This study is the third step of a project on the null controllability of the 1D heat equation. First, we show a boundary and internal results of controllability by a new approach using a linear, continuous, and surjective operator built from the solution of the heat system. Second, we improve the minimum time of null controllability of the 1D heat equation by using the notion of strategic zone actuators. So, we managed to improve the minimal time of null controllability to the 1D heat equation. In this study, the best minimum cost of null controllability has been estimated for the 1D heat equation based on the minimum controllability time calculated in the second step.

1. Introduction

One of the objectives of the theory of the control of partial differential equations of evolution is to be interested in the way of acting on dynamic systems. The exact controllability of distributed systems has attracted a lot of interest in recent years. Thanks to the pioneer Fattorinni-Russel [1] and Lions [2, 3] 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 and 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 internal or/and boundary controllability.

For hyperbolic problems, this method has given important results (Lions [2, 3], Niane [4], and Seck[5]).

Although when the controls have a small support (Niane [4], Guesmia [6], Glizer [7], Anguraj [8], and Seck et al [911]), it seems to be ineffective, even when for technical reasons the multiplier method does not give results.

As for the parabolic equations, there are the results of Russel [12] first. Later, G. Lebeau and el Robbiana [13] and Fursikov et al. [14] have proven with different methods which are very technical and long by using Carleman’s Inequalities, the exact null controllability of the heat equation.

So, the harmonic method is also ineffective for this kind of equations.

In this study, we explain how results on the cost of null controllability of the small-time heat equation can be used to reduce the cost of control.

Indeed, more recently, Khodja et al. [15, 16] and Lissy [17], in particular, Tusnack-Tenebaum [18], have shown that there is a minimal time of controllability below which null controllability is not achievable for a parabolic operator. From the work of Khodja et al. [16] and Lissy and Guéye [17], a minimum cost of null controllability associated with the minimum time of null controllability of Russel [12] was calculated.

Indeed, motivated by the works of Khodja [15], Tucsnak [18], and Lissy [17, 18] on the null controllability of the heat equation and the work of El Jai [19, 20] on the controllability use of strategic zone actuators, we managed, in this work, to improve the minimum time of null controllability to the 1D heat equation.

However, the restrictions and difficulties to establish the inequality of coercivity of the parabolic operator require to seek other internal control methods.

Thus, a mixed method combining the moment methods and the notion of strategic profile was used to find a better minimal time of null controllability of the 1D heat equation.

Naturally, once this result had been obtained on the minimum time of null controllability, we began the calculations and estimates to find a minimum control cost linked to this minimum time. This is how we established a best minimum cost of null controllability of the heat equation with an additional assumption on the strategy profile function.

These results open up broad prospects in this domain. Particularly, on semianalytical systems, the piecewise temperature-time distributions in solid bodies of regular shape were affected by a uniform surface heat flux using the line method (MOL) and the eigenvalue method. This method is also used in the numerical analysis of unstable thermal conduction in regular solid bodies including natural convection towards neighboring fluids, as well as for the calculation of spatio-temporal temperatures in simple bodies with cooling by thermal radiation using the method of digital lines.

2. Preliminary

2.1. Notations and Definition

Let ; the sequences of reals ; let us define now the setting that we will deal in the sequel and assume that

Definition 1. The condensation index of sequences is defined aswhere the function is defined by

2.2. Concept of Strategic Zone Actuators

A function square integrable is said strategic if it verifies for all , the solution of the heat equation:

Let be an interval of ; let A be the operator defined by

According to the spectral theory, see Lions [3], A admits a Hilbertian base of of eigenfunctions whose associated eigenvalues are rows in the ascending direction, where

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 haveWe have if and only ifTherefore, for that , it is sufficient that and .
So, if we take and , where , then is strategic.

Remark 1. Obviously, other strategic actuators can be built without great difficulty, see the work of Jai et al. [19, 20] and Seck and Ane [11].

2.3. Reminders on the Minimal Time of Null Controllability for the 1D Heat Equation

Letwhere , , and .

We know that system (9) admits a unique solution , see the work of Lions [3].

Verifying , , and , we have


We know from the work of Russel and Fatorrini [12], Lebeau and et Robbiano [13], and Fursikov et al. [14] that there exists a minimum time of control of the 1D heat equation.

