These last years, the exact controllability of distributed systems has been significantly enhanced by J. L. Lions   with the development of the Hilbert Uniqueness Methods (HUM). It is based essentially on the uniqueness properties of the homogeneous equation by a particular choice of controls, the construction of a Hilbert space and a continuous linear application of this Hilbert space in its dual which is, in fact, an isomorphism that establishes exact controllability.
For hyperbolic problems, this method yielded important results (Lions  ); although when the controls are small support (Niane , Seck   ), it seems not very effective, likewise when for technical reasons the multiplier method does not give satisfactory results.
As for the parabolic equations, there are the results of Imanuvilov-Fursikov  and G. Lebeau-L. Robbiano  who proved with different methods but very technical and long, the exact control of the Heat equation.
Also, the harmonic method is inoperative also for this kind of equations.
In this work, to circumvent certain constraints related to estimates in G. Lebeau’s work, we show that a new method which solves some of these difficulties. It is based Seck’s work; on criteria of surjectivity of a continuous linear operator of a Hilbert space in another construct directly from the problem of exact border controllability.
The criteria are of two types:
1) A surjectivity criterion that is a consequence of the properties of uniqueness (J. L. Lions);
2) A criterion of compactness that derives from the parabolic nature of the operator or the regularity of the control;
In both cases, these criteria are easier to verify than those of the Lions HUM method.
This method which we call Boundary Exact Controllability by Surjectivity and Compactness opens wide perspectives to the theory of the exact controllability in general, as well as to the theory of the exact controllability by actuators strategic zones and allows for the parabolic equations, from Schrödinger, plates, linearized Navier-Stockes to solve many questions thus opening up many perspectives.
2. Characterization of Exact Controllability
Indeed, we have the following result of functional analysis (see J. L. Lions and Ramdani  ) which will allow us to characterize the exact controllability of the heat equation.
For proof see also jeups 2012 Karim-Ramdani or J. L. Lions.
2.1. Exact Controllability Reminders
Let an open interval of . We put A to the operator defined by:
Lemma 1. Let E and H two Hilbert spaces and . Then, is surjective if and only if its adjoint is bounded below, i.e. there exists a constant such that
Lemma 2. The following assertions are equivalent:
1) The system (12) below is exactly controllable for .
2) The operator is inferiorly bounded, i.e. there exists such that
See Lions, El. Jai   or also Ramdani-Karim jeups 2012.
Definition 1. Let the operator defined below. We will say that the system defined by (12) that it is exactly controllable in time T if and only if is surjective.
2.2. Preliminary Results of Controllability
Definition 2. An integrable square function is called strategic if it satisfies, for all , the solution of heat equation
1) It suffices that the relation (5) be checked over an interval so that it is true over , because of the analycity of on , Brezis .
2) Here is an open bounded of , of regular border; is, a priori, the state space and T defines the time horizon considered for the exact controllability of the system (4).
Proposition 3. There are strategic actuators with support contained in any interval such that:
Proof. We can first notice that is strategic if and only if: .
Let such that and assume that: .
So, we have
We have if and only if
Therefore, for it is enough that:
So, if we take and where then is strategic.
Remark. Of course, other strategic actuators can be built without difficulty, see also El. Jai.
We define the Hilbert spaces that follow:
We equip with the following scalar product :
and, of the norm .
We know that is a Hilbert space; its dual is defined by:
We equip with the scalar product
and, of the norm .
We define the duality hook for by
Let , we notice the solution of the following heat equation
Let , we notice the solution of
3. Main Result of Boundary Exact Control
In the following, we want to establish an exact controllability result by the construction of a particular linear, continuous and surjective operator. Indeed, we want to solve the following problem:
for all in a space to be determined after, find such that if y is a solution of the homogeneous heat equation:
Thus, considering that and either , we have
and so we get on .
One can formally also see how the operator L can be constructed.
Multiply the Equation (12) by solution of the Equation (11), we have:
In order for the second member to make sense, it must be assumed that: . It suffices to assume that .
We know that:
Therefore, we deduce that
Remember that we had defined the following spaces:
and we equip it with the following scalar product:
and the natural norm of dual defined by
also provided with the scalar product and its natural norm.
With the previous notations, the formula (15) is written
By integrating by part by Green, we obtain
which allows us to define the operator
Remark. We can notice that: in equals
We thus define the operator L by
Using the relationship
Remark. The Lemma 1 and Lemma 2, responds to another philosophy than the usual one whose main hypothesis is the coercivity that assumes the verification of an estimate difficult to establish in explicit spaces.
So, we have the following boundary controllability result
Theorem 4 (Main result). For all , it exists such that if y is the solution of
Proof. First step:
By construction, . Indeed, the operator L is exactly defined by a formula (26); So just show that is strategic.
We have , after that
So is strategic: not degenerated.
Let’s show that is surjective?
We know that the operator L is defined by:
and his dual is
where K a constant defined by .
By the Lemma 1 and Lemma 2, we can deduce that L is surjective.
Third step: conclusion.
We know that is not degenerate, the operator L is surjective and, in addition, the operator is compact see also Seck.
Let now , it exists such that
Let , we have
Now, multiply the system (27) by integrating by parts
From where : which completes the proof.
4. Conclusion and Perspective
The exact controllability results by the HUM (Hilbert Uniqueness Method of Lions) method are not suitable for parabolic type operators. To get around these difficulties, in particular the coercivity hypothesis for the establishment of the inverse inequality, many have used Carlemann inequalities Fursikov-Imanuvilov, Lebeau-Robbiano, Khodja , Tusnack , ... But the length and heaviness of the calculations in the Carleman inequalities are dissuasive, from where this idea came to us to couple the notion of strategic zone actuators and the subjectivity-compactness of a linear operator which allowed us to have the exact controllability. And this technique opens up many perspectives for linear and semi-linear parabolic systems, of Schrodinger, of plates, ...
· The authors thank the referees in advance for their comments and suggestions.
· The authors thank the Dean of FASTEF ex ENS of the University Cheikh Anta Diop in Dakar and his assessor for their financial and moral support.
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 Lions, J.-L. (1988) Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Volume 8, Paris.
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 Khodja, F.A., De Teresa, L., Benabdallah, A. and Gonzlez-Burgos, M. (2014) Minimal Time for the Null Controllability of Parabolic Systems: The Effect of the Condensation Index of Complex Sequences. Journal of Functional Analysis, 267, 2077-2151.
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