T(x,y,u)T2M are denoted, respectively, by {∂/∂xi}, {∂/∂xi,∂/∂yi}, and {∂/∂xi,∂/∂yi,∂/∂ui}, for x∈M, (x,y)∈TM, and (x,y,u)∈T2M. For clarity, to distinguish the projections π1 of the tangent bundles TM and TTM, we use the notations πM:TM→M and πTM:TTM→TM. Moreover, the natural injection i1,1:T2M→TTM is locally defined by
(4)i1,1(xi;yi;ui)=∑i=1n‍(yi∂∂xi+ui∂∂yi).


## 2.2. Geometric Description of an Optimal Control Problem

LetB be a fiber bundle over M with projection τ:B→M. Consider also a vector field Π along the projection τ; that is, a smooth map Π:B→TM such that the diagram
(5)
is commutative, where πM represents the natural canonical projection from TM to M.Anoptimal control problem with state space M and control bundle B consists in finding the curves α:[0,T]→B of class C2, piecewise smooth and with T∈ℝ+, with fixed initial and final conditions in the state space, satisfying
(6)ddt(τ∘α)=Π∘α
and minimizing an integral functional ∫0T‍L(α(t))dt, where L is a smooth function L:B→ℝ called cost function.Equation (6) is known as the control system, while an integral curve of Π, that is, a curve α in B satisfying the control system, is called a trajectory of the control system. Note that if (xi) is a local coordinate system on M and (xi;ua) are natural coordinates on B (i=1,…,n, a=1,…,m, with n+m=dimB), then the control system is characterized by the system of differential equations
(7)x˙i=Πi(x,u),i=1,…,n,
for (x,u)∈B and where Πi represent the canonical local coordinates of Π on TM, that is, Π(x,u)=∑i=1n‍Πi(x,u)(∂/∂xi)|τ(x,u).The costate space of the system is the cotangent bundleT*M with natural canonical projection τM:T*M→M. The dynamics of the control system can be described by a symplectic or a presymplectic Hamiltonian system (see, e.g., [6–9]). Here, we are interested in the presymplectic description, and hence we should consider the presymplectic Hamiltonian system (𝒯,Ω,H) with the following. (i)
The total space𝒯=T*M×MB a fiber bundle over the manifold M with canonical projections pr1:T*M×MB→T*M and pr2:T*M×MB→B.(ii)
The canonical presymplectic formΩ on 𝒯 (i.e., a closed 2-form which may be degenerate) given by the pullback of the canonical symplectic form ω0 on T*M (i.e., a closed and nondegenerate 2-form) by the projection pr1, Ω=(pr1)*ω0. Locally,
(8)Ω=∑i=1n‍dxi∧dpi,where(xi;pi) are the natural local coordinates on T*M induced by (xi). Note that the kernel of Ω is locally given by ∂/∂ua=0.(iii)
The HamiltonianH:T*M×MB→ℝ defined by H=〈〈pr1,Π∘pr2〉〉-L∘pr2, where 〈〈·,·〉〉 represents the pairing duality of vectors and covectors on M. We get
(9)H(x,p,u)=∑i=1n‍piΠi(x,u)-L(x,u),for each(x,p,u)∈𝒯 (i.e., x∈M, p∈Tx*M and (x,u)∈B, with τ(x,u)=x).The dynamics of the presymplectic Hamiltonian system(𝒯,Ω,H) is determined by the vector field XH solution of the equation
(10)iXHΩ=dH.
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 α in B is a trajectory of the optimal control problem if there exists a lifting of α to the total space 𝒯 which is an integral curve of the vector field XH. Notice the following.  (i)
Locally,XH=∑i=1n‍(Ai(∂/∂xi)+Bi(∂/∂pi))+∑a=1m‍Ca(∂/∂ua), where Ai, Bi, and Ca are smooth functions on 𝒯. Hence, iXHΩ=∑i=1n‍(Aidpi-Bidxi).(ii)
On the other hand,dH=∑i=1n‍((∂H/∂xi)dxi+(∂H/∂pi)dpi)+∑a=1m‍(∂H/∂ua)dua.(iii)
So, (10) is equivalent to Ai=∂H/∂pi, Bi=-∂H/∂xi, and ∂H/∂ua=0, i=1,…,n, a=1,…,m.(iv)
Therefore, an integral curve ofXH, locally given by (xi;pi;ui), is such that
(11)x˙i=∂H∂pi,p˙i=-∂H∂xi,∂H∂ua=0,i=1,…,n,a=1,…,mThat is,XH is the vector field
(12)XH=∑i=1n‍(∂H∂pi∂∂xi-∂H∂xi∂∂pi)defined in the subsetW1={(x,p,u)∈𝒯:∂H/∂ua=0,a=1,…,m}.In the geometric framework, we have(13)W1={z∈𝒯:dH(z)(vz)=0,∀vz∈KerΩ(z)},
where KerΩ(z)={vz∈Tz𝒯:Ω(z)(vz,·)=0}. Indeed, since Ω is presymplectic, we have to consider the points of 𝒯 where (10) has solution. We assume that W1 is a submanifold of 𝒯. The dynamical vector field XH is determined by (iXHΩ-dH)|W1=0. However, the solution of that equation is not necessarily unique, and it is possible that there exist points on W1 where the solution vector field is not tangent to W1, and thus it does not necessarily induce a dynamics on W1. If it is the case, we construct a second constraint submanifold W2, that we assume to be a submanifold of W1, defined by the points on W1, where such a solution exists. But, again, it may happen, that we cannot guarantee the existence of a dynamics on W2, 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 Wf, where exists at least one vector field tangent to that submanifold and satisfying the dynamical equation. If the optimal control problem is regular, then Wf=W1.


