Positive and negative results on the internal controllability of parabolic equations coupled by zero and first order terms
This paper is devoted to studying the null and approximate controllability of two linear coupled parabolic equations posed on a smooth domain of () with coupling terms of zero and first orders and one control localized in some arbitrary nonempty open subset of the domain . We prove the null controllability under a new sufficient condition and we also provide the first example of a not approximately controllable system in the case where the support of one of the nontrivial first order coupling terms intersects the control domain .
Keywords: Controllability; Parabolic systems; Fictitious control method; Algebraic solvability.
MSC Classification: 93B05; 93B07; 35K40.
1.1 Presentation of the problem and main results
Let , let be a bounded domain of () of class and let be an arbitrary nonempty open subset of . Let , and . We consider the following system of two parabolic linear equations with variable coefficients and coupling terms of order zero and one
where is the initial condition and is the control.
The zero and first order coupling terms and are assumed (for the moment) to be in and in , respectively. For , the second order elliptic self-adjoint operator is given by
where the coefficients satisfy the uniform ellipticity condition
for a constant .
approximately controllable on the time interval if for every initial condition , every target and every , there exists a control such that the corresponding solution to System (1.1) satisfies
null controllable on the time interval if for every initial condition , there exists a control such that the corresponding solution to System (1.1) satisfies
It is well-known that if a parabolic system like (1.1) is null controllable on the time interval , then it is also approximately controllable on the time interval (this is an easy consequence of usual results of backward uniqueness for parabolic equations as given for example in ).
Since the case and in has already been studied in , we will always work under the following assumption.
There exists and a nonempty open subset of such that
Moreover, as we will see in Section 2, it is possible, with the help of appropriate changes of variables and unknowns (we lose a little bit of regularity on the coefficients though, see Section 2), to replace the coupling operator by the simpler coupling operator (where is the first direction in space), at least locally on some subset of .
Hence, without loss of generality, we can also work under the following assumption.
There exists a nonempty open subset of such that
For a nonempty set , let us denote by the subset of composed by the functions depending only on the variables . Let us now introduce the following condition, which will be crucial in our following results, and which is closely related to the particular form for the coupling term given in Assumption 1.2 (removing this assumption would make Condition 1.1 impossible to write down explicitly).
There exists a nonempty open set such that
Our first main result is the following:
Theorem 1 is stated and will be proved in the case of two coupled parabolic equations and one control. However, as in , it is possible to extend Theorem 1 to systems of parabolic equations controlled by controls for arbitrary . More precisely, consider the system
where is the initial data and is the control. Let us suppose that there exists , and a nonempty open subset of such that on . As explained in Section 2, we can suppose that the operator is equal to in . Assume that there exists an open set such that
Condition 1.1 is clearly technical since it does not even cover the case of constant coefficients proved in , the general case given in  (under some assumption on the control domain) or the one-dimensional result given in . However, Theorem 3 implies that one cannot expect the null controllability to be true in general without extra assumptions on the coefficients. We do not know what would be a reasonable necessary and sufficient condition on the coupling terms for the null controllability of System (1.1).
The second main result of the present paper is the following surprising result.
Let us assume that . Let be a nonempty regular open set satisfying . and consider a function satisfying
Then there exists such that the system
is not approximately controllable (hence not null controllable) on the time interval .
In other words, Theorem 2 tells us that for every control set strongly included in , there exists a potential for which approximate controllability of (2) does not hold, in any space dimension. We may improve a bit this result on the one-dimensional case, where we are able to obtain the following result, which expresses that for some well-constructed potential , that there exists one control domain on which System (3) is not approximately controllable (hence not null controllable) and another control domain on which System (3) is null controllable (hence approximatively controllable), highlighting the surprising fact that some geometrical conditions on the control domain has to be imposed in order to obtain a controllability result.