So,(1)System (9) is null controllable at any time (2)System (9) is not null controllable at any time

Example 1. Consider the following examples:Example 1: if , with  , in ; . in ; is dense in .Example 2: if the profile defined for any over an interval bythen is a function with strong rapid decay and by setting , the cost of the control, at the terminal instant. Let us poseAlso,We show by simple calculations that and .
In particular, if , then with but small, where is the constant defined above.

In the case , for in , there exists a unique optimal control bringing to 0, see the work of Lissy et al. [18].

The map , being linear continuous.

Definition 2. The norm of this operator is called the optimal null control cost at time T designated by .

So, by Definition 2, is infinimum of the constants such that in driving to at time T with

What is the behavior of when ?

One could expect that the cost is of the form

In the work of Seck et al. [11], we successful find a better control time (noted ) compared to that proposed by Khodja et al. [16], , which led us to the main result of this work.

3. Main Result on the Minimum Cost Linked to the Minimum Time Null Controllability of 1D Heat Equation

3.1. Fundamental Lemma

Lemma 1. If is a strategic actuator on , is a control and is strictly positive and real; for all , there exist  and  such that if is solution ofthen .

For proof, see the work of Seck et al. in [11].

Consider the heat equation with an internal strategic zone profile and a, internal control defined by

Let be a linear control operator; then, the previous equation (19) becomes

In the sequel of this study, the minimum time null controllability is denoted by .

Recall that is calculated and defined in the work of Seck et al. [11] as follows:

From the minimal time of null controllability of system (19), then we obtain with , where and .

Theorem 1 (main theorem). Let be an increasing function verifying morever as and a strategic profile); ; there exists:where is the infinimum of the constants such that in , there is a control driving to at time .

Remark 2. This theorem means that the cost of the control can increase arbitrarily fast as  .
This can be explained by the fact that, contrary to the usual case, the cost of the control depends not only on the behavior of at infinity but also on how it differs from its limit superior .

Proof. will be a positive constant independent of .
Let us fix , and we consider , and let be chosen later.
We define as follows: .
One readily verifies that there exists some positive constant C such thatWe consider the optimal control associated to this initial condition, which verifies by definition and estimates the control:By the moments’ method, we obtain :Applying for , we haveNow,We know thatAlso,Applying the Cauchy–Schwarz inequality, we deduce thatSo, we haveNow, let us consider any positive and increasing function such that when .
Such a function is necessarily bijective and we call its inverse.
Let us consider defined byso that we have the condensation index (see the works of Khodja [16] and Tusnack [19]) defined byIt is clear that (see the work of Seck [11])i.e.,wherewhere the function is defined bysince .
Then, we have to thank the works of Khodja [15] and Seck [10, 11]:Let us explain how to choose , and we assume that is close enough to .
Now, we choose in such a way thatwhich is always possible (at least for close enough to ), since is increasing and goes to at .
Indeed,Consequently,One then easily obtains the desired result by choosing in such a way thatBecause it is clear that if is positive, increasing, and converge to at , then is well defined at least for large enough which is sufficient for our purpose.

Remark 3. The assumptions on the function make it possible to bypass the coercivity of the parabolic operators, which is, moreover, difficult to establish for these kinds of operators

4. Conclusion and Perspectives

The theory of the control of distributed systems has experienced a meteoric rise in recent years, notably with the team of Lions and Coron [22] and, more recently, with Ammar Khodja, Tucsnak, and Guesmia. In particular, the controllability of the heat equation has been established since the mid-90s by Lebeau–Robbiano and Fursikov–Imanuvilov. In all these works and others more recent works of Tucsnack-Tenenbaum and Khodja et al., there is always a time from which the control is realizable (below , controllability is not feasible). Our aim was to find a better minimum time to carry out this control (knowing that we cannot control the heat equation all the time). Indeed, if , then the cost is better than (this is our main result).

The other objective of this work was to solve an optimal control problem. Thus, our modest contribution is improving the time (smaller time) and hence reducing the cost to achieve the control of the heat equation over a small support interval. Indeed, we knew that 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 as being the projection of 0 on this convex).

This method is no longer operative when the interval support is large or if there is a nonlinear term in the system.

We plan, in the near future, to generalize this result to the Schrödinger equation, to transport-type equations, and to linear dispersive parabolic systems.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.