## 3. Second-Order Variational Problem

From this section onwards,M is a Riemannian manifold with Riemannian metric 〈·,·〉. We denote the symmetric connection on M compatible with this metric by ∇ and the corresponding covariant derivative along a curve in M by DX/dt, where X is a vector field along the curve. Moreover, we denote the curvature tensor field by R.We are interested in the following second-order variational problem: find the curves that minimize(14)J(γ)=12∫0T‍〈D2γdt2,D2γdt2〉dt,
over the class 𝒞 of smooth curves γ:[0,T]→M satisfying the boundary conditions
(15)γ(0)=x0,γ(T)=xT,dγdt(0)=y0,dγdt(T)=yT,
where x0,xT∈M, y0∈Tx0M, yT∈TxTM, and T∈ℝ+. This problem was studied in 1989 by Noakes et al. on compact Lie groups [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
(16)D4γdt4+R(D2γdt2,dγdt)dγdt=0.
The solutions of this equation are called cubic polynomials on M.Recall that if a curveγ in M is locally represented by (xi), then the velocity vector field along γ is dγ/dt=∑i=1n‍x˙i(∂/∂xi)|γ(t) and the covariant acceleration of γ is given by
(17)D2γdt2=∑k=1n‍(x¨k+∑i,j=1n‍Γijkx˙ix˙j)∂∂xk|γ(t).
Here, Γijk are the Christoffel symbols defining the Riemannian connection, which can be obtained using the identity
(18)Γijk=12∑l=1n‍gkl(∂gjl∂xi+∂gli∂xj-∂gij∂xl),
where gij are the components of the Riemannian metric and [gij]1≤i,j≤n is the inverse matrix of the matrix [gij]1≤i,j≤n. Moreover, the lift γ2 to T2M of the curve γ is locally represented by (xi;x˙i;x¨i). Therefore, the action functional J:𝒞→ℝ of our problem can be written as J(γ)=∫0T‍L(γ2(t))dt, where L:T2M→ℝ is the Lagrangian of the problem. Locally, we have
(19)L(xi;yi;ui)=12∑i=1n‍giju-iu-j,
with u-k=uk+∑i,j=1n‍Γijkyiyj.Observe that the LagrangianL:T2M→ℝ of the problem is defined, for each [γ]02∈T2M, by
(20)L([γ]02)=12〈(K∘i1,1)([γ]02),(K∘i1,1)([γ]02)〉,
for i1,1:T2M→TTM the natural injection defined by (4) and K:TTM→TM the connection application locally given by
(21)K(∑i=1n‍Xi∂∂xi+∑i=1n‍Xi+n∂∂yi)=∑i=1n‍(Xn+i+∑j,k=1n‍ΓjkiyjXk)∂∂xi.The LagrangianL defines a dynamics on the third-order tangent space T3M since the Euler-Lagrange equation (16) can be interpreted as a vector field on T3M, whose integral curves are lifts to T3M of curves in M solutions of the Euler-Lagrange equation.


## 4. Second-Order Optimal Control Problem

The control system associated with the variational problem of the previous section is a control system of second-order onM. We now adapt for that situation the geometric description of Section 2.2.


### 4.1. Geometric Formulation of the Optimal Control Problem

The second-order control system onM that we are interested in is
(22)D2γdt2=∑i=1n‍ui∂∂xi|γ(t),
where γ:[0,T]→M is a curve in M and ui are real smooth functions called control functions. If γ is locally represented by (xi), then the control system is written as the set of differential equations dependent of the parameters ui(23)x¨i=ui-∑j,k=1n‍Γjkix˙jx˙k,i=1,…,n,
since D2γ/dt2 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 fieldΠ along the natural projection τ21:T2M→TM, that is, a smooth map Π:T2M→TTM such that the following diagram is commutative:
(24)
where πTM is the canonical projection. The state space is TM and T2M is the control bundle. If [γ]02∈T2M, we know that τ21([γ]02)=[γ]01∈TM and hence Π([γ]02)∈T[γ]01TM. Consequently, for real smooth functions Πi:T2M→ℝ, with i=1,…,n, the vector field Π can be expressed as Π([γ]02)=∑i=1nΠi([γ]02)(∂/∂xi)|[γ]01+∑i=1nΠi+n([γ]02)(∂/∂yi)|[γ]01. Along a curve α:[0,T]→T2M, we have
(25)Π∘α=∑i=1n‍(Πi∘α)∂∂xi|(τ21∘α)(t)+∑i=1n‍(Πi+n∘α)∂∂yi|(τ21∘α)(t).
Furthermore, if α is an integral curve of Π, then
(26)Now, ifα is locally represented by (xi;yi;ui), in order to describe the differential equations (23), we should consider (25) with
(27)Πi∘α=yi,Πi+n∘α=ui-∑j,k=1n‍Γjkiyjyk.
The variables (xi;yi) are called the state variables and ui are the control variables of our control problem.The optimal control problem consists in finding the curvesα:[0,T]→T2M of class C2, piecewise smooth, with fixed endpoints in the state space
(28)(τ21∘α)(0)=(x0,y0),(τ21∘α)(T)=(xT,yT),
satisfying the control system (d/dt)(τ21∘α)=Π∘α and minimizing the functional ∫0T‍G(α(t))dt, for G:T2M→ℝ the cost function defined by
(29)G([γ]02)=12〈K(Π([γ]02)),K(Π([γ]02))〉,
for each [γ]02∈T2M, where K is the connection application defined by (21). Notice that in local coordinates, the cost function is given by
(30)G(xi;yi;ui)=12∑i,j=1n‍gijuiuj.
The relation between (19) and (30) is L(xi;yi;ui-∑k=1n‍Γijkyiyj)=G(xi;yi;ui), or equivalently, L(xi;yi;ui)=G(xi;yi;u-i).


