1. Introduce the Problem
One of the objectives of the control theory of partial differential equations of evolution is to be interested in how to act on such dynamic systems. So, the exact controllability of distributed systems has attracted a lot of interest in recent years. And this thanks to one of the pioneers J.L. Lions   who developed the HUM method (Hilbert Uniqueness Methods). It is based essentially on the properties of uniqueness of the homogeneous equation by a particular choice of controls, the construction of a hilbert space and of 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 has given important results (Lions , Niane  , Seck et al.   ).
Although when the controls have a small support (Niane , Seck  ), it seems to be ineffective, even when for technical reasons, the multiplier method does not give results, see Niane .
As for the parabolic equations, there are the results of Russel  first; Later G. Lebeau-L. Robbiano  and Imanuvilov-Fursikov  who have proven with different methods but very technical and long by using Carleman’s Inequalies in the exact null controllability of the Heat equation.
Also, the harmonic method is also ineffective for this type of equation.
More recently, Khodja et al.  have shown that there is a minimal time T0 of controllability below which null controllability is not achievable for a parabolic operator. Thus, by Khodja , Tucsnak  and Avdonin , a means of calculating a minimum cost associated with this minimum time of null controllability has been established.
2. Problem Statement
In this work, to circumvent certain constraints linked to estimates in the work of G. Lebeau , Imanuvilov-Fursikov  notably the Carleman’s inequalities, we show that a new method solves some of these difficulties. It is based on a fusion of the moments method used by Khodja , Tucsnak , Avdonin  and the use of strategic actuators zones El. Jai   to solve the problem of null controllability of the heat equation with a minimum time controllability less than the minimal time of null controllability T0 provided by Khodja et al.  .
There are two types of criteria:
1) A criterion for constructing a functional space containing and its dual contained in thus making a pivotal space Brezis ;
2) A criterion of non-degeneration of a strategic zone profile which stems from the parabolic nature of the operator and the regularity of control Hörmander ;
In both cases these criteria allowed us to obtain a better minimum time of controllability.
This method opens wide perspectives to the theory of null controllability in general, as well as to the theory of exact controllability by zone strategic actuators and will allow for parabolic equations 1D (and 2D), Schrödinger, plates and of Navier-Stocks linearized to solve many questions thus opening many perspectives for the improvement of the minimum times of controllability.
3. Concept of Strategic Zone Actuators
3.1. Notations and Definition
Definition 1. A function square integrable is said strategic if it verify, for all , the solution y of the heat equation
Let an interval of , letA the operator defined by
According to the spectral theory, A admits a Hilbertian base of of eigenfunctions whose associated eigenvalues are rows in the ascending direction where
1) It suffices that the relation (2) is verified over an interval for it to be true on because of the analyticity of on .
2) Here is a bounded open of , of regular border; is, a priori, the state space and T define the time horizon considered for the exact controllability of the system (1).
Proposition 1. 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 posing that: .
Then, we have
We have if and only if
Therefore, for that it is sufficient that:
So, if we take and where then is strategic
Remark. Obviously, other strategic actuators can be built without great difficulty see Jai   .
3.2. Notations, Definition and Functional Spaces
Let , consider the following Hilberts spaces and their respective dual:
We equip with the following scalar product
and, the associeted norm .
The dual of is provided the scalar product
and, associeted norm .
If and we have:
Let us define now the setting that we will deal in the sequel and assume that
Definition 2. The condensation index of sequences is defined as
where the function E is defined by
To apply the moment method, let us define the concept of biorthogonal family.
Definition 3. Let be a real sequence and . We say that the family of functions is a biorthogonal family to the exponentials associated with if for any
Also assume a fundamental lemma we need in the sequel for the proof of the main result.
Lemma 2. See Khodja  or Tucsnak 
Let and let be a ordered sequence such that . Then, there exists a biorthogonal family to the exponentials associeted with such that for any there exists a constant such that
for k sufficiently large, where is the condensation index of the sequence .
4. Main Result of Null Controllability of the 1D Heat Equation
4.1. Main Theorem
Theorem 3. If is a strategic actuator on , a control and a strictly positive real; for all , there exist and such that if y is solution of
Proof. Let be the heat equation with an internal strategic zone profile and a control defined by:
Let be a linear control operator, then the previous Equation (19) becomes:
Then the solution of the previous Equation (20) is given by:
The Equation (19) is null controllable at time if which is equivalent to
Based on the definition of the following spaces previously defined:
and, if the solution , then we have and .
