Investigation of extremals in a generalized isoperimetric problem by means of higher-order conditions

Abstract

The problem of a material point going round maximum area on a plane in a fixed period of time, with its velocity vector varying in a compact set $U$, is considered. Under some assumptions on $U$ and with the help of higher-order conditions it is proved that multiply-traversing extremals do not even give a so-called 0-weak maximum (which would at the same time be a weak maximum in the case of continuous velocity). For simply traversing extremals providing an absolute maximum; a non-trivial result on the kind of the maximum is obtained. Furthermore, finite-dimensional and infinite-dimensional quadratic forms connected with the pointed-out generalizationof the isoperimetric problem are studied; their non-negativeness on the corresponding subspaces and some other properties are established.