Treffer: DISCONTINUOUS VARIATIONAL PROBLEMS

Variational problems are analyzed that specify jump discontinuities in the state variables. The solution extremals are required to jump when they reach a manifold of dimension n-q (n is the number of state variables). Two approaches to these problems are presented. The first approach is geometrical and therefore loses most of its practicality as the number of state variables is increased beyond two. However, it provides valuable diognostic insight. The boundary of the reachable set is regarded as a wavefront that is determined by wavelets. This principle is used to construct the wavefront just after the discontinuity and to determine the normal to its tangent plane. Of course the Lagrange multiplier vector is parallel to this normal. When q is equal to one there are at most two directions in which the extremal can be continued. The second approach is analytic and therefore much more flexible and powerful. The Lagrange multipliers are obtained from a set of nonlinear equations. They are underdetermined when q is greater than one. After the discontinuity the extremals fill an n-dimensional volume regardless of the value of q. (Author) ; Presented at the IBM Scientific Computing Symposium on Control Theory and Applications, Yorktown Heights, New York October 1964.