### 4.2. Presymplectic Hamiltonian System

The Hamiltonian description of our problem has the cotangent bundleT*TM as costate space. We consider the presymplectic system (𝒯,Ω,H) characterized as follows.  (i)
The total space is the bundle overTM given by
(31)𝒯=T*TM×TMT2M,where the fiber of𝒯 over a point (x,y)∈TM is T(x,y)*TM×(τ21)-1(x,y). The canonical projections are pr1:T*TM×TMT2M→T*TM and pr2:T*TM×TMT2M→T2M.(ii)
The presymplectic 2-formΩ on 𝒯 is defined by the pullback
(32)Ω=(pr1)*ω1,whereω1 denotes the canonical symplectic 2-form on T*TM.(iii)
The Hamiltonian functionH:𝒯→ℝ is defined by
(33)H=〈〈pr1,Π∘pr2〉〉-G∘pr2,whereΠ and G are defined, respectively, by (27) and (29), and 〈〈·,·〉〉 represents the pairing duality of vectors and covectors on TM.(iv)
The dynamical vector fieldXH:𝒯→T(𝒯) is the solution of the dynamical system
(34)iXHΩ=dH.We now apply the geometric algorithm of presymplectic systems to(𝒯,Ω,H). We first consider the submanifold W1 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 (xi) is a local coordinate system on M and (xi;yi;pi;qi) and (xi;yi;ui) represent, respectively, the natural local coordinates on T*TM and T2M, then (xi;yi;ui;pi;qi) is a coordinate system on the total space 𝒯=T*TM×TMT2M. In this context, it is obvious that Ω is expressed by
(35)Ω=∑i=1n‍(dxi∧dpi+dyi∧dqi),
and so KerΩ=span{∂/∂u1,…,∂/∂un}. It follows that W1 is locally defined by the constraints
(36)∂H∂ui=0,i=1,…,n.Note also that since our control system is affine on the controls, from (33) and (30), we get (∂2H)/(∂ui∂uj)=-(∂2G)/(∂ui∂uj)=-gij, 1≤i,  j≤n. As a consequence, the matrix
(37)[∂2H∂ui∂uj]1≤i,j≤n
is invertible, and this means that the system is regular at any point.ConsiderΩW1 the restriction to W1 of the presymplectic form Ω defined by (32).Proposition 1.
The 2-formΩW1 is symplectic.Proof.
Recall thatΩW1 is symplectic if and only if TzW1∩(TzW1)⊥={0}, for all z∈W1, where
(38)(TzW1)⊥={vz∈Tz𝒯:Ω(z)(vz,uz)=0,∀uz∈TzW1}.
Letz∈W1. Using the fact that W1 is defined by (36), we can conclude that ∂/∂uj∉TzW1, for j=1,…,n. Indeed, suppose that there exists a j such that ∂/∂uj∈TzW1; this means that d(∂H/∂ui)(∂/∂uj)=(∂2H)/(∂ui∂uj)=0, for i=1,…,n, but this is not true because of the invertibility of the matrix (37). Therefore,
(39)TzW1⊕KerΩ(z)=Tz𝒯.
To conclude the proof, we have just to observe that KerΩ(z)=(TzW1)⊥. It is sufficient to verify that (TzW1)⊥⊂KerΩ(z) since, by definition, the opposite inclusion always happens. If vz∈(TzW1)⊥, then Ω(z)(vz,uz)=0, for all uz∈TzW1. Furthermore, from ∂/∂ui∈KerΩ, we get Ω(z)(vz,∂/∂ui)=0, for all i=1,…,n. In this way, according to (39), we conclude that vz∈KerΩ(z), and hence KerΩ(z)=(TzW1)⊥. Consequently, as we have the direct sum (39), we get TzW1∩(TzW1)⊥={0}.The previous proposition assures us that(W1,ΩW1) 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 XW1 on W1 solution of the dynamical system (34) when restricted to W1, that is, such that
(40)iXW1ΩW1=dHW1,
where HW1 is the restriction of H to W1.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 H:𝒯→ℝ defined by (33):
(41)H(xi;yi;ui;pi;qi)=∑i=1n‍piyi+∑i=1n‍qi(ui-∑j,k=1n‍Γjkiyjyk)-12∑i,j=1n‍gijuiuj.
Then, the submanifold W1 is defined by
(42)qi=∑j=1n‍gijuj,i=1,…,n,
and this implies that the optimal controls are
(43)ui=∑j=1n‍gijqj,i=1,…,n.
So, we can consider the diffeomorphism
(44)φ:(T*TM,ω1)⟶(W1,ΩW1),(xi;yi;pi;qi)⟼(xi;yi;∑j=1n‍gijqj;pi,qi).
Observe that the inverse function of φ is the restriction to W1 of the projection pr1. It is easy to show that φ*ΩW1=ω1, which means that φ defines a symplectomorphism between the symplectic manifolds (T*TM,ω1) and (W1,ΩW1). This allows us to reduce the study of the dynamical system (40) on W1, to the study of the following system on T*TM:
(45)iXH1ω1=dH1,
where H1:T*TM→ℝ is defined by H1=HW1∘φ. Locally,
(46)H1(xi;yi;pi;qi)=∑i=1n‍piyi+∑i=1n‍qi(12∑l=1n‍gilql-∑j,k=1n‍Γjkiyjyk)
and the vector field XH1 on T*TM is the pushforward of XW1 by φ-1; that is,
(47)XH1=(φ-1)*XW1=Tφ-1∘XW1∘φ.
The solution vector field is determined according to
(48)XH1(xi;yi;pi;qi)=∑i=1n‍(∂H1∂pi∂∂xi+∂H1∂qi∂∂yi-∂H1∂xi∂∂pi-∂H1∂yi∂∂qi);
that is,
(49)XH1(xi;yi;pi;qi)=∑i=1n‍[yi∂∂xi+(∑j=1n‍gijqj-∑j,k=1n‍Γjkiyjyk)∂∂yi+∑j,k,l=1n‍qj(∂Γlkj∂xiylyk-12∂gjk∂xiqk)∂∂pi+(2∑j,k=1n‍qjΓikjyk-pi)∂∂qi].
As a consequence, the Hamiltonian equations are
(50)x˙i=yi,y˙i=∑j=1n‍gijqj-∑j,k=1n‍Γjkiyjyk,p˙i=∑j,k,l=1n‍qj(∂Γlkj∂xiylyk-12∂gjk∂xiqk),q˙i=2∑j,k=1n‍qjΓikjyk-pi,
for i=1,…,n.


