Null-controllability of linear parabolic-transport systems
Karine Beauchard, Armand Koenig, K\'evin Le Balc'h
TL;DR
This paper investigates the null-controllability of linear parabolic-transport systems on a one-dimensional torus, establishing conditions under which control is possible in optimal time based on spectral analysis.
Contribution
It introduces a unified framework for controllability of parabolic-hyperbolic systems, providing algebraic conditions and spectral analysis techniques for control feasibility.
Findings
Null-controllability achieved in optimal time with as many controls as equations.
Necessary and sufficient algebraic conditions for controllability when control acts only on specific components.
Negative results for small time controllability on high-frequency solutions.
Abstract
Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results. We consider constant coefficients heat-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one dimensional torus. We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as much controls as equations. When the control acts only on the transport (resp. parabolic) component, we prove an algebraic necessary and sufficient condition, on the coupling term, for the null controllability. The whole study relies on a careful spectral analysis, based on perturbation theory. The negative controllability…
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Null-controllability of linear parabolic-transport systems
Karine Beauchard, Armand Koenig, Kévin Le Balc’h Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France. email: [email protected]é Côte d’Azur, CNRS, LJAD, France. email: [email protected] Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France. email: [email protected]
Abstract
Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results.
We consider constant coefficients heat-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one dimensional torus.
We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as much controls as equations. When the control acts only on the transport (resp. parabolic) component, we prove an algebraic necessary and sufficient condition, on the coupling term, for the null controllability.
The whole study relies on a careful spectral analysis, based on perturbation theory. For high frequencies, the spectrum splits into a parabolic part and an hyperbolic part. The negative controllability result in small time is proved on solutions localized on high hyperbolic frequencies, that solve a pure transport equation up to a compact term. The positive controllability result in large time is proved by projecting the dynamics onto 3 eigenspaces associated to hyperbolic, parabolic and low frequencies; that defines 3 weakly coupled systems.
Keywords: parabolic-transport systems, null-controllability, observability.
MSC 2010: 93B05, 93B07, 93C20, 35M30.
Contents
- 1 Introduction
- 2 Preliminary results
- 3 Obstruction to the null-controllability in small time
- 4 Large time null-controllability
- 5 Hyperbolic control: coupling of order zero
- 6 Hyperbolic control: coupling of order one
- 7 Parabolic control
- A Pure transport solutions are not enough to disprove the observability inequality
1 Introduction
1.1 Parabolic-transport systems
We consider the linear control system
[TABLE]
where
- •
, , is a nonempty open subset of , , , , ,
- •
the state is ,
- •
the control is .
We assume
[TABLE]
Introducing the analogue block decomposition for the matrices and , the matrix and the function ,
[TABLE]
we see that the system (Sys) couples a transport equation on with a parabolic equation on
[TABLE]
We make the following hypothesis on the matrix
[TABLE]
We will prove later, with vector valued Fourier series and a careful spectral analysis, that for every and , there exists a unique solution of (Sys) (see Section 2.2.3). In this article, we are interested in the null controllability of (Sys).
Definition 1**.**
The system (Sys) is null-controllable on in time if for every , there exists a control supported on such that the solution of (Sys) satisfies .
We aim at
- •
identifying the minimal time for null controllability,
- •
controlling the system with a small number of controls ,
- •
understanding the influence of the algebraic structure on the above properties.
1.2 Statement of the results
1.2.1 Control on any component, minimal time
Our first result identifies the minimal time, when the control acts on each of the equations.
Theorem 2**.**
We assume that is a strict open subset of . We also assume (H.1)–(H.4) and that the control matrix is (and so ). We define111If is measurable, we note its Lebesgue measure.
[TABLE]
[TABLE]
and
[TABLE]
Then
- i)
the system (Sys) is not null-controllable on in time , 2. ii)
the system (Sys) is null-controllable on in any time .
In particular, when is an interval of and , then the minimal time for null controllability is .
Actually, the controls may be more regular than in Definition 1: we construct controls of the form where and .
The proof of Theorem 2 relies on a spectral decomposition: for high frequencies, the spectrum splits into a parabolic part and a hyperbolic part.
The negative result in time is expected, because of the transport component of the system, but its proof does require some care. Indeed, because of the coupling with a parabolic component, in general, there does not exist pure transport solutions to the system (Sys), concentrated outside (see Appendix A for more precision).
The proof of the positive result, in time relies on an adaptation, to systems with arbitrary size, of the strategy introduced by Lebeau and Zuazua [20] to control the system of linear thermoelasticity, that couples a scalar heat equation and a scalar wave equation. By projecting the dynamics onto appropriate eigenspaces, the system is decomposed into 3 weakly coupled systems. The first one behaves like a transport system, its controllability is handled by hyperbolic methods from [1]. The second one behaves like a parabolic system, its controllability is handled by the Lebeau-Robbiano method. The third one, associated to low frequencies, has finite dimension; its controllability is handled by a compactness/uniqueness argument.
The null controllability of the system (Sys) in time is an open problem.
1.2.2 Control on the hyperbolic component
Our second result concerns controls acting on the whole transport component, , but not on the parabolic component, . To get an aesthetic necessary and sufficient algebraic condition for null controllability, we also assume that the diffusion is given by , the coupling is realized exclusively by the transport term , i.e. and there is no zero order term in the parabolic dynamics, i.e. , which corresponds to the system
[TABLE]
By integrating with respect to the space variable the second equation of (4), we see that, for being steered to zero, an initial condition has to satisfy
[TABLE]
For any vector subspace of we denote by the vector subspace made of functions with zero mean value, i.e. .
Theorem 3**.**
We assume (H.1)–(H.4), , , , and . Let be defined by (3). The following statements are equivalent:
- •
For every and , there exists such that the solution of (4) satisfies .
- •
The couple of matrices satisfies the Kalman rank condition:
[TABLE]
With the same proof, similar statements can be proved for the following systems:
[TABLE]
with arbitrary initial conditions and Kalman rank condition on (see Section 5),
[TABLE]
with arbitrary initial conditions and Kalman rank condition on ,
[TABLE]
with initial conditions satisfying (5) and Kalman rank condition on .
The proof of the controllability of (4) uses 2 ingredients. The first ingredient is a strengthened version of Theorem 2 with smoother controls, more precisely, the associated observability inequality with observation of negative Sobolev norms of the parabolic component. The second ingredient is a cascade structure (or Brunovski form) of the system (4) ensured by the Kalman condition, to eliminate the observation of the parabolic component.
Proving an algebraic necessary and sufficient condition for null controllability of (Sys), involving both matrices , and is an open problem. In the context of parabolic systems, this difficulty already appeared, see [3] and [11].
1.2.3 Control on the parabolic component
Our third result concerns controls acting on the whole parabolic component, , but not on the hyperbolic component of the system, . To get an aesthetic necessary and sufficient condition for null controllability, we also assume that the coupling is realized exclusively by the transport term , i.e. , and there is no zero order term in the hyperbolic dynamics, i.e. . This corresponds to the system
[TABLE]
By integrating with respect to the space variable the first equation of (10), we see that, for being steered to zero, an initial condition has to satisfy
[TABLE]
i.e. .
We need to adapt the notion of null controllability, because null controllable initial conditions necessarily have a regular hyperbolic component. Indeed, in (10), the source term entering the hyperbolic equation on — that has to serve as an indirect control for — is smooth, because of the parabolic smoothing on . Such a smooth source term cannot steer to zero non-smooth initial conditions.
Theorem 4**.**
Let be an open interval of . We assume (H.1)–(H.4), , , , and . Let be defined by (3). The following statements are equivalent.
- •
For every and there exists such that the solution of (10) satisfies .
- •
The couple of matrices satisfies the Kalman rank condition:
[TABLE]
In Theorem 4, we assume that the open set of control is an interval because the proof uses [1, Lemma 2.6] (see Lemma 48 below). The generalization of this result to a general open set of is not known.
A similar statement can be obtained with the same proof, when , under Kalman rank condition on .
The proof of Theorem 4 follows essentially the same strategy as the one of Theorem 3: a strengthened version of Theorem 2 and a cascade structure ensured by Kalman condition. The regularity assumption on the hyperbolic component allows the elimination of the observation of the hyperbolic component.
After Theorem 4, two problems are still open:
- •
the characterization of null controllable initial conditions: it may be a larger space than , see Section 7,
- •
the algebraic necessary and sufficient condition for null controllability, involving both matrices and . In the context of parabolic systems, this difficulty already appeared, see [3] and [11].
1.3 Organization of the article
Section 2 is dedicated to preliminary results concerning the spectral analysis of on , the well posedness of (Sys) and the Hilbert uniqueness method.
In Section 3, we prove the negative null controllability result in time of Theorem 2.
In Section 4, we prove the positive null controllability result in time of Theorem 2.
In Section 5, we explain how to adapt this proof to get the null controllability in time of system (7). The interest of this section is to introduce the proof strategy of Theorem 3 and Theorem 4, in a less technical framework.
