# American Institute of Mathematical Sciences

March  2017, 9(1): 1-45. doi: 10.3934/jgm.2017001

## Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control

 Department of Mathematics, University of Michigan, 530 Church Street, 3828 East Hall, Ann Arbor, Michigan, 48109, USA

Received  June 2016 Revised  January 2017 Published  March 2017

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 function 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.

Citation: Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001
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Second order Skinner and Rusk formalism on Lie algebroids
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