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