Moreover, Heminna [3] gives the controllability for elasticity system with two controls: both tangential and normal, under the assumption of the wellposedness for the backward system, which is a key assumption for getting controllability.
Therefore, we get the controllability of (1.2) when the control is put on the moving endpoint.
Moreover, we discuss what are the main obstructions to overcome in order to get classical controllability result for this system.
In this paper, we utilize the compatibility of the potential with the natural working space Σ to get the controllability of the Schrödinger equation in Σ; and then employ the energy decomposition technique together with the continuity argument to derive the well-posedness of the perturbation equation and the continuous dependence of the solution on the initial data in the natural space Σ.
In the discrete setting of the finite difference method, we derive the observability inequality and get the exact controllability for the semi-discrete internally controlled wave equation in the one-dimensional case.
Therefore, we get the exact controllability of (2.1).
It has been recognized that to get into insides of controllability of the system it is indispensible to define a quantity to measure how the system is controllable.
So, many researchers have tried to get sufficient conditions guaranteeing the controllability results of various systems without involving the compactness of a semigroup.
Similarly, combining Corollary 3.1 with the results in [45], we get the following consequence of the controllability.
It is interesting to observe that it seems extremely difficult to use the above method to get some elementary results in the rapid approximate-controllability context.
Here we would like to give this proof in detail for two reasons: To get more (i.e., the cost of the approximate control) than the approximate controllability from the quantitative unique continuation property (Assumption 2); and to get some clues to the proof of Theorem 3.1.