## 4.1. Geometric Formulation of the Optimal Control Problem

The second-order control system onM that we are interested in is
(22)D2γdt2=∑i=1n‍ui∂∂xi|γ(t),
where γ:[0,T]→M is a curve in M and ui are real smooth functions called control functions. If γ is locally represented by (xi), then the control system is written as the set of differential equations dependent of the parameters ui(23)x¨i=ui-∑j,k=1n‍Γjkix˙jx˙k,i=1,…,n,
since D2γ/dt2 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 fieldΠ along the natural projection τ21:T2M→TM, that is, a smooth map Π:T2M→TTM such that the following diagram is commutative:
(24)
where πTM is the canonical projection. The state space is TM and T2M is the control bundle. If [γ]02∈T2M, we know that τ21([γ]02)=[γ]01∈TM and hence Π([γ]02)∈T[γ]01TM. Consequently, for real smooth functions Πi:T2M→ℝ, with i=1,…,n, the vector field Π can be expressed as Π([γ]02)=∑i=1nΠi([γ]02)(∂/∂xi)|[γ]01+∑i=1nΠi+n([γ]02)(∂/∂yi)|[γ]01. Along a curve α:[0,T]→T2M, we have
(25)Π∘α=∑i=1n‍(Πi∘α)∂∂xi|(τ21∘α)(t)+∑i=1n‍(Πi+n∘α)∂∂yi|(τ21∘α)(t).
Furthermore, if α is an integral curve of Π, then
(26)Now, ifα is locally represented by (xi;yi;ui), in order to describe the differential equations (23), we should consider (25) with
(27)Πi∘α=yi,Πi+n∘α=ui-∑j,k=1n‍Γjkiyjyk.
The variables (xi;yi) are called the state variables and ui are the control variables of our control problem.The optimal control problem consists in finding the curvesα:[0,T]→T2M of class C2, piecewise smooth, with fixed endpoints in the state space
(28)(τ21∘α)(0)=(x0,y0),(τ21∘α)(T)=(xT,yT),
satisfying the control system (d/dt)(τ21∘α)=Π∘α and minimizing the functional ∫0T‍G(α(t))dt, for G:T2M→ℝ the cost function defined by
(29)G([γ]02)=12〈K(Π([γ]02)),K(Π([γ]02))〉,
for each [γ]02∈T2M, where K is the connection application defined by (21). Notice that in local coordinates, the cost function is given by
(30)G(xi;yi;ui)=12∑i,j=1n‍gijuiuj.
The relation between (19) and (30) is L(xi;yi;ui-∑k=1n‍Γijkyiyj)=G(xi;yi;ui), or equivalently, L(xi;yi;ui)=G(xi;yi;u-i).


