Journal of the
Korean Mathematical Society JKMS

In this paper we formulate an optimal control problem governed by time-delay Volterra integral equations; the problem includes control constraints as well as terminal equality and inequality constraints on the terminal state variables. First, using a special type of state and control variations, we represent a relatively simple and self-contained method for deriving new necessary conditions in the form of Pontryagin minimum principle. We show that these results immediately yield classical Pontryagin necessary conditions for control processes governed by ordinary differential equations (with or without delay). Next, imposing suitable convexity conditions on the functions involved, we derive Mangasarian-type and Arrow-type sufficient optimality conditions.

Keywords: optimal control, time-delay, Volterra integral equation, necessary condition, Pontryagin minimum principle, Mangasarian-type, Arrow-type, sufficient optimality condition

MSC numbers: 49J22