Abstract
For a fixed end point problem in (n+1)-space with integrand function there are discussed certain necessary conditions that are satisfied by a minimizing arc having one or more corners. Firstly, there is clarified the type of minimum for which the continuity of the function
across corners is a necessary condition, since various authors have erroneously stated that this is a necessary condition for a weak relative minimum. There are discussed two methods by which one may establish the necessity of the continuity of this function in the case of a strong relative minimum. The more comprehensive method is that employed many years ago by the author in the study of discontinuous solutions for the non-parametric problem of Mayer. For the problem herein considered there is presented the second order condition involving the non-negativeness of the second variation along a non-singular extremaloid on the class of so-called generalized admissible variations vanishing at the end-values, including a discussion of conjugate points in the extended sense as introduced in the earlier paper on discontinuous solutions for the Mayer problem.
†This research was supported by the Air Force Office of Scientific Research office of Aerospace Research,United States Air Force, under Grant AFOSR-68-1398B. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.
†This research was supported by the Air Force Office of Scientific Research office of Aerospace Research,United States Air Force, under Grant AFOSR-68-1398B. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.
Notes
†This research was supported by the Air Force Office of Scientific Research office of Aerospace Research,United States Air Force, under Grant AFOSR-68-1398B. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.