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We are interested in high-order linear multistep schemes for timediscretization of adjoint equations arising within optimal control problems.First we consider optimal control problems for ordinary differential equationsand show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereasBDF methods preserve high--order accuracy. Subsequently we extend these resultsto semi--lagrangian discretizations of hyperbolic relaxation systems.Computational results illustrate theoretical findings.

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems