  In this paper we introduce a new procedure to solve nonlinear optimal control
problems with delays which exploits indirect methods combined with numerical
homotopy procedures. It is known that solving this kind of problems via
indirect methods (which arise from the Pontrya-gin Maximum Principle) is
complex and computationally demanding because their implementation is faced to
two main difficulties: the extremal equations involve forward and backward
terms, and besides, the related shooting method has to be carefully
initialized. Here, starting from the solution of the non-delayed version of the
optimal control problem, delays are introduced by a numerical continuation.
This creates a sequence of optimal delayed solutions that converges to the
desired solution. We establish a convergence theorem ensuring the continuous
dependence w.r.t. the delay of the optimal state, of the optimal control (in a
weak sense) and of the corresponding adjoint vector. The convergence of the
adjoint vector represents the most challenging step to prove and it is crucial
for the well-posedness of the proposed homotopy procedure. Two numerical
examples are proposed and analyzed to show the efficiency of this approach.
