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.

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 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.

2. Preliminary Results

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

2.1. 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 , , and , defined on an interval with , have contact of order at 0 if and for a local coordinate system on around , the derivatives of and up to order , included, coincide at 0.

The equivalence class determined by a curve is denoted by 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   -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 , where for and . If , the tangent bundle is identified with the manifold and for , is just the tangent bundle of , .

We have the canonical projections which define several different fibered structures on . Note that . Locally, .

Given a smooth curve in , the lift to of is a smooth curve in defined by , where . If is given locally by , then is locally represented by .

We can also consider natural injections , for .

Here, we are particularly interested in the second-order tangent bundle . We denote the canonical local coordinates on and by and , respectively. The natural bases of the tangent spaces , , and are denoted, respectively, by , , and , for , , and . For clarity, to distinguish the projections of the tangent bundles and , we use the notations and . Moreover, the natural injection is locally defined by

2.2. Geometric Description of an Optimal Control Problem

Let be a fiber bundle over with projection . Consider also a vector field along the projection ; that is, a smooth map such that the diagram (5) 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 of class , piecewise smooth and with , with fixed initial and final conditions in the state space, satisfying and minimizing an integral functional , where is a smooth function called cost function.

Equation (6) is known as the control system, while an integral curve of , 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 (, , with ), then the control system is characterized by the system of differential equations for and where represent the canonical local coordinates of 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., [6–9]). Here, we are interested in the presymplectic description, and hence we should consider the presymplectic Hamiltonian system with the following. (i) The total space a fiber bundle over the manifold with canonical projections and .(ii) The canonical presymplectic form on (i.e., a closed -form which may be degenerate) given by the pullback of the canonical symplectic form on (i.e., a closed and nondegenerate 2-form) by the projection , . Locally,  where are the natural local coordinates on induced by . Note that the kernel of is locally given by .(iii) The Hamiltonian defined by , where represents the pairing duality of vectors and covectors on . We get  for each (i.e., , and , with ).

The dynamics of the presymplectic Hamiltonian system 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 which is an integral curve of the vector field . Notice the following. (i) Locally, , where , , and are smooth functions on . Hence, . (ii) On the other hand, . (iii) So, (10) is equivalent to , , and , , . (iv) Therefore, an integral curve of , locally given by , is such that  That is, is the vector field  defined in the subset .

In the geometric framework, we have where . Indeed, since is presymplectic, we have to consider the points of where (10) has solution. We assume that is a submanifold of . 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 where the solution vector field is not tangent to , and thus it does not necessarily induce a dynamics on . If it is the case, we construct a second constraint submanifold , that we assume to be a submanifold of , defined by the points on , where such a solution exists. But, again, it may happen, that we cannot guarantee the existence of a dynamics on , 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 .

3. 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 of smooth curves satisfying the boundary conditions where , , , and . 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 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 and the covariant acceleration of is given by Here, are the Christoffel symbols defining the Riemannian connection, which can be obtained using the identity where are the components of the Riemannian metric and is the inverse matrix of the matrix . Moreover, the lift to of the curve is locally represented by . Therefore, the action functional of our problem can be written as , where is the Lagrangian of the problem. Locally, we have with .

Observe that the Lagrangian of the problem is defined, for each , by for the natural injection defined by (4) and the connection application locally given by

The Lagrangian defines a dynamics on the third-order tangent space since the Euler-Lagrange equation (16) can be interpreted as a vector field on , whose integral curves are lifts to of curves in 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 on . 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 on that we are interested in is where 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 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 along the natural projection , that is, a smooth map such that the following diagram is commutative: (24) where is the canonical projection. The state space is and is the control bundle. If , we know that and hence . Consequently, for real smooth functions , with , the vector field can be expressed as . Along a curve , we have Furthermore, if is an integral curve of , then (26)

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 of class , piecewise smooth, with fixed endpoints in the state space satisfying the control system and minimizing the functional , for the cost function defined by for each , 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) is , or equivalently, .

4.2. Presymplectic Hamiltonian System

The Hamiltonian description of our problem has the cotangent bundle as costate space. We consider the presymplectic system characterized as follows. (i) The total space is the bundle over given by  where the fiber of over a point is . The canonical projections are and .(ii) The presymplectic 2-form on is defined by the pullback  where denotes the canonical symplectic -form on .(iii) The Hamiltonian function is defined by  where and are defined, respectively, by (27) and (29), and represents the pairing duality of vectors and covectors on .(iv) The dynamical vector field is the solution of the dynamical system

We now apply the geometric algorithm of presymplectic systems to . We first consider the submanifold 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 defined by (32). If is a local coordinate system on and and represent, respectively, the natural local coordinates on and , then is a coordinate system on the total space . In this context, it is obvious that is expressed by and so . It follows that is locally defined by the constraints

Note also that since our control system is affine on the controls, from (33) and (30), we get , ,  . As a consequence, the matrix is invertible, and this means that the system is regular at any point.

Consider the restriction to of the presymplectic form defined by (32).

Proposition 1. The 2-form is symplectic.

Proof. Recall that is symplectic if and only if , for all , where
Let . Using the fact that is defined by (36), we can conclude that , for . Indeed, suppose that there exists a such that ; this means that , for , but this is not true because of the invertibility of the matrix (37). Therefore, To conclude the proof, we have just to observe that . It is sufficient to verify that since, by definition, the opposite inclusion always happens. If , then , for all . Furthermore, from , we get , for all . In this way, according to (39), we conclude that , and hence . Consequently, as we have the direct sum (39), we get .

The previous proposition assures us that 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 on solution of the dynamical system (34) when restricted to , that is, such that where is the restriction of to .

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 defined by (33): Then, the submanifold is defined by and this implies that the optimal controls are So, we can consider the diffeomorphism Observe that the inverse function of is the restriction to of the projection . It is easy to show that , which means that defines a symplectomorphism between the symplectic manifolds and . This allows us to reduce the study of the dynamical system (40) on , to the study of the following system on : where is defined by . Locally, and the vector field on is the pushforward of by ; that is, The solution vector field is determined according to that is, As a consequence, the Hamiltonian equations are for .