Exterior penalty method for nonlinear optimal control with distinct state and velocity delays

We consider nonlinear optimal control problems with distinct time delays in the state and its derivative. Using an exterior penalty method, we approximate the problem by unconstrained delayed variational problems and prove that their minimisers converge strongly to an optimal solution of the original problem under natural assumptions. A weaker convergence result holds in the affine–convex case. The approach is illustrated with engineering and control applications.