Then, in Section 6, we prove Theorem 3 and in Section 7, we prove Theorem 4.
1.4 Bibliographical comments
1.4.1 Wave equation with structural damping
We consider the 1D wave equation with structural damping and control
[TABLE]
where . This equation can be splitted in a system of the form (Sys) by considering ,
[TABLE]
i.e. (Sys) with , , ,
[TABLE]
Rosier and Rouchon [24] studied the equation (13) on a 1D-interval, , with a boundary control at and . This is essentially equivalent to take (13) with , Dirichlet boundary conditions at and , and a source term of the form , where is a fixed profile and is a scalar control. The authors prove that this equation is not controllable.
By Theorem 2, we extend this negative result to general controls (i.e. without separate variables) for periodic boundary conditions. Here, , , , the system (14) is not controllable even with an additional control in the second equation.
In [24], the authors prove that this system is not even spectrally controllable, because of an accumulation point in the spectrum. Indeed, by the moment method, a control that would steer the system from an eigenstate to another one would have a Fourier transform vanishing on a set with an accumulation point, which is not possible for an holomorphic function.
Martin, Rosier and Rouchon [21], studied the null-controllability of the equation (13) on the 1D torus, , with moving controls, i.e. with . By the change of variable , this is equivalent to study the null controllability of the system
[TABLE]
which has the form (Sys) with the same matrices , , as in (15) and
[TABLE]
In [21, Theorem 1.2], for , the authors prove that any initial data with can be steered to [math] in time by mean of a control .
By Theorem 3, we recover this positive null controllability result with a smaller minimal time and a weaker regularity assumption on the initial data for (13). This corresponds to an initial data for (16) because . Actually, Theorem 3 can be applied for in (16) but an easy adaptation of Theorem 3 gives the same result for every . We also prove the negative result in time . Here, , and .
The limitations in [21, Theorem 1.2] (regularity and time) are due to the use of controls with separate variables . The proof relies on the moment method and the construction of a biorthogonal family. A key point in both [21] and the present article is a splitting of the spectrum in one parabolic-type part, and one hyperbolic-type part.
Finally, Chaves-Silva, Rosier and Zuazua [7] study the multi-dimensional case of equation (13), , with Dirichlet boundary conditions and locally distributed moving controls . The control region is assumed to be driven by the flow of an ODE that covers all the domain within the alloted time . Then, the authors prove the null controllability of any initial data with a -control.
In the particular case with a motion with constant velocity, Theorem 3 gives the same minimal time for the null controllability and also the negative result in smaller time.
The proof strategy in [7] consists in proving Carleman estimates for the parabolic equation and the ODE in (14) with the same singular weight, adapted to the geometry of the moving support of the control.
As explained in [7, Section 5.2], the same construction cannot be used with periodic boundary conditions.
In the very recent preprint [15], the authors propose another construction of a weight, to get Carleman estimates for parabolic and transport equations in the torus (with the same weight). In the present article, we develop another strategy.
1.4.2 Wave-parabolic systems
Albano and Tataru [2] consider parabolic-wave systems with boundary control, where
- •
the coupling term in the wave equation is given by a second order operator with respect to ,
- •
the coupling term in the parabolic equation is given by a first order operator with respect to .
This large class contains the linear system of thermoelasticity
[TABLE]
where .
The authors of [2] prove the null controllability in large time of these systems, precisely in any time for the system (17). The proof relies on Carleman estimates for the heat and the wave equation with the same singular weight. This strategy inspired Chaves-Silva, Rosier and Zuazua [7].
Lebeau and Zuazua [20] prove the null-controllability of the linear system of thermoelasticity (17) with a locally distributed control in the source term of the wave equation, under the geometric control condition on . The method is based on a spectral decomposition. For high frequencies, the spectrum splits into a parabolic part and a hyperbolic part. Projecting the dynamics onto the parabolic/hyperbolic subspaces, the system is decomposed into 2 weakly coupled systems, the first one behaving like a wave equation, the second one like a heat equation. The wave equation is handled by using the microlocal techniques developped for the wave equation [4]. The parabolic equation is treated by using Lebeau and Robbiano’s method [19]. The low frequency part is treated by a compactness argument relying on a unique continuation property.
The proof of the positive controllability results in the present article is an adaptation, to coupled transport-parabolic systems of any size, of this approach, introduced for a wave-parabolic system. The transport equation is handled by using the results from Alabau-Boussouira, Coron and Olive [1].
The framework of systems (Sys) does not cover the system (17) because the order of the coupling terms is too high.
1.4.3 Heat equation with memory
Ivanov and Pandolfi [16] and after them Guerrero and Imanuvilov [14] consider the heat equation with memory
[TABLE]
In 1D, this equation can be splitted into a system of the form (Sys) by considering :
[TABLE]
i.e.
[TABLE]
In [16], the authors prove that the heat equation with memory term is not “null-controllable to the rest”. In [14], the authors prove that the scalar equation (18) is not null controllable (whatever ). Thus the system (19) is not null controllable.
Theorem 2 proves that, when Dirichlet boundary conditions are replaced by periodic boundary conditions, then system (19) is not null controllable, even with an additional control in the first equation.
1.4.4 1D-Linearized compressible Navier-Stokes equations
The compressible Navier-Stokes equation on the 1D torus writes
[TABLE]
where , are the density and velocity of the fluid. The state is and the control is . We consider a constant stationary state . The linearized system around the trajectory is
[TABLE]
This system is in the form (Sys) with
[TABLE]
By Theorem 2, the system (20) with two controls is null controllable in time and is not null controllable in time .
By Theorem 3, the system (20) with one control in the first line (i.e. ) is null controllable in time .
By Theorem 4, with one control in the second equation (i.e. ), any initial condition can be steered to zero in time by a control .
In [13], Ervedoza, Glass, Guerrero and Puel consider the (nonlinear) compressible Navier-Stokes equations on a bounded interval , without source term (i.e. ), but with a boundary control on both and at the two boundaries and . They prove the local controllability of this nonlinear system, around the trajectory , in appropriate functional spaces. A key ingredient is the controllability of the linearized system, which is proved to hold in time . Theorem 2 of the present article allows to recover the same result with interior control instead of boundary control, and also proves the negative result in small time.
In [9, Theorem 1.4], Chowdhury, Mitra, Ramaswamy and Renardy prove the null controllability of (20) with two controls in time , with spectral methods. Thus, Theorem 2 of the present article allows to recover the same result but with a better minimal time and also proves the negative result in time .
In [8, Theorem 1.3], Chowdhury and Mitra prove with moment methods that any initial condition with can be steered to zero in time by a control acting only on the second equation (i.e. ). In [9, Theorem 1.2], Chowdhury, Mitra, Ramaswamy and Renardy prove the same result for any initial conditions . Thus, Theorem 4 of the present article provides a smaller minimal time , for smoother initial conditions . It also proves the negative result in time .
2 Preliminary results
We want to understand the operator
[TABLE]
with domain
[TABLE]
where the derivatives are considered in the distributional sense . Throughout the article, we will note the function . We remark that applying to , where , we get
[TABLE]
Thus, if we define the following perturbation of
[TABLE]
then acts on the Fourier side as a multiplication by .
In Section 2.1, we apply the perturbation theory to the matrices near : the spectrum of splits into 2 parts: one close to zero that defines the hyperbolic component, one close to the spectrum of that defines the parabolic component. In Section 2.2, we deduce the dissipation of the parabolic component and the boundedness of the hyperbolic component. Thanks to these estimates, we prove the well-posedness of System (Sys). Finally, in Section 2.3, we recall the Hilbert Uniqueness Method.
2.1 Perturbation theory
If we want to understand the semigroup , we need to know the spectrum and the eigenvectors of . Here, we relate the spectral properties of to those of and , in the limit . This is instrumental in all the article. Our proofs are essentially self-contained, but the reader unfamiliar with the analytic perturbation theory in finite dimension may read [17, Ch. II §1 and §2].
For and , we define as the set of holomorphic functions in the complex disk with values in . Our first result is the following one.
Proposition 5**.**
There exist and a matrix-valued holomorphic function such that
- i)
P^{\mathrm{h}}(0)=\big{(}\begin{smallmatrix}I_{d_{1}}&0\\ 0&0\end{smallmatrix}\big{)}, 2. ii)
for all , is a projection that commutes with , 3. iii)
in the limit , .
Proof.
