Asymptotic method and transient terms in exact controls

There is a narrow but hidden link between optimal control theory and the so-called Tikhonov regularization method. In fact, the small coefficient representing the marginal cost of the control can be interpreted as the regularization parameter in a Tikhonov method as far as there exists an exact control. This strategy enables one to adjust the cost function in the optimal control model in order to define the exact control which minimizes a given functional involving both the control but also the state variables during the control process. The goal of this paper is to suggest a method which gives a simple way to characterize and compute the exact control corresponding to the minimum of a given cost functional as said above. It appears as an extension of the phase control which is a finite dimensional version of the HUM control of J.L. Lions but for partial differential equations.