Second-order variational problems on Lie groupoids and optimal control applications

Leonardo Colombo; David Martín de  Diego

doi:10.3934/dcds.2016064

Article Contents

2016, Volume 36, Issue 11: 6023-6064. Doi: 10.3934/dcds.2016064

This issue Previous Article Euler-Poincaré-Arnold equations on semi-direct products II Next Article Differential geometry of rigid bodies collisions and non-standard billiards

Second-order variational problems on Lie groupoids and optimal control applications

Leonardo Colombo^1, and
David Martín de Diego^2,

1.
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109
2.
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Calle Nicolás Cabrera 15, Campus UAM, Cantoblanco, Madrid, 28049

Received: June 2015

Revised: May 2016

Published: May 2017

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Abstract

In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in a Lagrangian submanifold. We also study the theory of reduction by symmetries and the corresponding Noether theorem.

Keywords:

Mathematics Subject Classification: Primary: 70G45; Secondary: 70Hxx, 49J15, 53D17, 37M15.

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