The spectrum of is continuous in (see [17, Ch. II §1.1]). Let us consider the “[math]-group” of eigenvalues, i.e. the set of eigenvalues that tend to [math] as . Then we note the sum of the projections onto the eigenspace222We stress that when we talk about “eigenspace”, we mean “generalized eigenspace” (or, in the terminology of Kato, algebraic eigenspace), i.e. the space of generalized eigenvectors. of associated with eigenvalues in the [math]-group along the other eigenspaces. Another way to define is to choose and small enough so that for , there is no eigenvalues of on the circle . Then, we define (see [17, Ch. II, Eq. (1.16)])
[TABLE]
In the terminology of Kato, is the “total projection for the [math]-group”. Then, according to [17, Ch. II §1.4], is the projection onto the sum of eigenspaces associated to eigenvalues of lying inside along the other eigenspaces. It is holomorphic in . For , the formula (25) that defines becomes
[TABLE]
Then, is the projection onto the eigenspace of associated to the eigenvalue [math] along the other eigenspaces (see [17, Ch. II §1.4]). So, according to the hypotheses (H.2–H.3) on the blocks of , P^{\mathrm{h}}(0)=\big{(}\begin{smallmatrix}I_{d_{1}}&0\\ 0&0\end{smallmatrix}\big{)}. This proves i).
According to the definition (25), commutes with . This proves ii). Then we have
[TABLE]
which, along with the holomorphy of , proves iii). ∎
We say that is the “projection on the hyperbolic branches”. We note , which we call the “projection on the parabolic branches”, and satisfies properties analog to :
Proposition 6**.**
The matrix-valued function is in and
- i)
P^{\mathrm{p}}(0)=\big{(}\begin{smallmatrix}0&0\\ 0&I_{d_{2}}\end{smallmatrix}\big{)}, 2. ii)
for all , is a projection that commutes with , 3. iii)
in the limit , .
We will need to split the hyperbolic branches further.
Proposition 7**.**
There exist and a family of matrix-valued holomorphic functions satisfying
- i)
for all and , is a non-zero projection that commutes with , 2. ii)
for all , and for all , 3. iii)
for every , there exists such that
[TABLE]
Remark 8*.*
For , the projection is holomorphic and thus continuous in . Therefore, the rank of , which is its trace, does not depend on (the even are similar, see [17, Ch. I, §4.6, Lem. 4.10]). In the same vein, the ranks of and do not depend on .
Proof.
The proof is essentially the “reduction process” of Kato [17, Ch. II §2.3]. According to Prop. 5, is holomorphic and . Then we define
[TABLE]
which is holomorphic in . Note that we have according to Kato [17, Ch. II Eq. (2.38)]
[TABLE]
Let us assume for the moment that [math] is not an eigenvalue of . Then, for , we define the total projection on the -group of eigenvalues of . Said otherwise, and according to the definition of , is the total projection on the -group of eigenvalues of . The projection is defined and holomorphic for small enough according to [17, Ch. II, §1.4].
Since for small enough, is the projection on some eigenspaces of associated with non-zero eigenvalues,
[TABLE]
with the last inclusion coming from the definition of . Thus is a subprojection of . Moreover, commutes with , so it commutes with . This proves Item i) in the case .
For , and are the projections on some sums of eigenspaces associated with different eigenvalues, so . Let us note for convenience . Then, for small, is the projection on all the eigenspaces of associated with non-zero eigenvalues. According to the definition of , this proves that is a subprojection of . Let us check that and have the same rank. This will prove that for all small enough, . The rank of , which is its trace, does not depend on . The same is true for . For , we have , so by using the fact that ,
[TABLE]
This proves that for all small enough, , and in turn finishes the proof of Item ii) in the case where .
If , then we add to for some . This amounts to adding to . This only shifts the eigenvalues of the restriction of to (but not of its restriction to ) by , while leaving the eigenprojections unchanged. Thus, choosing so that , we get the Items i) and ii) in the case .
We still need to prove the asymptotics of Item iii). Since is diagonalizable, so is . So, there is no nilpotent part in the spectral decomposition of . That is to say, for all ,
[TABLE]
Since is holomorphic, we have
[TABLE]
Finally, we multiply by to come back to , which gives us
[TABLE]
2.2 Estimates on Fourier components and well-posedness
2.2.1 Dissipation of the parabolic component
The goal of this section is the proof of the following result.
Proposition 9**.**
There exist such that for every , and ,
[TABLE]
Proof.
By using 6, for , we denote by the restriction of to the vector subspace , which is an endomorphism of .
By assumption (H.3), there exists such that . There exists an open disk in the complex plane such that and . Then, by continuity of the spectrum, for small enough, we have, for every , .
Step 1: Cauchy formula.
We prove the following equality between endomorphisms of
[TABLE]
where is the identity on . The right hand side is well defined because . Let us denote it by . Then
[TABLE]
By the Cauchy formula, thus . Moreover because all the eigenvalues of are inside (see [17, Ch. I, Problem 5.9]). Thus .
Step 2: Estimate. We deduce from (26) the following equality between endomorphisms of
[TABLE]
Note that, if is small enough, then the eigenvalues of are either inside (parabolic branch) or close to [math] (hyperbolic branch), for instance in . Thus \mathopen{}\mathclose{{}\left(\xi I_{d}-E(z)}\right) is invertible on for every and the above right hand side is well defined.
We deduce from (27) that
[TABLE]
The map (\xi,z)\in\partial\Omega\times\overline{D}(0,r)\mapsto\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left(\xi I_{d}-E(z)}\right)^{-1}P^{\mathrm{p}}(z)}\right| is continuous on a compact set thus bounded. Thus there exists a positive constant such that, for every and , \mathopen{}\mathclose{{}\left|\mathrm{e}^{-E(z)\tau}P^{\mathrm{p}}(z)}\right|\leq K\mathrm{e}^{-c\tau}. ∎
2.2.2 Boundedness of the transport component
The goal of this section is to prove the following result.
Proposition 10**.**
There exists such that for every , and ,
[TABLE]
Proof.
Let be as in 7, , , and . Taking into account that is stable by , we get
[TABLE]
Note that and commute because and commute and . Thus, by using that , we obtain
[TABLE]
where . We conclude for that
[TABLE]
with and K=\max\mathopen{}\mathclose{{}\left\{\sum_{\mu\in\operatorname{Sp}(A^{\prime})}\lvert P^{\mathrm{h}}_{\mu}(z)\rvert;\>z\in\overline{D}(0,r)}\right\}. ∎
2.2.3 Well-posedness
By gathering the results of the previous two subsubsections, we can prove that the heat-transport system (Sys) is well-posed. We define the Fourier coefficients of by
[TABLE]
We consider the operator defined by (21) and (22). By Bessel-Parseval identity and the fact that ,
[TABLE]
The goal of this section is to prove the following result.
Proposition 11**.**
* generates a semi-group of bounded operators on .*
This result will ensure well posedness of (Sys) in the following sense.
Definition 12**.**
Let , and . The solution of (Sys) is the function defined for by
[TABLE]
Moreover, satisfies the estimate
[TABLE]
where depends on but not on and . We will also note this solution.
Proof.
We deduce from 9 and 10 that for every , and ,
[TABLE]
where K=\max\mathopen{}\mathclose{{}\left\{K_{\mathrm{p}}\mathopen{}\mathclose{{}\left\lvert P^{\mathrm{p}}(\mathrm{i}x)}\right\rvert+K_{\mathrm{h}}\mathopen{}\mathclose{{}\left\lvert P^{\mathrm{h}}(\mathrm{i}x)}\right\rvert;x\in[-r,r]}\right\}.
For and we define
[TABLE]
By Bessel Parseval equality and (30) with , is a bounded operator on , because the number of such that is finite. The semi-group properties and are clearly satisfied. For , we have, by Bessel Parseval equality
[TABLE]
In the right hand side, each term of the series converges to zero when and, thanks to (30), is dominated for every and by
[TABLE]
which can be summed over because , see (28). By the dominated convergence theorem, the sum of the series converges to zero. ∎
Remark 13*.*
We can see from this proof that the semi-group is strongly continuous on any for any , i.e. we have
[TABLE]
2.3 Adjoint system and observability
The null-controllability of a linear system is equivalent to a dual notion called “observability”. We have the following general, abstract result (see [10, Lemma 2.48]).
Lemma 14**.**
Let , and be three Hilbert spaces. Let and be continuous linear maps. Then
[TABLE]
if and only if there exists such that
[TABLE]
From Lemma 14, see [10, Theorem 2.44], we deduce the following result.
Proposition 15**.**
Given , the system (Sys) is null-controllable on in time if and only if there exists such that for every , the solution to the equation
[TABLE]
satisfies
[TABLE]
Note that the solutions of the adjoint system (31) are of the form333When we write , it is to be understood as . We will use the same notation for etc.
[TABLE]
Moreover, we have a spectral theory for the adjoint system that is similar to Prop. 5–7. We just have to take the adjoint of each formulas of these Propositions.
Remark 16*.*
As for the semi-group (see Remark 13), the dual semi-group is strongly continuous on any for any , i.e. we have
[TABLE]
3 Obstruction to the null-controllability in small time
The goal of this section is to prove the first point of Theorem 2, by disproving the observability inequality (32) on an appropriate solution of the adjoint system (31). Intuitively, the lack of null-controllability in small time should come from the transport components. So, the idea is to construct approximate transport solutions. Note that in general, there is no exact transport solutions that are supported on a strict subset of (see Appendix A).