Consider the following system
There exists a coefficient such that:
Let us mention that Theorems 2 and 3 are the first negative result for the controllability of System (1.1) when the support of the first order coupling term intersects the control domain in the case of distributed controls. The authors want to highlight that the coupling operator is constant in the whole domain and nevertheless the system can be controllable or not following the localisation of the control domain, which is an unexpected phenomenon.
1.2 State of the art
Many models of interest involve (linear or non-linear) coupled equations of parabolic systems, notably in fluid mechanics, medicine, chemistry, ecology, geology, etc., and this explains why during the past years, the study of the controllability properties of linear or nonlinear parabolic systems has been an increasing subject of interest (see for example the survey ). The main issue is what is called the indirect controllability, that is to say one wants to control many equations with less controls than equations, by acting indirectly on the equations where no control term appears thanks to the coupling terms appearing in the system. This notion is fondamental for real-life applications, since in some complex systems only some quantities can be effectively controlled. Here, we will concentrate on the previous results concerning the null or approximate controllability of linear parabolic systems with distributed controls, but there are also many other results concerning boundary controls or other classes of systems like hyperbolic systems.
First of all, in the case of zero order coupling terms, the case of constant coefficients is now completely treated and we refer to  and  for parabolic systems having constant coupling coefficients (with diffusion coefficients that may depend on the space variable though) and for some results in the case of time-dependent coefficients. In the case of zero and one order coupling terms and constant coefficients, a necessary and sufficient condition in the case of equations and controls for constant coefficients is provided in  by the authors.
The case of space-varying coefficients remains still widely open despite many new partial results these last years. In the case where the support of the coupling terms intersects the control domain, a general result is proved in  for parabolic systems in cascade form with one control force (and possibly one order coupling terms). We also mention , where a result of null controllability is proved in the case of a system of two equations with one control force, with an application to the controllability of a nonlinear system of transport-diffusion equations. In the situation where the coupling regions do not intersect the control domain, the situation is still not very well-understood and we have partial results, in general under technical and geometrical restrictions, notably on the control domain (see for example , ,  and ). Let us mention that in this case, there might appear a minimal time for the null controllability of System (1.1) (see ), which is a very surprising phenomenon for parabolic equations, because of the infinite speed of propagation of the information.
Concerning the case of first order coupling terms, we mention  which gives some controllability results when the coefficient is equal to zero on the control domain. Let us also mention the recent work , which concerns the small systems in small dimension, that is to say and systems. The authors of  suppose that the control domain contains a part of the boundary . Recently, in , the first author studied a particular cascade system with space dependent coefficients and in dimension one thanks to the moment method, and obtained necessary and sufficient conditions on the coupling terms of order and for the null controllability. To conclude, let us also mention another result given in  by the authors, which provides a sufficient condition for null controllability in dimension one for space and time-varying coefficients under some technical conditions on the coefficients, which turns out to be exactly equivalent to Condition 1.1 under Assumption 1.2 (but with more regularity than in Assumption 1.1). Hence, Theorem 1 can be seen as a generalization in the multi-dimensional case of the one-dimensional result given in . For a more detailed state of the art concerning this problem, we refer to .
Hence, the present paper improves the previous results in the following sense:
In the previous results, it was surprising to have some very different sufficient conditions for the null controllability of System (1.1) in the case of first order coupling terms, for example on one hand constant coupling coefficients and on the other hand a region of control which intersects the boundary of the domain. Through the example of a not approximately controllable system given in Theorem 2 and 3, we can now better understand why such different conditions appeared since the expected general condition for the null controllability of System (1.1) with space and time-varying coefficients (i.e. it is sufficient that the control and coupling region intersect) may be false in general if .
2 Simplification of the coupling term
In this section, we will prove that it is possible to replace locally the coupling operator by , where is the first direction in space. This kind of simplification has already been used in [12, Lemma 2.6] for example, and we refer to this article for a more detailed proof (see also ). Let us remark that the regularities stated in Lemma 2.1 are higher than the one stated in Theorem 1 due to technical reasons appearing in the proofs of Lemmas 2.1 and 2.2.
