Global null-controllability and nonnegative-controllability of slightly superlinear heat equations

Global null-controllability and nonnegative-controllability of slightly superlinear heat equations

Abstract.

We consider the semilinear heat equation posed on a smooth bounded domain of with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset of . The goal of this paper is to prove the uniform large time global null-controllability for semilinearities where which is the case left open by Enrique Fernandez-Cara and Enrique Zuazua in 2000. It is worth mentioning that the free solution (without control) can blow-up. First, we establish the small-time global nonnegative-controllability (respectively nonpositive-controllability) of the system, i.e., one can steer any initial data to a nonnegative (respectively nonpositive) state in arbitrary time. In particular, one can act locally thanks to the control term in order to prevent the blow-up from happening. The proof relies on precise observability estimates for the linear heat equation with a bounded potential . More precisely, we show that observability holds with a sharp constant of the order for nonnegative initial data. This inequality comes from a new Carleman estimate. A Kakutani-Leray-Schauder’s fixed point argument enables to go back to the semilinear heat equation. Secondly, the uniform large time null-controllability result comes from three ingredients: the global nonnegative-controllability, a comparison principle between the free solution and the solution to the underlying ordinary differential equation which provides the convergence of the free solution toward in -norm, and the local null-controllability of the semilinear heat equation.

1. Introduction

Let , , be a bounded, connected, open subset of of class and be the outer unit normal vector to . We consider the semilinear heat equation with Neumann boundary conditions:

(1)

where .

Remark 1.1.

All our results stay valid for Dirichlet boundary conditions (see Section 7).

In (1), is the state to be controlled and is the control input supported in , a nonempty open subset of .
We assume that satisfies

(2)

In this case, solves (1) with and .
In the following, we will also assume that satisfies the restrictive growth condition

(3)

Under the hypothesis (3), blow-up may occur if in (1). Take for example with . The mathematical theory of blow-up for

(4)

was established in [24] and [25]. It was shown that blow-up

  • occurs globally in the whole domain if ,

  • is of pointwise nature if ,

  • is ’regional’, i.e., it occurs in an open subset of if .

See [26, Section 2 and Section 5] for a survey on this problem.
The goal of this paper is to analyze the null-controllability properties of (1).
Let us define . We recall two classical definitions of null-controllability.

Definition 1.2.

Let . The system (1) is

  • globally null-controllable in time if for every , there exists such that the solution of (1) satisfies .

  • locally null-controllable in time if there exists such that for every verifying , there exists such that the solution of (1) satisfies .

We have the following well-known local null-controllability result.

Theorem 1.3.

For every , (1) is locally null-controllable in time .

The proof of creftypecap 1.3 is a consequence of the (global) null-controllability of the linear heat equation with a bounded potential (due to Andrei Fursikov and Oleg Imanuvilov, see [23] or [21, Theorem 1.5]) and the small perturbations method (see [3, Lemma 6] and [1], [5], [30], [33], [40] for other results in this direction).

The following global null-controllability (positive) result has been proved independently by Enrique Fernandez-Cara, Enrique Zuazua (see [22, Theorem 1.2]) and Viorel Barbu under a sign condition (see [4, Theorem 2] or [6, Theorem 3.6]) for Dirichlet boundary conditions. It has been extended to semilinearities which can depend on the gradient of the state and to Robin boundary conditions (then to Neumann boundary conditions) by Enrique Fernandez-Cara, Manuel Gonzalez-Burgos, Sergio Guerrero and Jean-Pierre Puel in [19] (see also [13] for the Dirichlet case).

Theorem 1.4.

[19, Theorem 1]
We assume that (3) holds for . Then, for every , (1) is globally null-controllable in time .

Remark 1.5.

Historically, the first global null-controllability (positive) result for (1) with satisfying (3) was proved by Enrique Fernandez-Cara in [17] for and for Dirichlet boundary conditions.

The following global null-controllability (negative) result has been proved by Enrique Fernandez-Cara, Enrique Zuazua (see [22]).

Theorem 1.6.

[22, Theorem 1.1]
We set with and we assume that . Then, for every , there exists an initial datum such that for every , the maximal solution of (1) blows-up in time .

Remark 1.7.

Such a function does satisfy (3) for any because as . Then, creftypecap 1.6 shows that (1) can fail to be null-controllable for every under the hypothesis (3) with . creftypecap 1.6 comes from a localized estimate in that shows that the control cannot compensate the blow-up phenomena occurring in (see [22, Section 2]).

