Ill-Posedness of Uncontrollable Systems and Weakly Controllable Perturbed Systems

Precise definitions of well posedness and ill-posedness in the sense of Tikhanov and of Hadamard are presented for a general class of optimal control problems. Ill-posedness of "uncontrollable" systems is studied. The "regularization" method for optimal control problems is defined and it is shown that the "regularization" can not be applied to "uncontrollable" ill-posed problems, since the solution always remains "instable" in the sense of Tikhonov. Finally, direct relationship between "weak controllability" and ill-posedness is discussed.