Variational and optimal control problems with delayed argument

This paper deals with variational and optimal control problems with delayed argument and presents analogs of the classical necessary conditions for optimality for problems in (n + 1)-space. It is mainly concerned with the functional $$J(y) = \int_a^b {ft,y(t\user2{--}\tau ),y(t),\dot y(t\user2{--}\tau ),\dot y(t)] dt} $$ There are no side conditions; τ is a positive real number; andy is a continuous piecewise smooth vector function havingn components. The fundamental lemma of the calculus of variations is used in deriving an analog of the Euler equations. The usual construction is utilized in obtaining analogs of the Weierstrass and Legendre conditions. Also found is a fourth necessary condition involving the least proper value associated with a boundary value problem related to the second variation. A sufficient condition is obtained by the use of a simple expansion method. The last station of the paper outlines an extension of a maximal principle obtained by Hestenes to control problems which involve delays in both the state variable and the control variable.