Conjugate-point conditions for variational problems with delayed argument
Date of Original Version
A variational problem with delayed argument is investigated. The existing necessary conditions are first reviewed. New results, two conjugate-point conditions, are then derived for this problem. The method of proof is similar to that used by Bliss for the classical problem. An example shows that the two conditions are not equivalent, and that the first-order necessary conditions, the strengthened Legendre conditions, and the conjugate-point conditions do not in general constitute a set of sufficient conditions for the delay problem. It is shown that a special case, referred to as the separated-integrand problem, leads to considerable simplification of the results for the general problem. © 1974 Plenum Publishing Corporation.
Journal of Optimization Theory and Applications
Palm, W. J., and W. E. Schmitendorf. "Conjugate-point conditions for variational problems with delayed argument." Journal of Optimization Theory and Applications 14, 6 (1974): 599-612. doi:10.1007/BF00932963.