^{1, 2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

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.

In [

Riemannian cubic polynomials can be seen as a generalization of cubic polynomials to non-Euclidean spaces [

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 [

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 [

The organization of the paper is as follows. In Section

Consider a differentiable manifold

The tangent bundle of

Consider the following well-defined equivalence relation on the set of smooth curves in

We say that two smooth curves in

The equivalence class determined by a curve

We have the canonical projections

Given a smooth curve

We can also consider natural injections

Here, we are particularly interested in the second-order tangent bundle

Let

An

Equation (

The costate space of the system is the cotangent bundle

The total space

The canonical presymplectic form

where

The Hamiltonian

for each

The dynamics of the presymplectic Hamiltonian system

Locally,

On the other hand,

So, (

Therefore, an integral curve of

That is,

defined in the subset

In the geometric framework, we have

From this section onwards,

We are interested in the following second-order variational problem: find the curves that minimize

Recall that if a curve

Observe that the Lagrangian

The Lagrangian

The control system associated with the variational problem of the previous section is a control system of second-order on

The second-order control system on

From a geometric point of view, the control system can be described by a vector field

Now, if

The optimal control problem consists in finding the curves

The Hamiltonian description of our problem has the cotangent bundle

The total space is the bundle over

where the fiber of

The presymplectic 2-form

where

The Hamiltonian function

where

The dynamical vector field

We now apply the geometric algorithm of presymplectic systems to

Note also that since our control system is affine on the controls, from (

Consider

The 2-form

Recall that

Let

The previous proposition assures us that

We proceed with the analysis of the obtained system and an important simplification of our study. Using (

A trivial example of the optimal control problem discussed in the previous section is the case

The cost functional of the control problem is

The Hamiltonian

Let

Consider the optimal control problem of the previous section on

Let us analyze now the optimal control problem of the previous section on the upper half-plane model of the hyperbolic plane:

In this case, we have the cost function

The dynamics of the problem is described by a presymplectic system on

The work of L. Abrunheiro was supported in part by