The Reflection Principle in the Control Problem of the Heat Equation

We consider the control problem for the generalized heat equation for a Schrödinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts and the corresponding control cost does not exceed the one on the whole domain. As an application, we obtain null-controllability results for the heat equation on half-spaces, orthants, and sectors of angle π/2n. As a byproduct, we also obtain explicit control cost bounds for the heat equation on certain triangles and corresponding prisms in terms of geometric parameters of the control set.


Introduction and Main Result
Let be open, , and let be measurable with a symmetric matrix for almost every . Suppose, in addition, that there exist 1 2 0 such that   1  2  2  2 a.e. (1.1) and let be real-valued. We denote by and the Dirichlet and Neumann realizations of the differential expression Michela Egidi michela.egidi@ruhr-uni-bochum.de Albrecht Seelmann albrecht.seelmann@math.tu-dortmund.de as self-adjoint lower semibounded operators on 2 defined via their quadratic forms with form domain 1 0 and 1 , respectively. For details of this construction, we refer the reader to the discussion in Section 4 below.
Let 0, be a measurable subset, and . We consider the heat-like system is called control cost.
For bounded domains , there is already a rich literature on null-controllability results for system (1.2) with being the Laplace or a Schrödinger operator with an open or measurable control set , see, e.g., [2,9,15,19,27]. On the other hand, null-controllability of these systems on unbounded domains has been an issue of growing interest only recently, see for example [7,9,19] and the references therein. As far as bounds on the control cost are concerned, one classically asks for the type of time dependency, which has been largely explored in [5,10,11,16,18,19,21,27], see also the references therein. However, the dependency of the control cost on geometric parameters of and has only recently been addressed in [6,7,9,19], see also [17] for previous results. Especially the work [19] focused on these dependencies, which have been exploited in several asymptotic regimes there in order to discuss homogenization.
The aim of the present paper is to prove null-controllability results for the above situation on some new unbounded domains, such as sectors of certain angles, half-spaces, and orthants, with bounds on the associated control cost that are fully or partially consistent with the known ones on cubes and the full space. As a byproduct, we also obtain explicit control cost bounds in terms of only geometric parameters of the control set for the corresponding system on some bounded domains with flat boundary that were not accessible before. We believe that the lack of the control cost's dependency on the domain may be an effect of the flatness of its boundary. However, it is not our current scope to explore this matter in depth, but merely to add examples to support this conjecture. All these results are discussed in detail in Section 2 below as applications of a reflection principle which we now describe.
The key idea in our considerations is to relate system (1.2) for certain domains to a corresponding symmetrized system on a larger domain, where null-controllability results are available. In order to make this precise, let be the reflection with respect to the first coordinate, that is, 1 1 2 . We suppose that is contained in the half-space 0 1 such that 0 1 for some open set with 0 1 and which is symmetric with respect to the reflection , that is, . In particular, we have , cf. Figure 1. Observe that by construction is for almost every a symmetric matrix satisfying (1.1) with the same constants 1 2 .
Let the self-adjoint operators on 2 associated with the differential expression be defined analogously to above. Set , and consider the corresponding symmetrized system , and control set . The notions of nullcontrollability and control cost for this system carry over verbatim.
The main result of the present paper now reads as follows.
The restriction to the reflection symmetry with respect to the hyperplane 0 1 in Theorem 1.1 is not essential. Indeed, by rotating the whole system, we can deal with reflection symmetries with respect to any hyperplane in . This way, the above theorem allows us to infer null-controllability on the respective parts of a domain with a reflection symmetry if the system on the whole domain is null-controllable. Theorem 1.1 is, in fact, an instance of a much more general result, which is formulated in Theorem 3.1 below. The proof of this abstract result heavily relies on an extension relation, which in the situation of Theorem 1.1 takes the form (1.6) where 2 2 corresponds to the extension of functions by symmetric or antisymmetric reflection, respectively. Since here is a multiple of an isometry, this essentially means that the subspace Ran reduces the operator , cf. Remark 3.5 below. With this in mind, the extension relation (1.6) transfers to operators related to and by functional calculus, such as fractional Laplacians, yielding an analogue of Theorem 1.1 also for such operators; cf. Lemma 3.3 and Remark 4.2 (a) below. More generally, an extension relation of the form (1.6) can be formulated with respect to every reducing subspace of , opening the way to null-controllability of subsystems corresponding to reducing subspaces, even in a general abstract framework, see Remark 3.5 below. In this regard, our abstract result provides a flexible and convenient way to derive new results from existing ones independently of how the existing results have originally been proved.
The rest of the paper is organized as follows. In Section 2, we discuss the nullcontrollability results and control cost bounds announced above for the case where is the identity matrix as consequences of Theorem 1.1. Here, we will make explicit use of recent results from [9,19] for cubes and the full space. The assumption on the essential boundedness of the potential is tailored towards these applications but it is not essential for the general argument. In Section 3, we prove the abstract result discussed above, which is the core of the proof of Theorem 1.1 and yields applications in broader settings, see, e.g., Remarks 2.11, 4.2, and 4.3 below. Section 4 deals with the proof of Theorem 1.1 based on the abstract result by establishing the extension relation (1.6). Finally, Appendix A provides an integration by parts formula used in Section 4.