Proof of the lack of null-controllability in time .
Step 1: Construction of approximate transport solutions.
Let with minimum absolute value (i.e. ). Let be such that the solution of the transport equation on , with initial condition , has its support is disjoint from . Such a solution exists because .
To exploit the spectral asymptotics of the previous section, that are valid in the high-frequency limit, we want a high-frequency version of . To that end, for we consider the polynomial and . Since is the image of by a differential operator, we have . If we note and , we have . In particular, for , .
In summary, satisfies the following properties
- •
is non-zero,
- •
has no components along frequencies less than ,
- •
the support of is a subset of the support of .
In particular, the last property implies that the solution of , with initial condition is such that is disjoint from .
We adopt the notations of 7. Let . We define
[TABLE]
By 7, acts as on the range of . So the definition of can be written alternatively as
[TABLE]
So, according to the representation of the solutions of the adjoint system (Eq. (33)), solves the parabolic-transport system (31). On the other hand, solves the transport equation . We will prove that in the limit , is an approximation of .
Step 2: Approximation of the exact solution by the transport solution.
According to Parseval’s identity, we have for every
[TABLE]
Then, according to the holomorphy of and , and the fact that , we have locally uniformly with respect to
[TABLE]
Now, reminding that for , , we deduce that
[TABLE]
Thanks to Parseval’s identity we rewrite it as
[TABLE]
Step 3: Conclusion.
By the triangle inequality, we have for
[TABLE]
Then, since the support of does not intersect , the first term of the right-hand side is zero, and according to the inequality (36), we have uniformly in
[TABLE]
Integrating this estimate for , we get the following upper bound on :
[TABLE]
To disprove the observability inequality, we also need a lower bound of . According to Parseval’s identity, we have
[TABLE]
Since is in the range of , for large enough, we have |\mathrm{e}^{TR_{\mu}^{\mathrm{h}}\mathopen{}\mathclose{{}\left(\frac{\mathrm{i}}{n}}\right)^{*}}P_{\mu}^{\mathrm{h}}\mathopen{}\mathclose{{}\left(\frac{\mathrm{i}}{n}}\right)^{\!*}\varphi_{0}|\geq c>0. Then, since for , as soon as is large enough,
[TABLE]
Comparing the lower bound (38) and the upper bound (37), we see that the observability inequality (32) (with ) cannot hold. ∎
4 Large time null-controllability
The goal of this section is to prove the point of Theorem 2. An adapted decomposition of is introduced in Section 4.1. The control strategy is presented in Section 4.2. Projecting the dynamics onto the parabolic/hyperbolic subspaces, the system is decomposed into 2 weakly coupled systems, the first one behaving like a transport equation, the second one like a heat equation. The transport equation is handled in Section 4.3 by using the methods developped in [1]. The parabolic equation is treated in Section 4.4 by adapting the Lebeau-Robbiano method [19] to systems with arbitrary size. The low frequency part is treated by a compactness argument and a unique continuation property in Section 4.5.
In the whole Section 4, the parameter is assumed to be small enough so that Propositions 5, 6, 7, 9 and 10 hold.
4.1 An adapted decomposition of
Proposition 17**.**
Let be such that . We have the following decomposition
[TABLE]
where
[TABLE]
Moreover the projections , , and defined by
[TABLE]
are bounded operators on .
Proof.
The function is continuous thus there exists such that, for every , . Let . We deduce from
[TABLE]
and Bessel-Parseval identity that the series \sum P^{\mathrm{p}}\mathopen{}\mathclose{{}\left(\frac{\mathrm{i}}{n}}\right)\hat{f}(n)e_{n} converges in . Using , we get the decomposition
[TABLE]
with convergent series in . This proves . The sum is direct because is orthogonal and when . The linear mappings and are orthogonal projections, thus bounded operators on . We deduce from Bessel-Parseval identity and (43) that is a bounded operator on and so is . ∎
The operator defined in (21) maps into thus we can define an operator on by and . Moreover, generates a -semi-group of bounded operators on and . For the same reasons, we can define an operator on by and , that generates a -semi-group of bounded operators on : . Finally, we can define an operator on by and , that generates a -semi-group of bounded operators on : .
Proposition 18**.**
The operator generates a group of bounded operators on . The operator generates a group of bounded operators on
Proof.
We just need to check that defines a bounded operator of and when . It is clear for because it has finite dimension. For , one may proceed as in the proof of 11, noticing that the estimate of 10 is valid for any . ∎
For the duality method, we will need the dual decomposition of (39), i.e.
[TABLE]
By using the definitions of and in (41) and (42) and the fact that is an Hilbert basis of , we get
[TABLE]
[TABLE]
Moreover,
[TABLE]
and the spaces , and are stable by .
4.2 Control strategy
Let be as in (3) and be such that
[TABLE]
In this section, we consider controls of the form
[TABLE]
where
[TABLE]
[TABLE]
The control is intended to control the hyperbolic component of the system and the control the parabolic component.
The control strategy for system (Sys) consists in
- •
first proving the null controllability in time in a subspace of with finite codimension,
- •
then using a unique continuation argument, to get the full null controllability.
The first step of this strategy is given by the following statement.
Proposition 19**.**
There exists a closed subspace of with finite codimension and a continuous operator
[TABLE]
that associates with each a pair of controls such that
[TABLE]
By “continuous operator”, we mean that, for every , the map is continuous: there exists such that
[TABLE]
The proof strategy of 19 consists in splitting the problem in two parts:
- •
for any initial data and parabolic control , steer the hyperbolic high frequences to zero at time (20),
- •
for any initial data and hyperbolic control , steer the parabolic high frequences to zero at time (21).
Proposition 20**.**
If (in Eq. (40–41)) is large enough, there exists a continuous operator
[TABLE]
such that for every ,
[TABLE]
Proposition 21**.**
If is large enough, there exists a continuous operator
[TABLE]
such that for every ,
[TABLE]
Admitting that 20 and 21 hold, we can now prove 19.
Proof.
We observe that the relation holds if the two following equations are simultaneously satisfied
[TABLE]
If we set
[TABLE]
then solving system (52) is equivalent to
[TABLE]
The operator is compact on because it takes values in . Thus, by Fredhlom’s alternative (see [6, Thm. 6.6]), there exist and continuous linear forms on such that the equation (53) has a solution if and only if
[TABLE]
Under these conditions (54), the equation (53) has a solution given by a continuous map defined on the closed vector subspace of defined by
[TABLE]
Then belongs to . We get the conclusion with
[TABLE]
20 is proved in Section 4.3. 21 is proved in Section 4.4. The unique continuation argument to control the low frequencies is presented in Section 4.5.
4.3 Control of the hyperbolic high frequencies
The goal of this subsection is to prove 20. We remind that and that the control satisfies (50).
4.3.1 Reduction to an exact controllability problem
The goal of this paragraph is to transform the null-controllability problem of 20 into an exact controllability problem associated with an hyperbolic system. Precisely, we will get 20 as a corollary of the following result.
Proposition 22**.**
If (in Eq. (40–41)) is large enough, then, for every , there exists a continuous operator
[TABLE]
such that for every ,
[TABLE]
22 will be proved in Section 4.3.2. Now, we prove 20 thanks to 22.
Proof of 20.
Let . We have to find such that
[TABLE]
or, equivalently,
[TABLE]
According to the well-posedness of the system (Sys) and the continuity of the projection (Definition 12 and Proposition 17), the linear map
[TABLE]
is continuous from into , equipped with the -norm. Since is supported in by (50), we have
[TABLE]
As pointed out in 18, is well-defined for all . Therefore, by using (57) and (58), (56) is equivalent to
[TABLE]
We get the conclusion with
[TABLE]
4.3.2 Exact controllability of the hyperbolic part
The goal of this section is to prove 22. By the Hilbert Uniqueness Method, 22 is equivalent to the following observability inequality (it is an adaptation of [10, Thm. 2.42]).
Proposition 23**.**
If is large enough, there exists a constant such that for every , the solution of (31) satisfies
[TABLE]
where denotes the first components of .
Proof.
Let . By using the definition (46) of , decomposes as follows
[TABLE]
Then, the solution of (31) is
[TABLE]
Let .
Step 1: We prove the existence of , independent of , such that
[TABLE]
where and
[TABLE]
By using i) and iii) of 7, we have
[TABLE]
which leads to
[TABLE]
where
[TABLE]
By regularity of , Bessel-Parseval identity and (64) there exists , independent of , such that
[TABLE]
By (65), the function defined by
[TABLE]
solves
[TABLE]
We introduce the solution of
[TABLE]
Using the Duhamel formula for system (69) and the estimate (67), we obtain
[TABLE]
where is independent of . The time is the minimal time for the observability of the system (70) on (see for instance [1, Theorem 2.2]). Indeed, for any ,
[TABLE]
Since , there exists , independent of , such that
[TABLE]
By the triangular inequality, (68) and (71), we deduce that
[TABLE]
which ends the first step.