When the nonlinear term is dissipative, i.e., for every , then blow-up cannot occur. Furthermore, such a nonlinearity produces energy decay for the uncontrolled equation, therefore naively one may be led to believe that it can help in steering the solution to zero in arbitrary short time. The results of Sebastian Anita and Daniel Tataru show that this is false, more precisely that for ‘strongly’ superlinear one needs a sufficiently large time in order to bring the solution to zero. An intuitive explanation for this is that the nonlinearity is also damping the effect of the control as it expands from the controlled region into the uncontrolled region (see [3]).

Theorem 1.8.

[3, Theorem 3]
We set with and we assume that . Then, there exist , such that for every , , there exists such that the solution to (1) satisfies .

Remark 1.9.

In particular, for such a as in creftypecap 1.8, (1) is not globally null-controllable in small time . creftypecap 1.8 is due to pointwise upper bounds on the solution of (1) which are independent of the control (see [3, Section 3]).

2. Main results

2.1. Small-time global nonnegative-controllability

We introduce a new concept of controllability.

Definition 2.1.

Let . The system (1) is globally nonnegative-controllable (respectively globally nonpositive-controllable) in time if for every , there exists such that the solution of (1) satisfies

(5)

The first main result of this paper is a small-time global nonnegative-controllability result for (1).

Theorem 2.2.

We assume that (3) holds for and for (respectively for ). Then, for every , (1) is globally nonnegative-controllable (respectively globally nonpositive-controllable) in time .

Remark 2.3.

creftypecap 2.2 is almost sharp because it does not hold for according to creftypecap 1.8. The case where as is open.

Remark 2.4.

creftypecap 2.2 does not treat the case with because of the sign condition.

2.2. Large time global null-controllability

The second main result of this paper is the following one.

Theorem 2.5.

We assume that (3) holds for , for or for and . Then, there exists sufficiently large such that (1) is globally null-controllable in time .

Remark 2.6.

creftypecap 2.5 proves that creftypecap 1.6 is almost sharp. Indeed, let us take with , then by creftypecap 2.5, there exists sufficiently large such that (1) is globally null-controllable in time . In particular, one can find a localized control which prevents the blow-up from happening. The case is open.

Remark 2.7.

creftypecap 2.5 does not treat the case with because of the sign condition.

Remark 2.8.

The small-time global null-controllability of (1) remains open when (3) holds for .

2.3. Proof strategy of the small-time global nonnegative-controllability

We will only prove the global nonnegative-controllability result. The nonpositive-controllability result is an easy adaptation.
The proof strategy of creftypecap 2.2 will follow Enrique Fernandez-Cara and Enrique Zuazua’s proof of creftypecap 1.4 (see [22]).
The starting point is to get some precise observability estimates for the linear heat equation with a bounded potential for nonnegative initial data. More precisely, we show that observability holds with a sharp constant of the order for nonnegative initial data (see creftypecap 4.4 below). This is done thanks to a new Carleman estimate in (see creftypecap 4.9 below). This leads to a nonnegative-controllability result in in the linear case with an estimate of the control cost of the order which is the key point of the proof (see creftypecap 4.1 below).
We end the proof of creftypecap 2.2 by a Kakutani-Leray-Schauder’s fixed-point strategy. The idea of taking short control times to avoid blow-up phenomena is the same as in [22] and references therein. More precisely, the construction of the control follows two steps. The first step consists in steering the solution of (1) to in time with an appropriate choice of the control. Then, the two conditions: and the dissipativity of in imply that the free solution of (1) (with ) defined in stays nonnegative and bounded by using a comparison principle (see Section 5).

2.4. Proof strategy of the large time global null-controllability

We will only treat the case where for . The other case, i.e., for is an easy adaptation.
The proof strategy of creftypecap 2.5 is divided into three steps.
First, for every initial data , one can steer the solution of (1) in time (for instance) to a nonnegative state by using creftypecap 2.2.
Secondly, we let evolve the system without control and we remark that

with independent of and when . This kind of argument has already been used by Jean-Michel Coron in the context of the Burgers equation (see [10, Theorem 8]).
Finally, by using the second step, for sufficiently large, belongs to a small ball of centered at , where the local null-controllability holds (see creftypecap 1.3). Then, one can steer to with an appropriate choice of the control.

