Title: Second-order constrained variational problems on Lie algebroids: applications to optimal control,
Abstract: The aim of this work is to study, from an intrinsic and geometric point of
view, second-order constrained variational problems on Lie algebroids, that is,
optimization problems defined by a cost functional which depends on
higher-order derivatives of admissible curves on a Lie algebroid. Extending the
classical Skinner and Rusk formalism for the mechanics in the context of Lie
algebroids, for second-order constrained mechanical systems, we derive the
corresponding dynamical equations. We find a symplectic Lie subalgebroid where,
under some mild regularity conditions, the second-order constrained variational
problem, seen as a presymplectic Hamiltonian system, has a unique solution. We
study the relationship of this formalism with the second-order constrained
Euler-Poincaré and Lagrange-Poincaré equations, among others. Our study is
applied to the optimal control of mechanical systems.