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IMA Journal of Mathematical Control and Information Advance Access published online on June 8, 2009
IMA Journal of Mathematical Control and Information, doi:10.1093/imamci/dnp009
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Sufficiency by a direct method in the variable state problem of calculus of variations: singular extremals

Gerardo Sánchez Licea{dagger}

Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, México, DF 04510, México

{dagger} Email: gslicea{at}apolo.acatlan.unam.mx

Received on April 8, 2008; Revision received November 3, 2008. Accepted on March 8, 2009

A direct sufficiency proof for singular and non-singular extremals for the parametric variable state problem of Bolza in the calculus of variations is presented. This technique is self-contained in the sense that it makes no use of the classical concepts of conjugate or focal points, fields of extremals or certain matrix Riccati inequalities. Moreover, we also study the non-parametric variable state problem of Bolza and derive sufficient conditions for singular and non-singular extremals for weak and strong local minima.

Keywords: calculus of variations; singular extremals; strengthened condition of Weierstrass; sufficiency for local minima.


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