3. Parabolic equations: Well-posedness and regularity

The goal of this section is to state well-posedness results, dissipativity in time in -norm, maximum principle and - estimates for linear parabolic equations. We also give the definition of a solution to the semilinear heat equation (1). The references of these results only treat the case of Dirichlet boundary conditions but the proofs can be easily adapted to Neumann boundary conditions.

3.1. Well-posedness

We introduce the functional space

(6)

which satisfies the following embedding (see [15, Section 5.9.2, Theorem 3])

(7)

Linear parabolic equations

Definition 3.1.

Let , and . A function is a solution to

(8)

if for every ,

(9)

and

(10)

The following well-posedness result in holds for linear parabolic equations.

Proposition 3.2.

Let , and . The Cauchy problem (8) admits a unique weak solution . Moreover, there
exists such that

(11)

The proof of creftypecap 3.2 is based on Galerkin approximations, energy estimates and Gronwall’s argument (see [15, Section 7.1.2]).
We also have the following classical -estimate for (8).

Proposition 3.3.

Let , and . Then the solution of (8) belongs to and there exists such that

(12)

The proof of creftypecap 3.3 is based on Stampacchia’s method (see the proof of [28, Chapter 3, Paragraph 7, Theorem 7.1]).
Let us also mention the dissipativity in time of the -norm of the heat equation with a bounded potential.

Proposition 3.4.

Let , and . Then, there exists such that the solution of (8) with , satisfies for every ,

(13)

The proof of creftypecap 3.4 is based on the application of the variational formulation (9) with a cut-off of and a Gronwall’s argument.

Nonlinear parabolic equations

We give the definition of a solution of (1).

Definition 3.5.

Let , . A function is the solution of (1) if for every ,

(14)

and

(15)

The uniqueness of a solution to (1) is an easy consequence of the fact that is locally Lipschitz because .

3.2. Maximum principle

We state the maximum principle for the heat equation.

Proposition 3.6.

Let , and . Let and be the solutions to

(16)

Then, we have the comparison principle

(17)

The proof of creftypecap 3.6 is based on the comparison principle for smooth solutions of (16) (see [41, Theorem 8.1.6]) and a regularization argument.
We state a comparison principle for the semilinear heat equation (1) without control .

Proposition 3.7.

Let , . We assume that there exist a subsolution and a supersolution in of (1), i.e., (respectively ) satisfies (14), (15) replacing the equality by the inequality (respectively by the inequality ). Moreover, we suppose that and are ordered in the following sense

Then, there exists a (unique) solution of (1). Moreover, satisfies the comparison principle

(18)

For the proof of creftypecap 3.7, see [41, Corollary 12.1.1].

3.3. - estimates

We have the well-known regularizing effect of the heat semigroup.

Proposition 3.8.

[8, Proposition 3.5.7]
Let , and be the solution to (8) with . Then, there exists such that for every , we have

(19)

4. Global nonnegative-controllability of the linear heat equation with a bounded potential

4.1. Statement of the result

Let . We consider the heat equation with a bounded potential

(20)

and the following adjoint equation

(21)

The goal of this section is to prove the following theorem.

Theorem 4.1.

For every , (20) is globally nonnegative-controllable in time . More precisely, for every , there exists , with

(22)

such that for every , there exists such that

(23)

and

(24)
Remark 4.2.

Actually, by looking carefully at the proof of creftypecap 4.1 (see Section 4.5 below), we can see that the control in creftypecap 4.1 can be chosen constant in the time and the space variables.

Remark 4.3.

It is well-known that (20) is globally nonnegative-controllable in time because it is globally null-controllable in time (see [20, Theorem 2]) but the most interesting point is the cost of nonnegative-controllability given in creftypecap 4.1. In particular, the exponent of the term will be the key point to prove creftypecap 2.2 (see Section 5).

4.2. A precise - observability inequality for the linear heat equation with bounded potential and nonnegative initial data

The proof of creftypecap 4.1 is a consequence of this kind of observability inequality.

Theorem 4.4.

For every , there exists of the form (22) such that for every , the solution to (21) satisfies

(25)

An immediate corollary of creftypecap 4.4 is this observability inequality - that we state to discuss it below, but that will not be used in the present article.

Corollary 4.5.

For every , there exists of the form (22) such that for every the solution to (21) satisfies

(26)

It is well-known that null-controllability in is equivalent to an observability inequality in for every (see [9, Theorem 2.44]). The main idea behind creftypecap 4.5 is the fact that nonnegative-controllability in is a consequence of an observability inequality in for every (see Section 4.5).

Remark 4.6.