Step 2: We prove the existence of , independent of , such that
[TABLE]
Taking into account that the projection commutes wih we deduce from (62) that for any ,
[TABLE]
thus,
[TABLE]
because, for , . By using the regularity of , Bessel-Parseval identity and (64), we obtain independent of such that
[TABLE]
We deduce from Step 1, the triangular inequality and the previous estimate that
[TABLE]
which ends Step 2.
Step 3: Conclusion.
For every , we have thus
[TABLE]
u Using (62), the triangular inequality, Step 2 and the previous inequality, we obtain
[TABLE]
From this estimate and the compact embedding , a classical compactness-uniqueness argument gives the observability inequality (60) (see for instance [12, Lemma 2.1 and Remark 2.2]).
Indeed, by Peetre’s lemma (see [22, Lemma 3]), we have from (74) that
[TABLE]
has finite-dimension. Moreover, from [22, Lemma 4], to prove (60), we only need to show that is reduced to zero. First, by definition, we remark that decreases as increases. By a small perturbation of , we may therefore assume that for small, in which case is stable by where is the restriction of to . Then, if is not reduced to zero, it contains an eigenfunction of , i.e. a function of the form where , and X=P^{\mathrm{h}}\mathopen{}\mathclose{{}\left(\frac{\mathrm{i}}{n}}\right)X. By definition of , the first components of that eigenfunction vanishes on i.e. , or equivalently . Thus
[TABLE]
where does not depend on . For a large enough choice of , this is impossible. ∎
4.4 Control of the parabolic high frequencies
The goal of this subsection is to prove 21. We recall that and are chosen such that and the control is such that (49) and (50) hold.
The strategy is the following one: identify the equation satisfied by the last components of the parabolic equation (31) with the help of the asymptotics of 7, then construct smooth controls by adapting the Lebeau-Robbiano’s method to systems.
In this section, for every vector , we will note its first components and its last components.
4.4.1 Reduction to a null-controllability problem
The goal of this paragraph is to transform the null-controllability problem of 21 into a null-controllability problem associated to a parabolic system. Precisely, we will prove that 21 is a consequence of the following result.
Proposition 24**.**
If is large enough, then for every , there exists a continuous operator
[TABLE]
such that for every ,
[TABLE]
24 will be proved thanks to an adaptation of Lebeau-Robbiano’s method in Section 4.4.4, after two sections of preliminary results. Now we prove 21 thanks to 24.
Proof of 21.
Let . We have to find such that
[TABLE]
or equivalently,
[TABLE]
In view of the support of the controls (Eq. (50)), the equality (76) is equivalent to
[TABLE]
or
[TABLE]
By using Definition 12 and Proposition 17, we see that the mapping is continuous from into . Thus we get the conclusion with
[TABLE]
4.4.2 Equation satisfied by the parabolic components of the free system
We begin by proving that if is in then we can compute the first components of from the last . This will allow us to write an uncoupled equation for these components.
Proposition 25**.**
If is small enough, there exists a matrix such that for every ,
[TABLE]
Moreover, is holomorphic in and .
Proof.
We write
[TABLE]
Since is a projection, is in if and only if
[TABLE]
In particular, if , then . And since P^{\mathrm{p}}(0)^{\ast}=\big{(}\begin{smallmatrix}0&0\\ 0&I_{d_{2}}\end{smallmatrix}\big{)} (see Proposition 6), , and so, if is small enough, and is invertible.
In that case, . This proves that the map
[TABLE]
is one-to-one. But the rank of does not depend on (Remark 8), and so it is always . So the previous map is bijective. We note the first component of its inverse. Note that we have . Then, if , we have
[TABLE]
To prove the converse, note that the inverse of is . ∎
Increasing if necessary, we may assume that for , is well-defined. Then, we define the (bounded) operator from to by
[TABLE]
Then, according to the definition of , we have the following corollary that allows us to compute the first components from the last .
Corollary 26**.**
For every (the space of functions with no components along frequencies less than ), we have the equivalence .
The Corollary 26 makes it easy to write an equation on the last components of the adjoint system (31) if the initial condition is in .
Proposition 27**.**
We define the operator by
[TABLE]
Let and . Then, for all , and satisfies the following equation
[TABLE]
Proof.
The function satisfies the system
[TABLE]
If we take the last components of this system, we get, in ,
[TABLE]
But for all , , so, according to Corollary 26, . Substituting this inside the equation (82) gives the stated equation (81). ∎
4.4.3 Smooth control of a finite number of parabolic vectorial components
For we introduce
[TABLE]
[TABLE]
and the projection defined by
[TABLE]
which is a bounded operator on (compostion of the bounded operator with an orthogonal projection). The goal of this section is to prove the following result.
Proposition 28**.**
There exists such that, for every and , there exists a linear map444The space means that the function is supported on where is a compact subset of , and all the derivatives vanish on at time and .
[TABLE]
such that, for every and
[TABLE]
[TABLE]
Proof.
Let . Throughout this proof, we will note the matrices defined by
[TABLE]
Step 1:
We prove that satisfies if and only if solves the following moments problem in
[TABLE]
and .
We first recall that, if is a projection operator on and , then
[TABLE]
because .
As a consequence, the relation is equivalent to
[TABLE]
where is the scalar product of and
[TABLE]
For , we denote by the solution of the adjoint system (31). Then, by 27, , where and
[TABLE]
where the first 2 scalar products are in and the last one is in . By Corollary 26, the assertion (85) is equivalent to
[TABLE]
where and . By considering with and , we obtain
[TABLE]
The previous property is equivalent to
[TABLE]
or, equivalently,
[TABLE]
which proves (84).
Step 2: Solving the moment problem.
We look for a solution of the moment problem (84) of the form
[TABLE]
where and is a scalar function with an appropriate support. More precisely, let
- •
be an open subset such that and such that on ,
- •
such that and
[TABLE]
For instance, we may consider such that for . Indeed, for every , the change of variable gives
[TABLE]
where . The function takes its minimal value at and thus, by Laplace’s method (see [23, Chapitre 9, Théorème VI.1]),
[TABLE]
which proves (87) for a large enough constant .
Then we choose . We also look for of the form
[TABLE]
The construction of will use the following algebraic result.
Lemma 29**.**
There exists such that, for every and the matrix in , defined by blocks by
[TABLE]
is invertible and
[TABLE]
where is the hermitian norm on .
Remark 30*.*
For instance, when , then is given by
[TABLE]
For with block decomposition
[TABLE]
where for every , we have
[TABLE]
Thus .
Proof of Lemma 29.
The proof relies on the following spectral inequality, due to Lebeau and Robbiano (see [19] and also [18, Thm. 5.4]):
[TABLE]
By summing the components, the same inequality holds when is a vector, , and denotes the hermitian norm on .
Let and written by blocks with . Then, by using the definition of , , the properties of and the above spectral inequality in vectorial form, we obtain
[TABLE]
There exists such that, for every , . Then,
[TABLE]
Then, by considering and Y=\exp\mathopen{}\mathclose{{}\left(-k^{2}(T-t)E_{2}(k)}\right)X_{k}, we obtain
[TABLE]
Therefore, by using the change of variable and (87), we get
[TABLE]
The above relation, valid for any proves that any eigenvalue of is positive, thus is invertible. Moreover, for any , the vector satisfies
[TABLE]
Thus
[TABLE]
This gives the conclusion with \mathcal{C}=\max\mathopen{}\mathclose{{}\left\{C_{1}C_{0};\ C_{1}+C_{0}\sqrt{2c}}\right\}. ∎
Now, let us come back to the proof of Proposition 28. For such a control of the form given by equations (86) and (88), the moment problem (84) writes
[TABLE]
or equivalently with the notations of Lemma 29. Thus, it is sufficient to take . By the definition of in (84), and Bessel-Parseval identity there exists independent of such that
[TABLE]
Thus, by Lemma 29
[TABLE]
Step 3: Estimates on .
Let . By (86) and the definition of , there exists such that
[TABLE]
For any such that we have,
[TABLE]
By Bessel-Parseval identity, we have
[TABLE]
By working as in the proof of 9, we obtain, for large enough, positive constants such that
[TABLE]
By (90),
[TABLE]
This provides a constant independant of such that
[TABLE]
and (91) gives the expected estimate on in . ∎
4.4.4 Lebeau-Robbiano’s method
The goal of this section is to prove 24. Let . We fix and . For , we set , where is such that . Let . We define
[TABLE]
where is the control operator introduced in 28. By construction and therefore, by 9
[TABLE]
By the semi-group property proved in 11, there exists positive constants and such that
[TABLE]
Then, according to the triangle inequality and Cauchy-Schwarz inequality,
[TABLE]
and by 28
[TABLE]
Thus
[TABLE]
By setting
[TABLE]
we get
[TABLE]
It is easy to see that there exists such that . Thus and more precisely there exists positive constants such that
[TABLE]
Moreover, from (92),
[TABLE]
We set , , …, . We have as . Then, for any , we define the control
[TABLE]
Then, because all its derivatives vanish at times . Thus .
By (93), thus is the limit, in , as , of the sequence . As a consequence, is the limit in of the sequence . Finally,
[TABLE]
By 28, for any ,
[TABLE]
This concludes the proof of 24. ∎
4.5 Control of the low frequencies
The goal of this subsection is to prove Theorem 2. Let where is defined in (3). Then, there exists such that (48) holds. Let and be as in Proposition 19.
Without loss of generality, we may assume that by the following procedure. Let be a complement of in . Then is a complement of in555If , we write it as , and in turn we decompose along the sum : . So . This proves that . Moreover, if , since , we have , which is . So the sum is direct. , and we extend to by setting for every .
Implicitly, is equipped with the topology of the -norm. The operator is defined in Definition 12.
We introduce the vector subspace of defined by
[TABLE]
Step 1: We prove that is a closed subspace of with finite codimension.
For , the function belongs to , thus
[TABLE]
is well defined in by 18. Then, is a compact operator on because it has finite rank. By the Fredholm alternative, is a closed subspace of and there exists a closed subspace of , with finite codimension in , such that is a bijection from to . Note that is also a closed subspace with finite codimension in .
For any , by using that and (94), we obtain
[TABLE]
thus
[TABLE]
This proves that contains , which is a closed subspace with finite codimension in . Therefore, there exists a finite dimensional subspace of such that . This gives the conclusion of Step 1.
Step 2: We prove that, up to a possibly smaller choice of , there exists such that for every .
When , by extending controls defined on by zero on , we see that . Thus, the map is decreasing and takes integer values. As a consequence the discontinuities on are isolated. If is not such a discontinuity point, then there exists such that for every . In case is such a discontinuity point, one may replace by a smaller value, still such that , for which this holds.
Step 3: We prove that \mathopen{}\mathclose{{}\left(\mathrm{e}^{-t{\mathcal{L}^{*}}}\mathcal{F}_{T}^{\perp}}\right)^{\perp}\subset\mathcal{F}_{T} for every .
Let and be such that for every . Then for every , i.e. . By Step 1, is a closed subspace of thus . Therefore . By definition of , this implies that . By Step 2, we get , which ends the proof of Step 3.
Step 4: We prove that is left invariant by , i.e. for every .
The subspace is closed in because it has finite dimension. Thus \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathrm{e}^{-t{\mathcal{L}^{*}}}\mathcal{F}_{T}^{\perp}}\right)^{\perp}}\right)^{\perp}=\mathrm{e}^{-t{\mathcal{L}^{*}}}\mathcal{F}_{T}^{\perp} and we deduce from Step 3 that, for every , . Taking into account that , we obtain for every . By the semi-group property, this equality holds for every .
Step 5: We prove the existence of such that any
can be written
[TABLE]
Let be the restriction of the semigroup to , i.e. . Then where is a matrix such that for every . But then . The Kernel decomposition theorem applied to , and the structure of the generalized eigenspaces of gives the conclusion of Step 4.
Step 6: We prove that any element of can be steered to in an arbitrary short time
, i.e. for every and , there exists such that . By the Hilbert Uniqueness Method, it is sufficient to prove an observability inequality for . By using the finite-dimensionality of , it is equivalent to prove that the following unique continuation property holds: if with in , then . By using the spectral inequality of Lebeau-Robbiano, i.e. (89) and (95), we readily get the result.
Step 7: Conclusion.
Step 5 implies the controllability of the system in any time . As is an arbitrary time such that , this concludes the null-controllability in any time . ∎
By a duality argument, we obtain the following result, that will be used in the next sections.
Corollary 31**.**
For every and , there exists such that, for every the solution of the adjoint system (31) satisfies
[TABLE]
where .
We will use the following standard lemma that gives a canonical isometry between and .
Lemma 32**.**
Let be an open subset of or a compact manifold (possibly with boundary). Let and be the inclusion map.666We recall that is the closure of for the -norm, and that is the dual of with respect to the pivot space . The map extends to a bijective isometry from to .
Proof.
The map is defined on by
[TABLE]
Thus, for evey ,
[TABLE]
where we used the definition of as the dual of with respect to the pivot space (see for instance [25, Sec. 2.9]). Since is dense in , this proves that extends by continuity to .
This extension is an isometry from onto its range. As such it is injective and its range is closed. To prove it is bijective, we check that its range is dense, i.e. that its orthogonal is zero.
If is orthogonal to , then, according to the definition of (Eq. (96)) is orthogonal in to . But is dense in , so . Thus . ∎
Proof of Corollary 31.
We apply the duality Lemma 14 with
[TABLE]
where is the solution to the system (Sys) with initial data and control , and
[TABLE]
where is the solution to the system (Sys) with initial data and control . The null-controllability result proved above is equivalent to the inclusion , thus to the existence of such that for every ,
[TABLE]
We compute the adjoint operators of and thanks to the duality relation between the solution of (Sys) and the solution of the adjoint system (31):
[TABLE]
First, we have . To compute , we introduce the input-output operator , where is the solution of (Sys) with initial condition and right-hand side . By (98), is the restriction of to . We have , where stands for the inclusion map . Thus, according to Lemma 32, the right-hand side of the inequality (97) is
[TABLE]
which gives the conclusion. ∎
5 Hyperbolic control: coupling of order zero
The goal of this section is to prove the following result on the system
[TABLE]
Theorem 33**.**
We assume (H.1)–(H.4), , , , and . Let be defined by (3). The following statements are equivalent.
- •
The system (100) is null controllable in any time .
- •
The couple of matrices satisfies the Kalman rank condition:
[TABLE]
The interest of this theorem is that its proof is essentially the same as the one of Theorems 3 and 4 (that will be done in the next sections) but it is less technical.
In Section 5.1, we prove that the Kalman condition (101) is necessary for the null controllability of System (100). In Section 5.2, we prove that the Kalman condition (101) is sufficient for the null controllability of System (100), first in the case (i.e. with one hyperbolic line in the system) where the cascade structure (or Brunovski form) is easy to handle, then in the general case which is more delicate to write.
5.1 The Kalman condition is necessary
If the null controllability property for (100) holds, then, by considering the Fourier components of the solution and the control, we obtain the null controllability, for every , of the system
[TABLE]
with state and control . This requires the null controllability of the control system
[TABLE]
with state and control , i.e. the Kalman rank condition (see for instance [10, Thm. 1.16])
[TABLE]
that can equivalently be written in the form (101).
5.2 The Kalman condition is sufficient
In this section, we explain how to complete the proof of Theorem 2 to prove that the Kalman rank condition (101) implies the null controllability of (100) in time , in Theorem 33.
First, we treat the case then we generalize to the case .
From now and until end of this subsection, will denote positive constants which will vary from line to line. For and , we denote by the -th component of a vector .
5.2.1 The case of one hyperbolic component:
By using the Hamilton-Cayley’s theorem, we know that there exist such that
[TABLE]
By using the Kalman condition (101), the matrix defined as follows
[TABLE]
is invertible. We set
[TABLE]
From (102), (103), (104), we check that we have the following relations
[TABLE]
The function solves
[TABLE]
The system (105) is a “cascade system”. Indeed, roughly speaking the control directly controls the component , the component indirectly controls the component in the second equation through the coupling term , the component indirectly controls the component in the third equation through the coupling term , … the component indirectly controls the component in the last equation through the coupling term .
The adjoint system of (105) is
[TABLE]
From Corollary 31, we know that for every , the solution of (106) satisfies
[TABLE]
By using the fact that is a companion matrix, see (104), we have that for every , the -th equation of (106) is
[TABLE]
Then we deduce
[TABLE]
Here, we have used in particular that
[TABLE]
and
[TABLE]
Then, we deduce from (107) and (108) that
[TABLE]
By using the fact that is the first vector of the canonical basis of , see (104), the first equation of (106) is
[TABLE]
Then, we obtain that
[TABLE]
So, we deduce from (109) and (110) the observability inequality
[TABLE]
in the case . This concludes the proof of Theorem 33 in the case by duality.
5.2.2 The case of several hyperbolic components:
In this section, we deal with the general problem of null-controllability of (100). To this aim, we introduce the -th column of the matrix (), i.e.
[TABLE]
From the Kalman rank condition (101), we construct an adapted basis of .
Lemma 34**.**
There exist and sequences and with , such that
[TABLE]
is a basis of . Moreover, for every , with , there exist (, ) such that
[TABLE]
For a proof of this lemma, see [3, Lemma 3.1].
Let the basis of provided by Lemma 34 and be the matrix whose columns are the elements of , i.e.
[TABLE]
Let us observe that the basis has been constructed in such a way that (111) is satisfied.
Let the matrices and , , be defined by
[TABLE]
We set
[TABLE]
From (111), (113) and (112), by denoting P_{i}\coloneqq\mathopen{}\mathclose{{}\left(K_{21}^{l_{i}}|K_{22}K_{21}^{l_{i}}|\dots|K_{22}^{s_{i}-1}K_{21}^{l_{i}}}\right), we have
[TABLE]
Then, we obtain
[TABLE]
where is the -element of the canonical basis of with . In the following, we will also use the notation .
We argue as in the previous subsection. We perform the same change of variable , we consider the solution of the adjoint system
[TABLE]
From Corollary 31, we recall that the solution of (115) satisfies
[TABLE]
Then, we use the coupling terms in the system (115) in order to get rid of the term in the right hand side of the inequality (116).
From the cascade form of the matrix , see (113), more precisely from the cascade form of the block matrix and the form of the matrices , see (112), the equations of the adjoint system (115) are
[TABLE]
To simplify, we will denote by , , the space .
We deduce successively from (5.2.2) with , the following estimates
[TABLE]
So, we have for every and ,
[TABLE]
Then, by using (113) and (114), we have . Consequently, the -th equation of the adjoint system (115) is
[TABLE]
Then, we obtain
[TABLE]
By gathering (118) and (119), we obtain
[TABLE]
By using that , we finally deduce from (120) and (116) the observability inequality
[TABLE]
This concludes the proof of Theorem 33 in the case by duality.
6 Hyperbolic control: coupling of order one
The goal of this section is to prove Theorem 3. The requirement of the Kalman rank condition (6) for null-controllability is an adaptation of the proof given in Section 5.1. Now, we explain how to complete the proof of Theorem 2 to prove that the Kalman condition is sufficient for null controllability. We set
[TABLE]
We only give the proof in the case . The case is an easy adaptation of the case and the arguments already presented for coupling terms of order zero in Section 5.2.2.
6.1 A special observability inequality
The goal of this section is to prove the following observability inequality.
Proposition 35**.**
There exists such that for every , the solution of the adjoint system (31) satisfies
[TABLE]
In order to prove 35, by a duality argument, it is sufficient to establish the following null-controllability result.
Proposition 36**.**
For every , there exists such that .
Proof of the equivalence between 35 and 36.
We apply Lemma 14 with
[TABLE]
where is the solution to the system (Sys) with initial data and control , and
[TABLE]
where is the solution to the system (Sys) with initial data and control . Note that by integrating the second equation of the system (1), we see that a control of the form cannot change the mean of the parabolic component. This justifies that and do indeed take values in .
The null-controllability result of 36 is equivalent to the existence of such that for every ,
[TABLE]
We have . We claim that the right-hand side of the inequality (123) satisfies
[TABLE]
where . This will prove that the inequality (123) is exactly the observability inequality (122).
We write as
[TABLE]
where is the input-output operator introduced in the proof of Corollary 31, is seen as an unbounded operator on with domain , and is the inclusion map (see Lemma 32). Note that while written this way looks like an unbounded operator (because is), we have , so that the composition of operators above is indeed a continuous operator. So, we have
[TABLE]
Since is an isometry between and (see Lemma 32), this proves the relation (124). ∎
First, we show that the null-controllability result of 36 is true at the high-frequency level, i.e. we prove the following adaptation of 19.
Proposition 37**.**
There exists a closed subspace of with finite codimension and a continuous operator
[TABLE]
that associates with each a pair of controls such that
[TABLE]
In order to prove 37, it is enough to prove 21 with parabolic control of the form . Thus, by using Section 4.4.1, it is sufficient to show the following adaptation of 24.
Proposition 38**.**
If is large enough, then for every , there exists a continuous operator
[TABLE]
that associates with each a control such that
[TABLE]
Proof.
Let and be such that . Note that is well-defined because . We know from 24 that there exists such that the solution of (Sys) with initial data and control satisfies
[TABLE]
Then, by setting and by applying to the system (Sys) satisfied by , we deduce that is the solution of (Sys) with initial data and control and satisfies
[TABLE]
because and commute.
We get the conclusion of the proof of 38 with the continuous operator \underline{\mathcal{U}}_{T}^{{\mathrm{p}},\blacklozenge}(f_{0})=\underline{\mathcal{U}}_{T}^{\mathrm{p}}\mathopen{}\mathclose{{}\left(f_{0}^{\blacklozenge}}\right) where is the operator defined in 24. ∎
Secondly, we have to show that the null-controllability result of 36 is true at the low frequency-level, as we have already shown for Theorem 2 in Section 4.5. All the steps of Section 4.5 remain unchanged except the Step . Indeed, the unique continuation argument transforms into: if with in then thanks to the spectral inequality of Lebeau-Robbiano (89), that is to say, because .
This concludes the proof of 36 thus the proof of 35. ∎
6.2 The case of one hyperbolic component:
By the Hamilton-Cayley’s theorem, there exist such that
[TABLE]
By using the Kalman condition (6), the matrix defined as follows
[TABLE]
is invertible. By setting
[TABLE]
we check that we have the following relations
[TABLE]
Then, by setting , we have
[TABLE]
The system (127) is a “cascade system” with coupling terms of order one in the spatial variable.
The adjoint system of (127) is
[TABLE]
We know from 35 that the solution of (128) satisfies
[TABLE]
By using the fact that is a companion matrix, see (104), for every , the -th equation of (128) is
[TABLE]
Then, by applying to (130) with , we get that there exists such that
[TABLE]
therefore we have
[TABLE]
Then, we deduce from (129) and (131) that
[TABLE]
By using the fact that is the first vector of the canonical basis of , see (126), the first equation of (128) is
[TABLE]
Then, we obtain that
[TABLE]
So, we deduce from (132) and (133) the observability inequality
[TABLE]
which concludes the proof of Theorem 3 in the case . ∎
7 Parabolic control
The goal of this section is to prove Theorem 4 and to illustrate the necessity of a regularity assumption on the initial condition.
7.1 A regularity assumption is necessary
We consider for the system
[TABLE]
i.e. , , ,
[TABLE]
that satisfies (H.3),(H.4) and the Kalman condition (12) because . By Theorem 4, any initial condition is null controllable. The following statement illustrates that
- •
a regularity assumption on is necessary for the null controllability
- •
the one given by Theorem 4 is sufficient but may not be necessary.
Proposition 39**.**
An initial condition is null controllable with if and only if .
Remark 40*.*
Similar problems of regularity between initial data and control have already been noticed in the context of transport systems, see [1, Remark 5].
Proof.
In the proof, we use the notation .
The change of variable
[TABLE]
leads to
[TABLE]
The null controllability of with control is equivalent to the null controllability of with control . On Fourier components, equation (135) gives the ordinary differential equations
[TABLE]
Let . The solution writes
[TABLE]
[TABLE]
thus the relation is equivalent to the moment problem
[TABLE]
Note that the assumption implies for every thus the null controllability of this component does not require any condition on the control .
Necessary condition:
We assume null controllable with a control and we prove that . By the Bessel-Parseval equality and Cauchy-Schwarz inequality,
[TABLE]
thus .
Sufficient condition:
We assume and we construct a control that steers this initial condition to [math].
Let be the Grammian matrix, in , of the family where and , i.e. for every . Then is invertible for every (because it is the Grammian matrix of a linearly independent family) and, when ,
[TABLE]
thus there exists such that, for every , . We take
[TABLE]
where
[TABLE]
Then, by Bessel-Parseval equality, we have for various positive constants depending on ,
[TABLE]
Thus . Note that the moment problem (137) can equivalently be written
[TABLE]
Thus, by (138), solves (137). ∎
7.2 Proof of Theorem 4
The Kalman rank condition (12) is a necessary condition for null-controllability of (10) by the same arguments as in Section 5.1. Thus we only explain how to complete the proof of Theorem 2 to prove that it is a sufficient condition for null-controllability of (10). We introduce the space
[TABLE]
equipped with the scalar product of and the space
[TABLE]
equipped with the scalar product of .
The null-controllability of the system (10) in with control of the form is equivalent to the following observability inequality: for every , there exists such that, for every , the solution of the adjoint system (31) satisfies
[TABLE]
where is made of the last components of .
Proof of the equivalence between the null-controllability in and the observability inequality (142).
We apply the duality Lemma 14 with
[TABLE]
[TABLE]
Note that the mean value of the first components is indeed zero. The null-controllability result in is equivalent to the inclusion , thus to the existence of a constant such that for every
[TABLE]
We compute the adjoint operators of and thanks to the duality relation between the solution of (Sys) and the solution of the adjoint system (31):
[TABLE]
First, is the restriction of the -last components of to . Then, by (144) and Lemma 32 (working as in the proof of Corollary 31), the left-hand side of (143) is
[TABLE]
Thus the inequality (143) is indeed the observability inequality (142). ∎
By using the strategy developed in Section 6, we claim that, in the case , it is sufficient to prove the following result in order to prove the observability inequality (142).
Proposition 41**.**
For every , there exists such that for every , the solution of the adjoint system (31) satisfies
[TABLE]
The observability inequality (145) has to be compared to the observability inequality (122) in Section 6. Roughly speaking, the term comes from the fact that we will perform steps of elimination, each of them “costs” one derivative (instead of two in Section 6.2) because we will use transport equations which are of order one in time and space (instead of parabolic equations which are of order two in space variable). The last step of elimination “costs” two derivatives because we will use a heat equation which is of order one in time and two in space. This explains the number derivatives. By adapting the arguments of Section 5.2.2, we can also treat the case .
In order to prove 41, by duality (a simple adaptation of the proof that 35 and 36 are equivalent), it is sufficient to establish the following null-controllability result.
Proposition 42**.**
For every , there exists such that .
The proof of this result is an adaptation of the proof of Theorem 2:
- •
we prove that parabolic high frequencies are null-controllable,
- •
we prove that hyperbolic high frequencies are null-controllable,
- •
we combine these two propositions to prove that high frequencies are null-controllable,
- •
we finally deal with low frequencies.
For the first point, we just need a special case of the corresponding result that was used in the proof of Theorem 2, i.e. 21.
Proposition 43**.**
If is large enough, there exists a continuous operator
[TABLE]
that associates with any a control such that
[TABLE]
Proof.
Proposition 43 is a consequence of Proposition 21 because is continuously embedded in and for every . ∎
For the second point, we will prove the following adaptation of 20.
Proposition 44**.**
If is large enough, there exists a continuous operator
[TABLE]
that associates with any a control such that
[TABLE]
While the ideas of the proof are the same as for 20, the proof of this Proposition is technically more delicate, as we have to build regular controls, and, on the observability side, deal with the (slightly impractical) and norms. We postpone the proof to the next subsection. For now, let us assume 44 holds true, and finish the proof of Theorem 4.
We now combine Propositions 43 and 44 with the Fredholm alternative, as in the proof of 19, to prove that high frequencies are null-controllable. That is to say, we get the following adaptation of 19.
Proposition 45**.**
There exists a closed subspace of with finite codimension and a continuous operator
[TABLE]
that associates with each a pair of controls such that
[TABLE]
The last step consists in showing that the null-controllability result of 42 is true at the low frequency-level, as we have already shown for Theorem 2 in Section 4.5. All the steps of Section 4.5 remain unchanged except the Step 6. Indeed, the unique continuation argument transforms into: if with in then thanks to the spectral inequality of Lebeau-Robbiano (89), that is to say, because .
This concludes the proof of 42 thus the proof of 41. ∎
7.3 Proof of 44
The proof of 44 is an adaptation of the one of 20, with the following changes:
- •
we deal with the fact that we want a control of the form ,
- •
we adapt the duality argument to take into account the regularity of the controls that we want (it involves some norms),
- •
we adapt all the inequalities to replace the relevant norms by norms,
- •
to build regular controls of the simple transport equation , we use [1].
Step 1: reduction to an exact controllability problem.
We claim that in order to prove 44, we only have to prove the following exact controllability result.
Proposition 46**.**
If is large enough, then for every , there exists a continuous operator
[TABLE]
that associates with any , a control such that
[TABLE]
Indeed, by the choice of support in time of the controls, and by the reversibility of (see Section 4.3.1 for the details), the relation (146) is equivalent to
[TABLE]
Note that functions in have zero mean (see the definition of Eq. (42)). Thus, is invertible on , and its inverse is, on the Fourier side, the multiplication by . Moreover, the operator commute with and the semi-group . So the relation (146) is equivalent to
[TABLE]
So, if 46 holds, we may choose (assuming it makes sense)
[TABLE]
Thus, to end this first step, we just have to check that the right-hand side of (148) is indeed in .
The projection has range , and sends to itself, as do . So belongs to .
The group sends every into itself (see Remark 13). Since is just the multiplication on the Fourier side by , the operator also sends every into itself. Thus, we just have to check that because sends into .
- •
The function belongs to by hypothesis, so also belongs to (see Remark 13).
- •
The parabolic control belongs to by hypothesis, thus for almost every , belongs to and thus
[TABLE]
This concludes this first step.
Step 2: Observability inequality associated to the controllability problem of 46.
Let
[TABLE]
be the restriction of to and
[TABLE]
where stands for the map . Note that and are continuous.
The controllability problem of 46 is equivalent to the inclusion . Therefore, according to the duality Lemma 14, it is equivalent to the following inequality: there exists such that for every , . Since is a projection on , since is the restriction of the first components of to , and since is an isometry between and ,777See Lemma 32, and also recall that because has no boundary . this inequality reads: there exists such that for every , the solution of the adjoint system (31) satisfies
[TABLE]
where are the first components of .
Let . For the remaining of this proof, we use the notations of Section 4.3.2, and in particular we introduce the decompositions (61) and (62). In the following arguments, the constants do not depend on .
Step 3: We prove the observability inequality (149) assuming that, for every , there exists such that the solution of (70) satisfies
[TABLE]
We will prove (150) in Step 3.
We proceed as in the proof given in Section 4.3.2. By the explicit expression (66) of and Bessel-Parseval identity, there exists independent of such that
[TABLE]
Using the Duhamel formula, we obtain that the function defined by (68) satisfies
[TABLE]
By (150), the triangular inequality, (68) and (152), we deduce that
[TABLE]
Using Bessel-Parseval identity and the decomposition (73), we obtain
[TABLE]
We deduce from (153), the triangular inequality and (154) that
[TABLE]
Taking into account that , we get888Remark that if is a matrix and , then . Indeed, noting the duality between and , we have for every , , and taking the supremum over , we do have .
[TABLE]
Using (62), the triangular inequality and the previous two estimates, we obtain
[TABLE]
Proceeding as in the end of the proof given in Section 4.3.2, the inequality (155), together with a compactness-uniqueness argument, end Step 2.
Step 4: We prove that the solution of (70) satisfies (150)
. By duality, it is actually enough to prove the following exact-controllability result.
Proposition 47**.**
Let and . For every , there exists such that the solution of
[TABLE]
satisfies .
To prove 47, we will use the following lemma, which is an easy adaptation of [1, Lemma 2.6].
Lemma 48**.**
Let and . Then, there exists small enough and a cut-off function with
[TABLE]
such that, for every ,
[TABLE]
Remark 49*.*
We assumed that the function is extended by -periodicity in the spatial variable.
Now, we give the proof of 47 thanks to Lemma 48.
Proof of 47.
We take the control
[TABLE]
We easily check that the control belongs to by using the support of (157), and the regularity of the three functions , and . Let be the solution of (156) with initial data and control defined in (159). We just have to check that satisfies . We write the solution along the characteristic, that is to say
[TABLE]
By integrating in space between [math] and and by using the defintion of (158), we obtain
[TABLE]
then which concludes the proof of 47. ∎
This ends the proof of 46.∎
Appendix A Pure transport solutions are not enough to disprove the observability inequality
Proposition 50**.**
Let us assume that the matrix
[TABLE]
has rank , or, equivalently, that there is no eigenvector of in the kernel of (see for instance [5, Lemma 1]). Let and . There exists 999With the help of 7, we could even prove that can be chosen indepentantly of . such that every solution of the adjoint system (31) of the form satisfies .
This statement shows that, for a dense set of matrices pure transport solutions of the adjoint system (31) cannot be used to disprove the observability inequality (32), and thus the null controllability of (Sys).
Proof.
Let us note the set of solutions of the adjoint system (31) of the form . Remark that according to the expression (33) of the solutions of the adjoint system, the relation is equivalent to
[TABLE]
We claim that is finite dimensional. Indeed, if it is infinite dimensional, then, according to the relation (160), there is infinitely many such that is an eigenvalue of . Let be an associated sequence of eigenvectors, chosen such that . Since the unit sphere of is compact, we may assume that converges to some with . Then we have
[TABLE]
And since , we must have and . But this is in contradiction with the hypothesis of the Proposition. Therefore is finite dimensional.
So, according to the description (160) of , there exists such that every solution of the adjoint system (31) of the form has no frequencies higher than : . But finite linear combination of exponentials have the unique continuation property.101010For instance because they are entire functions, and entire functions have the unique continuation property. So the expressions and both define a norm on . Since is finite dimensional, these two norms are equivalent. This proves the claimed inequality. ∎
Acknowledgments:
The second author would like to thank his Ph. D. advisor, Gilles Lebeau, for numerous discussions and in particular for pointing us [20]. He also thanks Émmanuel Trélat for helping us with some duality arguments.
The second author was partially supported by the ERC advanced grant SCAPDE, seventh framework program, agreement no. 320845.
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