Margarida Camarinha

On the geometry of Riemannian cubic polynomials

Abstract We continue the work of Crouch and Silva Leite on the geometry of cubic polynomials on Riemannian manifolds. In particular, we generalize the theory of Jacobi fields and conjugate points and present necessary and sufficient optimality conditions

Elastic Curves as Solutions of Riemannian and Sub-Riemannian Control Problems

Abstract. We consider the nonlinear dynamic interpolation problem on Riemannian manifolds and, in particular, on connected and compact Lie groups. Basically we force the dynamic variables of a control system to pass through specific points in the configuration space, while minimizing a certain energy function, by a suitable choice of the controls. The energy function we consider depends on the velocity and acceleration along trajectories. The solution curves can be seen as generalizations of the classical splines in tension for the Euclidean space. The relations with sub-Riemannian optimal control problems are explained.

Hamiltonian Structure of Generalized Cubic Polynomials

Abstract We present a Hamiltonian formulation of a second order variational problem on a Riemannian manifold (Q, ), which gives rise to generalized cubic polynomials on Q, and explore the possibility of writing down the extremal solutions of that problem as a flow in the space ∪ q ϵ Q T q Q ⊕ T q * Q ⊕ T q * Q . For that we utilize the connection ∇ on Q, corresponding to the metric . We exhibit the extremal equations in Hamiltonian form and identify the correct symplectic form. In general the results depend upon a choice of frame for TQ, but for the special situation when Q is a Lie group G with Lie algebra G, our results are global and the flow reduces to a flow on G × g ×g* ×g*.

SOME APPLICATIONS OF QUASI-VELOCITIES IN OPTIMAL CONTROL

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinematic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelerations. The formulation of the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently [20]. We also provide several examples to illustrate our construction.

Second order optimality conditions for a higher order variational problem on a Riemannian manifold

In this paper, we derive second order optimality conditions for a higher order variational problem on a general Riemannian manifold, which can be viewed as an extension of the minimizing acceleration problem in Euclidean space and yields the geometric generalization of the classical cubic polynomials. This continues the work initiated by Crouch and Silva Leite (1995). In particular, we define conjugate points and prove a necessary and sufficient condition for optimality, in the absence of such singularities.

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the Lie group and its Lie algebra. Using these control geometric tools, the relation between the Hamiltonian approach developed here and the known variational one is analyzed. After making explicit the left trivialized system, we use the technique of Marsden–Weinstein reduction to remove the symmetries of the Hamiltonian system. In view of the reduced dynamics, we are able to guarantee, by means of the Lie–Cartan theorem, the existence of a considerable number of independent integrals of motion in involution.

Optimal control of affine connection control systems from the point of view of Lie algebroids

The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.

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

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