## 4.2. Presymplectic Hamiltonian System

The Hamiltonian description of our problem has the cotangent bundleT*TM as costate space. We consider the presymplectic system (𝒯,Ω,H) characterized as follows.  (i)
The total space is the bundle overTM given by
(31)𝒯=T*TM×TMT2M,where the fiber of𝒯 over a point (x,y)∈TM is T(x,y)*TM×(τ21)-1(x,y). The canonical projections are pr1:T*TM×TMT2M→T*TM and pr2:T*TM×TMT2M→T2M.(ii)
The presymplectic 2-formΩ on 𝒯 is defined by the pullback
(32)Ω=(pr1)*ω1,whereω1 denotes the canonical symplectic 2-form on T*TM.(iii)
The Hamiltonian functionH:𝒯→ℝ is defined by
(33)H=〈〈pr1,Π∘pr2〉〉-G∘pr2,whereΠ and G are defined, respectively, by (27) and (29), and 〈〈·,·〉〉 represents the pairing duality of vectors and covectors on TM.(iv)
The dynamical vector fieldXH:𝒯→T(𝒯) is the solution of the dynamical system
(34)iXHΩ=dH.We now apply the geometric algorithm of presymplectic systems to(𝒯,Ω,H). We first consider the submanifold W1 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 (xi) is a local coordinate system on M and (xi;yi;pi;qi) and (xi;yi;ui) represent, respectively, the natural local coordinates on T*TM and T2M, then (xi;yi;ui;pi;qi) is a coordinate system on the total space 𝒯=T*TM×TMT2M. In this context, it is obvious that Ω is expressed by
(35)Ω=∑i=1n‍(dxi∧dpi+dyi∧dqi),
and so KerΩ=span{∂/∂u1,…,∂/∂un}. It follows that W1 is locally defined by the constraints
(36)∂H∂ui=0,i=1,…,n.Note also that since our control system is affine on the controls, from (33) and (30), we get (∂2H)/(∂ui∂uj)=-(∂2G)/(∂ui∂uj)=-gij, 1≤i,  j≤n. As a consequence, the matrix
(37)[∂2H∂ui∂uj]1≤i,j≤n
is invertible, and this means that the system is regular at any point.ConsiderΩW1 the restriction to W1 of the presymplectic form Ω defined by (32).Proposition 1.
The 2-formΩW1 is symplectic.Proof.
Recall thatΩW1 is symplectic if and only if TzW1∩(TzW1)⊥={0}, for all z∈W1, where
(38)(TzW1)⊥={vz∈Tz𝒯:Ω(z)(vz,uz)=0,∀uz∈TzW1}.
Letz∈W1. Using the fact that W1 is defined by (36), we can conclude that ∂/∂uj∉TzW1, for j=1,…,n. Indeed, suppose that there exists a j such that ∂/∂uj∈TzW1; this means that d(∂H/∂ui)(∂/∂uj)=(∂2H)/(∂ui∂uj)=0, for i=1,…,n, but this is not true because of the invertibility of the matrix (37). Therefore,
(39)TzW1⊕KerΩ(z)=Tz𝒯.
To conclude the proof, we have just to observe that KerΩ(z)=(TzW1)⊥. It is sufficient to verify that (TzW1)⊥⊂KerΩ(z) since, by definition, the opposite inclusion always happens. If vz∈(TzW1)⊥, then Ω(z)(vz,uz)=0, for all uz∈TzW1. Furthermore, from ∂/∂ui∈KerΩ, we get Ω(z)(vz,∂/∂ui)=0, for all i=1,…,n. In this way, according to (39), we conclude that vz∈KerΩ(z), and hence KerΩ(z)=(TzW1)⊥. Consequently, as we have the direct sum (39), we get TzW1∩(TzW1)⊥={0}.The previous proposition assures us that(W1,ΩW1) 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 XW1 on W1 solution of the dynamical system (34) when restricted to W1, that is, such that
(40)iXW1ΩW1=dHW1,
where HW1 is the restriction of H to W1.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 H:𝒯→ℝ defined by (33):
(41)H(xi;yi;ui;pi;qi)=∑i=1n‍piyi+∑i=1n‍qi(ui-∑j,k=1n‍Γjkiyjyk)-12∑i,j=1n‍gijuiuj.
Then, the submanifold W1 is defined by
(42)qi=∑j=1n‍gijuj,i=1,…,n,
and this implies that the optimal controls are
(43)ui=∑j=1n‍gijqj,i=1,…,n.
So, we can consider the diffeomorphism
(44)φ:(T*TM,ω1)⟶(W1,ΩW1),(xi;yi;pi;qi)⟼(xi;yi;∑j=1n‍gijqj;pi,qi).
Observe that the inverse function of φ is the restriction to W1 of the projection pr1. It is easy to show that φ*ΩW1=ω1, which means that φ defines a symplectomorphism between the symplectic manifolds (T*TM,ω1) and (W1,ΩW1). This allows us to reduce the study of the dynamical system (40) on W1, to the study of the following system on T*TM:
(45)iXH1ω1=dH1,
where H1:T*TM→ℝ is defined by H1=HW1∘φ. Locally,
(46)H1(xi;yi;pi;qi)=∑i=1n‍piyi+∑i=1n‍qi(12∑l=1n‍gilql-∑j,k=1n‍Γjkiyjyk)
and the vector field XH1 on T*TM is the pushforward of XW1 by φ-1; that is,
(47)XH1=(φ-1)*XW1=Tφ-1∘XW1∘φ.
The solution vector field is determined according to
(48)XH1(xi;yi;pi;qi)=∑i=1n‍(∂H1∂pi∂∂xi+∂H1∂qi∂∂yi-∂H1∂xi∂∂pi-∂H1∂yi∂∂qi);
that is,
(49)XH1(xi;yi;pi;qi)=∑i=1n‍[yi∂∂xi+(∑j=1n‍gijqj-∑j,k=1n‍Γjkiyjyk)∂∂yi+∑j,k,l=1n‍qj(∂Γlkj∂xiylyk-12∂gjk∂xiqk)∂∂pi+(2∑j,k=1n‍qjΓikjyk-pi)∂∂qi].
As a consequence, the Hamiltonian equations are
(50)x˙i=yi,y˙i=∑j=1n‍gijqj-∑j,k=1n‍Γjkiyjyk,p˙i=∑j,k,l=1n‍qj(∂Γlkj∂xiylyk-12∂gjk∂xiqk),q˙i=2∑j,k=1n‍qjΓikjyk-pi,
for i=1,…,n.