Likewise B is in and is written:
basis of eigenfunctions.
Then the Equation (20) becomes:
Therefore the solution becomes:
(20) is null controllable at time if and only if which means that
We have , and (3.11) becomes:
Let’s do the following variable change (to have the backward problem):
we have then and with a bi-orthogonal family of in which satisfy the condition:
this is to say
Therefore by estimation, Khodja  and Tucsnak , we have
where a constant depending only on and (3.12) becomes
If admits a bi-orthogonal family , then (Kronecker symbol), which finally gives
now the system (20) is null controllable if and if and only if because ( is strategic on I).
Let’s take a look at norm of ?
And we had according to the theorem 4.1 of Khodia ,
denotes the associated interpolation function (37)
then the inequality (3.16) becomes:
Now let’s pose
the minimal time of null controllabilty of system (20), then we obtain with .
so, (20) is null controllable if and only if and . o
4.2. Controllability on the FT Space
The spaces et have been defined previously; and the same calculations will be repeated on these spaces.
Remark. We can thus notice that by construction:
iii) What we can summarize on the following diagram (see Figure 1).
Taking back the following system (20):
By setting as a linear control operator and we resume the calculations on the spaces and ; then the previous Equation (20) becomes:
Then the solution of Equation (20) is done:
Knowing that The Equation (20) is null controllable at time in if which equals
If the solution , we have et . Likewise B is in and is written:
So (20) becomes:
Hence the solution becomes:
Figure 1. Pivot space L2.
(20) is null controllable at time if and only if
We have , and (2.23) becomes:
Let’s change the variable
and then we have with a bi-orthogonal family of in which satisfy the condition:
with where was the constant of inequality (2.15) because we have by taking the same calculations, we end up with:
So the system (20) is null controllable if
that is to say if and .
Remark. 1) There is no uniqueness of the control profile bringing the system of the initial condition to the final state (the set of strategic profiles is a closed affine subspace: we can naturally choose a norm control minimal on as being the projection of 0 on this convex).
2) In Khodja et al. , it has been shown that there is a minimal time and that if , the Equation (20) is null controllable (i.e. otherwise not controllable).
3) In this theorem 3, we show that there is a minimal time to this i.e. for which we have null controllability.
4) This result of theorem 3.1 was obtained under the condition that the strategic profile zone .
5) Indeed, if a profile is strategic over an interval then (where C a constant) is still strategic.
6) Thereby the set is not empty.
5. Conclusions and Perspectives
In the literature, the controllability of the heat equation has been established since the mid-90s by Lebeau and Fursikov. In all these works and others more recent, there is always a time from which the control is realizable. Our aim was to find a better minimum time to carry out this control.
So, in this work, we were based on the work of Khodja  et al. and Tusnack  et al. to find a better time to achieve the null controllability of the 1-D heat equation. This goal was achieved with another simpler approach and the addition of a strategic profile assumption.
Another work is being finalized to find a minimum cost linked to this minimum time to obtain the null controllability of the heat equation.
 Lions, J.-L. (1988) Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Recherches en Mathématiques Appliquées (Research in Applied Mathematics), Vol. 9, Perturbations, Masson, Paris.
 Niane, M.T. (1990) Régularité, contrôlabilité exacte et contrôlabilité spectrale de l’équation des ondes et de l’équation des plaques vibrantes,Thèse de Doctorat d’Etat, Université Cheikh Anta Diop de Dakar, Sénégal.
 Russel, D.L. and Fattorini, H.O. (1971) Exact Controllability for Linear Parabolic Equation in One Space Dimension. Archive for Rational Mechanics and Analysis, 43, 272-292.
 Khodja, F.A., Benabdallah, A., Gonzàlez-Burgo, M. and Teresa, L.D. (2013) Minimal Time for the Null Controllability of Parabolic Systems: The Effect of the Condensation Index of Complex Sequences.
 Khodja, F.A., De, T.L., Benabdallah, A. and González-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.
 Tucsnak, M. and Tenenbaum, G. (2007) New Blow-Up Rates for Fast Controls of Schrödinger and Heat Equations. Journal of Differential Equations, 243, 70-100.