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In this paper we introduce a new procedure to solve nonlinear optimal controlproblems with delays which exploits indirect methods combined with numericalhomotopy procedures. It is known that solving this kind of problems viaindirect methods (which arise from the Pontrya-gin Maximum Principle) iscomplex and computationally demanding because their implementation is faced totwo main difficulties: the extremal equations involve forward and backwardterms, and besides, the related shooting method has to be carefullyinitialized. Here, starting from the solution of the non-delayed version of theoptimal control problem, delays are introduced by a numerical continuation.This creates a sequence of optimal delayed solutions that converges to thedesired solution. We establish a convergence theorem ensuring the continuousdependence w.r.t. the delay of the optimal state, of the optimal control (in aweak sense) and of the corresponding adjoint vector. The convergence of theadjoint vector represents the most challenging step to prove and it is crucialfor the well-posedness of the proposed homotopy procedure. Two numericalexamples are proposed and analyzed to show the efficiency of this approach.

Solving nonlinear optimal control problems with state and control delays by shooting methods combined with numerical continuation on the delays