Abstract: In this paper we present necessary conditions for optimal control problems with time delays. The dynamic constraint is formulated as a control delay differential equation, with time delays occuring in both state and control variables. We allow the dependence of the data on the state variables to be nonsmooth, and the necessary conditions are expressed in terms of set-valued subgradients, in place of conventional derivatives. In the problems considered, the end-time is included in the decision variables. The fact that the end-time is a choice variable is accommodated by a new kind of transversality condition. While nonsmooth necessary conditions have earlier been derived for optimal control problems with time delays, for the most part this has been in the framework of controlled differential inclusions with time delays. By contrast, the necessary conditions of this paper cover general problems involving general, nonsmooth, controlled delay differential equations.
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