7. Some remarks and conclusion

Many issues have not been addressed here, notably problems with constraints on control and, above all, on state. Constraints on control are generally dealt with directly using a suitable optimization method. Constraints on state can often be taken into account using penalty methods (see  ).

Nonlinear problems raise many difficulties, if only for the existence (not to mention uniqueness) of optimal control. This requires special techniques, notably compactness, which are standard in the study of nonlinear partial differential equations. On the other hand, the calculation of directional derivatives — we have seen some examples — is generally done without...