## 5. Examples

### 5.1. Optimal Control Problem on the Euclidean Spaceℝn

A trivial example of the optimal control problem discussed in the previous section is the caseM=ℝn. The tangent space of ℝn at an arbitrary point can be identified with ℝn, and the Riemannian metric on ℝn is the Euclidean one. By means of the canonical basis {ei} of ℝn, the components of the metric are gij=ei·ej=δij and the Christoffel symbols are all null.The cost functional of the control problem is(51)G:ℝ3n⟶ℝ,G(xi;yi;ui)=12∑i=1n‍(ui)2,
and the control system, described by a vector field Π:ℝ3n→ℝ4n along the projection τ21:ℝ3n→ℝ2n, is locally given by
(52)x˙i=yi,y˙i=ui,i=1,…,n.The HamiltonianH:ℝ5n→ℝ of the presymplectic system that describes the dynamics of the problem is
(53)H(xi;yi;ui;pi;qi)=∑i=1n‍[piyi+qiui-12(ui)2].
Therefore, the constraint equations ∂H/∂ui=0, i=1,…,n, defining the symplectic manifold W1 are ui=qi, i=1,…,n. Consequently, the Hamiltonian H1 defined on the symplectic manifold ℝ4n becomes
(54)H1(xi;yi;pi;qi)=∑i=1n‍[piyi+12(qi)2].
From (48), we know that XH1(xi;yi;pi;qi)=∑i=1n‍[yi(∂/∂xi)+qi(∂/∂yi)-pi(∂/∂qi)] is the solution vector field on ℝ4n and the corresponding Hamiltonian equations are
(55)x˙i=yi,y˙i=qi,p˙i=0,q˙i=-pi,
for i=1,…,n. Note that these equations give the equations d4xi/dt4=0, i=1,…,n, and so the curve γ in ℝn locally represented by (xi) is such that d4γ/dt4=0. This corresponds to (16), since, on the Riemannian manifold ℝn, the covariant derivative along a curve is the usual derivative along a curve in ℝn and the curvature tensor is null. We have obtained the equation of cubic polynomials on the Euclidean space as we would expect.


### 5.2. Optimal Control Problem on the SphereS2

LetS2={x∈ℝ3:∥x∥=a} be the sphere of radius a with the usual spherical coordinates x1=θ and x2=φ. S2 is equipped with the Riemannian metric ds2=a2sin2φdθ2+a2dφ2, and so g12=g21=0, g22=a2, and g11=a2sin2φ. Using (18), we obtain the Christoffel symbols: Γ121=Γ211=cotφ, Γ112=-sinφcosφ, and all the others are null.Consider the optimal control problem of the previous section onM=S2. The state space is TS2, the bundle of controls is T2S2, and the costate space is T*TS2. We represent the corresponding coordinates on these spaces by (θ,φ,yθ,yφ), (θ,φ,yθ,yφ,u1,u2), and (θ,φ,yθ,yφ,pθ,pφ,qθ,qφ), respectively. The cost function of the control problem is
(56)G:T2S2⟶ℝ,G(θ,φ,yθ,yφ,u1,u2)=12[a2(u1)2sin2φ+a2(u2)2],
and the control system is represented by
(57)θ˙=yθ,φ˙=yφ,y˙θ=u1-2yθyφcotφ,y˙φ=u2+(yθ)2sinφcosφ.
The Hamiltonian is defined on the presymplectic manifold T*TS2×TS2T2S2 by
(58)H(θ,φ,yθ,yφ,u1,u2,pθ,pφ,qθ,qφ)=pθyθ+pφyφ+qθ(u1-2yθyφcotφ)+qφ(u2+(yθ)2sinφcosφ)-12[a2(u1)2sin2φ+a2(u2)2].
We can verify that the constraint equations ∂H/∂ui=0,  i=1,2, which define the symplectic manifold W1, are
(59)qθ-a2u1sin2φ=0,qφ-a2u2=0,
and consequently the controls are given by
(60)u1=1a2qθcosec2φ,u2=1a2qφ.
Therefore, the Hamiltonian defined on T*TS2 is written as
(61)H1(θ,φ,yθ,yφ,pθ,pφ,qθ,qφ)=pθyθ+pφyφ+12a2(qθ)2cosec2φ+12a2(qφ)2+qφ(yθ)2sinφcosφ-2qθyθyφcotφ.
Hence, according to (48), the solution vector field XH1 on T*TS2 is given by
(62)XH1(θ,φ,yθ,yφ,pθ,pφ,qθ,qφ)=yθ∂∂θ+yφ∂∂φ+(1a2qθcosec2φ-2yθyφcotφ)∂∂yθ+[1a2qφ+(yθ)2sinφcosφ]∂∂yφ+[1a2(qθ)2cosec2φcotφ-2qθyθyφcosec2φ-qφ(yθ)2+2qφ(yθ)2sin2φ1a2]∂∂pφ+(2qθyφcotφ-2qφyθsinφcosφ-pθ)∂∂qθ+(2qθyθcotφ-pφ)∂∂qφ.
The Hamiltonian equations are
(63)θ˙=yθ,φ˙=yφ,y˙θ=1a2qθcosec2φ-2yθyφcotφ,y˙φ=1a2qφ+(yθ)2sinφcosφ,p˙θ=0,p˙φ=1a2(qθ)2cosec2φcotφ-2qθyθyφcosec2φ-qφ(yθ)2+2qφ(yθ)2sin2φ,q˙θ=2qθyφcotφ-2qφyθsinφcosφ-pθ,q˙φ=2qθyθcotφ-pφ.


### 5.3. Optimal Control Problem on the Hyperbolic Planeℋ2

Let us analyze now the optimal control problem of the previous section on the upper half-plane model of the hyperbolic plane:(64)ℋ2={x1+ix2∈ℂ:x2>0}.
The Riemannian metric on ℋ2 is ds2=[d(x1)2+d(x2)2]/(x2)2, and so its components are g11=g22=1/(x2)2 and g12=g21=0. From (18), we get Γ121=Γ211=Γ222=-Γ112=-1/x2 and Γ111=Γ221=Γ122=Γ212=0.In this case, we have the cost function(65)G:T2ℋ2⟶ℝ,G(x1,x2,y1,y2,u1,u2)=12(u1)2+(u2)2(x2)2
and the control system
(66)x˙1=y1,x˙2=y2,y˙1=u1+2y1y2x2,y˙2=u2+(y2)2-(y1)2x2.The dynamics of the problem is described by a presymplectic system onT*Tℋ2×Tℋ2T2ℋ2 with Hamiltonian defined by
(67)H(x1,x2,y1,y2,u1,u2,p1,p2,q1,q2)=p1y1+p2y2+q1(u1+2y1y2x2)+q2[u2+(y2)2-(y1)2x2]-12(u1)2+(u2)2(x2)2.
Therefore, the symplectic manifold W1 is locally defined by
(68)q1-u1(x2)2=0,q2-u2(x2)2=0,
and so the controls are such that
(69)u1=q1(x2)2,u2=q2(x2)2.
As a consequence, the Hamiltonian H1 is defined on T*Tℋ2 by
(70)H1(x1,x2,y1,y2,p1,p2,q1,q2)=p1y1+p2y2+12(x2)2(q1)2+12(x2)2(q2)2+2q1y1y2x2+q2(y2)2-(y1)2x2.
Moreover, the corresponding Hamiltonian equations are
(71)x˙1=y1,x˙2=y2,y˙1=(x2)2q1+2y1y2x2,y˙2=(x2)2q2+(y2)2-(y1)2x2,p˙1=0,p˙2=2q1y1y2(x2)2+q2(y2)2-(y1)2(x2)2-x2(q1)2-x2(q2)2,q˙1=-2q1y2x2+2q2y1x2-p1,q˙2=-2q1y1x2-2q2y2x2-p2.


## 5.1. Optimal Control Problem on the Euclidean Spaceℝn

A trivial example of the optimal control problem discussed in the previous section is the caseM=ℝn. The tangent space of ℝn at an arbitrary point can be identified with ℝn, and the Riemannian metric on ℝn is the Euclidean one. By means of the canonical basis {ei} of ℝn, the components of the metric are gij=ei·ej=δij and the Christoffel symbols are all null.The cost functional of the control problem is(51)G:ℝ3n⟶ℝ,G(xi;yi;ui)=12∑i=1n‍(ui)2,
and the control system, described by a vector field Π:ℝ3n→ℝ4n along the projection τ21:ℝ3n→ℝ2n, is locally given by
(52)x˙i=yi,y˙i=ui,i=1,…,n.The HamiltonianH:ℝ5n→ℝ of the presymplectic system that describes the dynamics of the problem is
(53)H(xi;yi;ui;pi;qi)=∑i=1n‍[piyi+qiui-12(ui)2].
Therefore, the constraint equations ∂H/∂ui=0, i=1,…,n, defining the symplectic manifold W1 are ui=qi, i=1,…,n. Consequently, the Hamiltonian H1 defined on the symplectic manifold ℝ4n becomes
(54)H1(xi;yi;pi;qi)=∑i=1n‍[piyi+12(qi)2].
From (48), we know that XH1(xi;yi;pi;qi)=∑i=1n‍[yi(∂/∂xi)+qi(∂/∂yi)-pi(∂/∂qi)] is the solution vector field on ℝ4n and the corresponding Hamiltonian equations are
(55)x˙i=yi,y˙i=qi,p˙i=0,q˙i=-pi,
for i=1,…,n. Note that these equations give the equations d4xi/dt4=0, i=1,…,n, and so the curve γ in ℝn locally represented by (xi) is such that d4γ/dt4=0. This corresponds to (16), since, on the Riemannian manifold ℝn, the covariant derivative along a curve is the usual derivative along a curve in ℝn and the curvature tensor is null. We have obtained the equation of cubic polynomials on the Euclidean space as we would expect.