Addendum
After completion of the present work, the current authors obtained a family of so-called spectral inequalities in the recent article [8], which together with [19,Theorem 2.8] can be used to reproduce the results from Section 2.1.

Applications
We here discuss the applications of Theorem 1.1 mentioned in the previous section for the particular case where each is the identity matrix, that is, . These applications draw upon null-controllability results from the works [9,19] on cubes and the full space. As considered there, we also take control sets of the form , where is some measurable set with certain geometric properties, namely a thick set or an equidistributed set. We treat the two types of sets in the subsections below separately.
The general strategy for our applications is to build from the given set a symmetric set with respect to the reflection such that . After having verified corresponding geometric properties of , this new set yields a suitable control set for the symmetrized system on , where a null-controllability result is available. Applying Theorem 1.1 then gives the desired null-controllability result for the original system on . In summary, our main task here is to establish the geometric properties of the symmetrized set from those of the given set .
Recall that by rotation of the whole system Theorem 1.1 can be applied for a reflection with respect to any hyperplane in . Since the Laplacian is rotation invariant, only the potential then gets rotated, but remains essentially bounded. In this context, we remind the reader that denotes the reflection with respect to the hyperplane 0 1 . Furthermore, for the rest of this section, we use the following notation: where denotes the Lebesgue measure. In this case, is also referred to as -thick to emphasise the parameters.
The above definition has played a crucial role in a recent development on the nullcontrollability of the heat equation on . In [9], see also [28], it is shown that thickness of is a necessary and sufficient condition for the heat equation (2.1) on to be null-controllable. Moreover, in [9] the authors give an explicit estimate of the control cost in terms of the thickness parameters of the set and the time . In the same paper, the authors also consider the heat equation (2.1) on the cube with control set , where is -thick with for all 1 . In this case, they show that nullcontrollability holds in any time 0 with a bound on the control cost of the same form as for the full space case and independent of the scale .
In [19], these bounds have been strengthened in both time and geometric parameters dependency and the authors have shown that they are close to optimality in certain asymptotic regimes, see [19,Section 5]. For both cases , they can be written as 1 where 0 is a universal constant and 1 1¨¨¨, see [19,Theorem 4.9]. We show below that system (2.1) on half-spaces, positive orthants, and sectors of angle 2 , 2, is null-controllable with a control cost bound fully or partially consistent with (2.2); as a consequence, certain asymptotic properties of (2.2) are inherited, see Remark 2.10 below. Moreover, we obtain null-controllability and explicit control cost bounds for the same system on isosceles right-angled triangles and corresponding prisms. It is worth to note that although the results for the listed unbounded domains are new 1 , the heat equation on half-spaces and positive orthants with Dirichlet boundary conditions could also be treated with the techniques from [26] (see also Remark 2.6 below), while the novelty of the result on triangles and corresponding prisms is the explicitness of the control cost bound on the model parameters.
In order to enter the setting of the main theorem, let and with some -thick set , and consider the system with 0 2 and 2 0 2 . We need the following easy geometric lemma, parts of which are already contained in [9].
(a) Let 0 1 . Then, the set is -thick with 2 and . Then, is a -thick set with parameters

Let
. There is such that 0 1 or 0 1 , cf. Figure 2(a). In the first case, we have 1 .
In the second case, we have which completes the proof of part (a). For part (b), we observe that can be obtained from 0 by successive reflection with respect to all coordinate axes. In this regard, part (b) follows by analogous arguments as in part (a), cf. Figure 2(b); for more details, see also the proof of [9,Theorem 4].
Finally, we prove part (c). Let . Then, there exists such that or . This follows from the fact that contains the cylinder 2 0 2 1 2 2 Ś 3 0 for some , where 2 0 stands for the Euclidean 2-dimensional ball centred at zero with radius , and such a cylinder contains a parellelepiped of type or a reflection , or both, cf. Figure 3(c). The claim now follows by analogous calculations as in part (a), taking into account that . In this regard, the angle 8 constitutes a kind of worst case, whereas especially the angle 4 could have been handled in a slightly more efficient way. We refrained from doing so for the sake of simplicity.  Proof Choose the 2 2 -thick set as in Lemma 2.2 (b). Then, the heat equation (2.3) on is null-controllable in any time 0 from the control set . The associated bound on the control cost now reads as in (2.5). Since the set is symmetric with respect to every coordinate axis, the claim now follows by successively applying Theorem 1.1 for the reflections with respect to the coordinate axes. This step by step leads to null-controllability of the system (2.1) on 0 , 1 , with control set 0 . Each control cost does not exceed the one from the previous step and, thus, the corresponding bound reads as in (2.5). Taking into account that 0 0 , the final step then proves the claim.