It is interesting to mention that (26) holds with of the form

(27)

for every (see [20, Theorem 2]). The exponent of the term is the key point to prove creftypecap 1.4. Note that the optimality of the exponent has been proved by Thomas Duyckaerts, Xu Zhang and Enrique Zuazua in the context of parabolic systems in even space dimensions and with Dirichlet boundary conditions (see [14, Theorem 1.1] and also [44, Theorem 5.2] for the main arguments of the proof). creftypecap 4.5 shows that we can actually decrease the exponent to the exponent for nonnegative initial data. In some sense, we can make the connection between the recent preprint of Camille Laurent and Matthieu Léautaud who disprove the Miller’s conjecture about the short-time observability constant of the heat equation in the general case and show that the conjecture holds true for nonnegative initial data by using Li-Yau estimates (see [29] and [32]).

Remark 4.7.

In the context of the wave equation in one space dimension, the (optimal) constant of observability inequality for the linear wave equation with a bounded potential is actually (see [42, Theorem 4]) which leads to the exact controllability of the semilinear wave equation in large time for semilinearities satisfying (3) with (see [42, Theorem 1] and also [7, Problem 5.5] for the presentation of the related open problem in the multidimensional case). Roughly speaking, as an ordinary differential argument would indicate, this constant of observability inequality is very natural because the wave operator is of order two in the time and the space variables. Then, by analogy and by taking into account that the heat operator is of order one in the time variable and of order two in the space variable, one could rather expect a constant of obervability inequality of the order or which seem to be more intuitive than the term .

4.3. A new Carleman estimate

The goal of this section is to establish a Carleman estimate for nonnegative initial data (see creftypecap 4.9 below). First, we introduce some classical weight functions for proving Carleman inequalities.

Lemma 4.8.

Let be a nonempty open subset. Then there exists such that in , in , and in .

A proof of this lemma can be found in [9, Lemma 2.68].
Let be a nonempty open set satisfying and let us set

(28)
(29)

for , where is the function provided by creftypecap 4.8 for this and is a parameter.
We have the following new Carleman estimate.

Theorem 4.9.

There exist two constants and , such that,

(30)

for every , the nonnegative solution of (21) satisfies

(31)
Proof.

Unless otherwise specified, we denote by various positive constants varying from line to line which may depend on , but independent of the parameters and .
We introduce other weights which are similar to and

(32)
(33)

The following estimates

(34)

will be very useful for the proof.
Let . The general case comes from an easy density argument by using the fact that is dense in for the topology.
The solution of (21) is nonnegative by applying the maximum principle given in creftypecap 3.6 with and .
We define

The proof is divided into five steps:

  • Step 1: We integrate over an identity satisfied by .

  • Step 2: We get an estimate which looks like to (31) up to some boundary terms.

  • Step 3: We repeat the step 1 for .

  • Step 4: We repeat the step 2 for .

  • Step 5: We sum the estimates of the step 2 and the step 4 to get rid of the boundary terms.

Remark 4.10.

The ‘trick’ of the proof to get rid of the boundary terms is inspired by the proof of the usual Carleman estimate for Neumann boundary conditions due to Andrei Fursikov and Oleg Imanuvilov (see [23, Chapter 1] and also [20, Appendix]).

Step 1: An identity satisfied by . We readily obtain that

(35)

where

(36)
Remark 4.11.

The starting point, i.e., the identity (35) is the same as in the classical proof developed by Andrei Fursikov and Oleg Imanuvilov in [23] (see also [21, Proof of Lemma 1.3] or [31, Section 7]). But, from now, the proof strategy of the -Carleman estimate is very different from the usual one of the -Carleman estimate. Indeed, we will focus on the fourth right hand side term of (36)

It is nonnegative because is nonnegative and it is of order two in the parameter whereas the seventh right hand side term of (36)

is of order in the parameter . This comparison suggests to integrate the identity (35) in order to obtain (31) for and as defined in (30).

We integrate (35) over

(37)

Note that all the terms in (37) are well-defined. Indeed, by using and the parabolic regularity in to (21) (see [12, Theorem 2.1]), we deduce that then .
Step 2: Estimates for . As a consequence of the properties of (see creftypecap 4.8), we have

(38)

which yields

(39)

By combining (37) and (39), we have

(40)

We have the following integration by parts

(41)
(42)
(43)

where .
From (40), (41), (42), (43), we have

(44)

By using the first two lines of (34) and , we have

(45)

By combining (44) and (45), we get

(46)