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Geometric constrained variational calculus I: Piecewise smooth extremals

Enrico Massa

Dipartimento di Matematica, Università di Genova, Via Dodecaneso, 35, 16146 Genova, Italy

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Danilo Bruno

Department of Advanced Robotics, Istituto Italiano di Tecnologia, Via Morego, 30, 16163 Genova, Italy

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Gianvittorio Luria

DIPTEM Sez. Metodi e Modelli Matematici, Università di Genova, Piazzale Kennedy, Pad. D, 16129 Genova, Italy

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, and

Enrico Pagani

Dipartimento di Matematica, Università di Trento, Via Sommarive, 14, 38050 Povo di Trento, Italy

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https://doi.org/10.1142/S0219887815500619Cited by:7 (Source: Crossref)

Abstract

A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in Hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints.

Keywords:

AMSC: 49K15, 70F25

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