Singularities in a variational problem with an inequality

Boris Garfinkel and Gregory Thomas McAllister, Jr.

Vol. 16 (1966), No. 2, 273–283

DOI: 10.2140/pjm.1966.16.273

Abstract

The subject of the paper is the variational problem of Lagrange with an inequality in the form (a) ϕ(x,y) ≧ 0 or (b) ϕ(x,y,y′) ≧ 0. The question of existence and uniqueness of the continuation of a minimizing arc is investigated at points of the boundary ϕ = 0. Various phenomena, including splitting of extremals, dead-end, entry into, and exit from the boundary, are treated and the conditions for their occurrence are derived. The nature of the continuation is related to the “index” associated with an extremal.

An appendix extends the results to a control problem of the Mayer type.

Mathematical Subject Classification

Primary: 49.00

Milestones

Received: 6 May 1964

Published: 1 February 1966