Let for every and . Suppose that Assumption 1.1 is verified. Then, there exist a nonempty open subset of , a positive real number and a -diffeomorphism from to an open set that keeps invariant and such that if we call and , then there exist a matrix , a vector and coefficients such that locally on one has
Proof of Lemma 2.1
Let us consider some open hyper-surface of class included in on which , where is the normalized outward normal on (this can always be done since on and is at least continuous), small enough such that it can be parametrized by a local diffeomorphism
where is a nonempty open set. We call . Let us consider some extension of (that exists thanks to the regularity of and ) that we denote by . Using the Cauchy-Lipschitz Theorem, we infer that for every , there exists a unique global solution to the Cauchy Problem
Since is continuous and on , we deduce that there exists some such that for every and every . We define
Then, by the inverse mapping theorem, is a -diffeomorphism from to with . Let us call and , then it is clear that
and hence we obtain (2.1) and the regularities wished for the new coefficients by writing down the equation verified by .
Let us now perform a second useful reduction.
There exists an open subset of and a function such that for some constant and if
then there exists some coefficients and such that locally on one has
Proof of Lemma 2.2
Let us consider some such that for some constant , and consider the change of unknowns
Using equation (2.1), we infer that verifies
where and . Hence, if we choose satisfying and in , which is always possible, then and verify (2.2) and we have and .
3 Proof of Theorem 1
3.1 Strategy : Fictitious control method
The fictitious control method has already been used for instance in , , ,  and . Roughly, the method is the following: we first control the equations with two controls (one on each equation) and we try to eliminate the control on the last equation thanks to algebraic manipulations locally on the control domain. For more details, see for example [19, Section 1.3]. Let us be more precise and decompose the problem into three different steps:
Analytic Problem: Null controllability by two forces
Find a solution in an appropriate space to the control problem by two controls
where the controls and are regular enough and with a support strongly included in (remind that was introduced in Condition 1.1). Solving Problem (3.1) is easier than solving the null controllability on the time interval of System (1.1), because we control System (3.1) with one control on each equation. The important point is that the control has to be regular enough, so that it can be differentiated a certain amount of times with respect to the space and/or time variables (see the next section about the algebraic resolution).
Let . Suppose that and for every and . Then there exists two constants and such that for every initial condition one can find a control verifying moreover for which the solution to System (3.1) is equal to zero at time and the following estimate holds:
The controllability of parabolic systems with regular controls is nowadays well-known. For a proof of Proposition 3.1, one can adapt the strategy developed in [13, 14, 15, 25] where the authors prove the controllability of parabolic systems with controls thanks to the fictitious control method and the local regularity of parabolic equations. For more details, we refer to [20, Chap. I, Sec. 2.4]. It is also possible to use the Carleman estimates (see for instance  and [19, Section 2.3]), however this will impose the coefficients of System (3.1) to be regular in the whole space (and would require higher regularity on ).
Algebraic Problem: Null controllability by one force
For given with strictly included in , find , in an appropriate space, satisfying the following control problem:
with strictly included in , which impose the initial and final data and the boundary conditions. We recall that is equal to in . We will solve this problem using the notion of algebraic resolvability of differential systems, which is based on ideas coming from [26, Section 2.3.8] and was already used in some different contexts in , ,  or . The idea is to write System (3.1) as an underdetermined system in the variables and and to see as a source term. More precisely, we remark that System (3.1) can be rewritten as
The goal in Section 3.2 will be then to find a partial differential operator satisfying
Thus to solve control problem (3.1), it suffices to take
3.2 Algebraic solvability of the linear control problem
The goal of this section is to solve algebraic problem (3.2). We will use the following lemma:
Let be a nonempty open subset of () and let . Consider two differential operators and defined for every by
where, for , . If for every where