## 5.2. Optimal Control Problem on the SphereS2

LetS2={x∈ℝ3:∥x∥=a} be the sphere of radius a with the usual spherical coordinates x1=θ and x2=φ. S2 is equipped with the Riemannian metric ds2=a2sin2φdθ2+a2dφ2, and so g12=g21=0, g22=a2, and g11=a2sin2φ. Using (18), we obtain the Christoffel symbols: Γ121=Γ211=cotφ, Γ112=-sinφcosφ, and all the others are null.Consider the optimal control problem of the previous section onM=S2. The state space is TS2, the bundle of controls is T2S2, and the costate space is T*TS2. We represent the corresponding coordinates on these spaces by (θ,φ,yθ,yφ), (θ,φ,yθ,yφ,u1,u2), and (θ,φ,yθ,yφ,pθ,pφ,qθ,qφ), respectively. The cost function of the control problem is
(56)G:T2S2⟶ℝ,G(θ,φ,yθ,yφ,u1,u2)=12[a2(u1)2sin2φ+a2(u2)2],
and the control system is represented by
(57)θ˙=yθ,φ˙=yφ,y˙θ=u1-2yθyφcotφ,y˙φ=u2+(yθ)2sinφcosφ.
The Hamiltonian is defined on the presymplectic manifold T*TS2×TS2T2S2 by
(58)H(θ,φ,yθ,yφ,u1,u2,pθ,pφ,qθ,qφ)=pθyθ+pφyφ+qθ(u1-2yθyφcotφ)+qφ(u2+(yθ)2sinφcosφ)-12[a2(u1)2sin2φ+a2(u2)2].
We can verify that the constraint equations ∂H/∂ui=0,  i=1,2, which define the symplectic manifold W1, are
(59)qθ-a2u1sin2φ=0,qφ-a2u2=0,
and consequently the controls are given by
(60)u1=1a2qθcosec2φ,u2=1a2qφ.
Therefore, the Hamiltonian defined on T*TS2 is written as
(61)H1(θ,φ,yθ,yφ,pθ,pφ,qθ,qφ)=pθyθ+pφyφ+12a2(qθ)2cosec2φ+12a2(qφ)2+qφ(yθ)2sinφcosφ-2qθyθyφcotφ.
Hence, according to (48), the solution vector field XH1 on T*TS2 is given by
(62)XH1(θ,φ,yθ,yφ,pθ,pφ,qθ,qφ)=yθ∂∂θ+yφ∂∂φ+(1a2qθcosec2φ-2yθyφcotφ)∂∂yθ+[1a2qφ+(yθ)2sinφcosφ]∂∂yφ+[1a2(qθ)2cosec2φcotφ-2qθyθyφcosec2φ-qφ(yθ)2+2qφ(yθ)2sin2φ1a2]∂∂pφ+(2qθyφcotφ-2qφyθsinφcosφ-pθ)∂∂qθ+(2qθyθcotφ-pφ)∂∂qφ.
The Hamiltonian equations are
(63)θ˙=yθ,φ˙=yφ,y˙θ=1a2qθcosec2φ-2yθyφcotφ,y˙φ=1a2qφ+(yθ)2sinφcosφ,p˙θ=0,p˙φ=1a2(qθ)2cosec2φcotφ-2qθyθyφcosec2φ-qφ(yθ)2+2qφ(yθ)2sin2φ,q˙θ=2qθyφcotφ-2qφyθsinφcosφ-pθ,q˙φ=2qθyθcotφ-pφ.


## 5.3. Optimal Control Problem on the Hyperbolic Planeℋ2

Let us analyze now the optimal control problem of the previous section on the upper half-plane model of the hyperbolic plane:(64)ℋ2={x1+ix2∈ℂ:x2>0}.
The Riemannian metric on ℋ2 is ds2=[d(x1)2+d(x2)2]/(x2)2, and so its components are g11=g22=1/(x2)2 and g12=g21=0. From (18), we get Γ121=Γ211=Γ222=-Γ112=-1/x2 and Γ111=Γ221=Γ122=Γ212=0.In this case, we have the cost function(65)G:T2ℋ2⟶ℝ,G(x1,x2,y1,y2,u1,u2)=12(u1)2+(u2)2(x2)2
and the control system
(66)x˙1=y1,x˙2=y2,y˙1=u1+2y1y2x2,y˙2=u2+(y2)2-(y1)2x2.The dynamics of the problem is described by a presymplectic system onT*Tℋ2×Tℋ2T2ℋ2 with Hamiltonian defined by
(67)H(x1,x2,y1,y2,u1,u2,p1,p2,q1,q2)=p1y1+p2y2+q1(u1+2y1y2x2)+q2[u2+(y2)2-(y1)2x2]-12(u1)2+(u2)2(x2)2.
Therefore, the symplectic manifold W1 is locally defined by
(68)q1-u1(x2)2=0,q2-u2(x2)2=0,
and so the controls are such that
(69)u1=q1(x2)2,u2=q2(x2)2.
As a consequence, the Hamiltonian H1 is defined on T*Tℋ2 by
(70)H1(x1,x2,y1,y2,p1,p2,q1,q2)=p1y1+p2y2+12(x2)2(q1)2+12(x2)2(q2)2+2q1y1y2x2+q2(y2)2-(y1)2x2.
Moreover, the corresponding Hamiltonian equations are
(71)x˙1=y1,x˙2=y2,y˙1=(x2)2q1+2y1y2x2,y˙2=(x2)2q2+(y2)2-(y1)2x2,p˙1=0,p˙2=2q1y1y2(x2)2+q2(y2)2-(y1)2(x2)2-x2(q1)2-x2(q2)2,q˙1=-2q1y2x2+2q2y1x2-p1,q˙2=-2q1y1x2-2q2y2x2-p2.



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*Source: 243621-2013-08-26.xml*