  This paper provides necessary conditions of optimality for optimal control
problems with time delays in both state and control variables. Different
versions of the necessary conditions cover fixed end-time problems and, under
additional hypotheses, free end-time problems. The conditions improve on
previous available conditions in a number of respects. They can be regarded as
the first generalized Pontryagin Maximum Principle for fully non-smooth optimal
control problems, involving delays in state and control variables, only special
cases of which have previously been derived. Even when the data is smooth, the
conditions advance the existing theory. For example, we provide a new
`two-sided' generalized transversality condition, associated with the optimal
end-time, which gives more information about the optimal end-time than the
`one-sided' condition in the earlier literature. But there are improvements in
other respects, relating to the treatment of initial data, specifying past
histories of the state and control, and to the unrestrictive nature of the
hypotheses under which the necessary conditions are derived.
