Computation of optimal trajectories for delay systems: An optimize-then-discretize strategy for general-purpose NLP solvers

We propose an “optimize-then-discretize” approach for the numerical solution of optimal control problems for systems with delays in both state and control. We first derive the optimality conditions and an explicit representation of the gradient of the cost functional. Then, we use explicit discretizations of the state/costate equations and employ general-purpose Non-Linear Programming (NLP) solvers, in particular Conjugate Gradient or Quasi-Newton schemes, to easily implement a descent method. Finally, we prove convergence of the algorithm to stationary points of the cost, and present some numerical simulations on model problems, including performance evaluation.