Marcela Zuccalli

Optimal control of underactuated mechanical systems: A geometric approach

In this paper, we consider a geometric formalism for optimal control of underactuated mechanical systems. Our techniques are an adaptation of the classical Skinner and Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics of the optimal control problem. These developments will allow us to develop a new class of geometric integrators based on discrete variational calculus.

Higher-order discrete variational problems with constraints

An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamiltons principle of critical action. This family of geometric integrators is called variational integrators. In this paper, we derive new variational integrators for higher-order Lagrangian mechanical system subjected to higher-order constraints. From the discretization of the variational principles, we show that our methods are automatically symplectic and, in consequence, with a very good energy behavior. Additionally, the symmetries of the discrete Lagrangian imply that momentum is conserved by the integrator. Moreover, we extend our construction to variational integrators where the Lagrangian is explicitly time-dependent. Finally, some motivating applications of higher-order problems are considered; in particular, optimal control problems for explicitly time-dependent underactuated systems and an interpolation problem on Riemannian manifolds.

A geometric approach to discrete connections on principal bundles

This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and relationships. An existence result for discrete connections on principal bundles equipped with appropriate Riemannian metrics is proved.

Lagrangian reduction of discrete mechanical systems by stages

In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introduce a notion of symmetry groups for objects of LP_d and introduce a reduction procedure that is closed in the category LP_d. Furthermore, under some conditions, we show that the reduction in two steps (first by a closed normal subgroup of the symmetry group and then by the residual symmetry group) is isomorphic in LP_d to the reduction by the full symmetry group.

Noncentral Extensions as Anomalies in Classical Dynamical Systems

Abstract A two cocycle is associated to any action of a Lie group on a symplectic manifold. This allows to enlarge the concept of anomaly in classical dynamical systems considered by F Toppan in [J. Nonlinear Math. Phys. 8, Nr. 3 (2001), 518–533] so as to encompass some extensions of Lie algebras related to noncanonical actions.

On Lie algebra extensions in a symplectic framework

It is shown that the construction carried out by Carinena and Ibort [J. Math. Phys. 29, 541–545 (1988)] involving nonsymplectic actions of Lie groups gives rise to “true” noncentral extensions of the corresponding Lie algebras.