Remark 2.6
Control cost estimates on the half-space and the positive orthant can alternatively be obtained by means of the exhaustion approach studied in [26]. In this case, the corresponding bounds will not incorporate any change in the thickness parameters as the above bounds do. However, the considerations from [26] currently allow to apply this approach only in the case of Dirichlet boundary conditions.  Proof This is proved analogously to the case 2 in Proposition 2.7 with and the corresponding control cost bound from (2.2).

Remark 2.9
In the recent work [6], it has been shown that the system (2.1) on the strip 1 is null-controllable in any time 0 if and only if is a thick set (which can be arbitrarily changed outside the strip), and an explicit control cost bound has been provided.
With similar arguments as in the propositions above, one can infer null-controllability results for system (2.1) on the product spaces , and equality is attained if, for instance, agrees with , which corresponds to the case of full control on . The upper bounds on obtained above can be compared to this lower bound in the asymptotic regime where 0 while remains fixed. This means that becomes more and more "well-distributed" within and is sometimes also called the homogenization limit, see, e.g., [19,Section 5]  (b) Recent developments on null-controllability for parabolic equations in Banach spaces [3,12] together with the abstract considerations in Section 3 also open the way towards similar results for certain strongly elliptic differential operators with constant coefficients on , 1 ; a bound on the associated control cost on of a form close to (2.2) is given in [3, Theorem 2.3], see also [12,Corollary 4.6]. We briefly show how to infer a corresponding result for the -Laplacian on the half-space 0 1 in Remark 4.2 (b) below. Below we extend the above result to sectors of angle 2 , 2, isosceles rightangled triangles, and corresponding prisms. A treatment of some related product spaces would also be possible, but we will not do this here for simplicity. In the framework of the main theorem, let 0 , 2 , and with some 2 -equidistributed set for . As in the previous subsection, we use the above-mentioned results for the system for 0 0 0 (2.10) with 0 2 and 2 0 2 to infer null-controllability and related control cost bounds on the desired domains.

Null-controllability from Equidistributed Sets
Analogously to Lemma 2.2 in the case of thick sets, we first need the following geometric consideration.
Proof We need to show that for every cube 4 4 , , there exists such that or . To this end, we observe that for every there exists such that or 3 3 , cf. Figure 3(a). In the first case, we are done with . In the second case, the cube 3 3 contains a ball of radius 2 , which, in turn, contains the reflection of a cube with sides of length 2 parallel to coordinate axes, cf. Figure 3(b). This cube must contain at least one cube of the form with , for which 3 3 . This concludes the proof.
Remark 2.14 If 4, it is possible to slightly strengthen the above lemma. Indeed, in this case the resulting set is 2 -equidistributed since the reflection 4 maps cells of the lattice in 4 to cells of the lattice in 4 and vice versa, and each cube 2 2 , , contains at least one cell of the lattice that belongs either to 4 or to 4 . However, for the sake of simplicity we opted for a unified statement valid for every reflection .  which has the same asymptotic behaviour for 0 as the lower bound 1 .

Abstract Result
In this section, we prove a general version of Theorem 1.1 for more abstract control systems, which in principle allows one to apply this result, for instance, also to other types of differential operators than discussed here so far, such as fractional Laplacians, second order elliptic operators, and magnetic Schrödinger operators. that is, if the mild solution to (3.1) vanishes at time ; see, e.g., [20] for the notion of a mild solution to abstract Cauchy problems. The associated control cost in time 0 is then defined as sup 0 1 inf 0 satisfies 3.2 .
The main result of this section is as follows.  , the claimed estimate for the control cost is immediate. This completes the proof.

Remark 3.2 Since the extension relation (3.3)
is only used to establish the semigroup relation (3.6), the conclusion of Theorem 3.1 still holds if (3.6) is directly taken as an assumption.
We refrained from doing so because the extension relation between the operators gives a nice understanding of the interplay between the systems (3.1) and (3.5).
In the particular case where and are lower semibounded self-adjoint operators on Hilbert spaces and , respectively, relation (3.6) for the semigroups can be extended to far more general functions of the operators. The following result formulates this explicitly. It is of particular interest, for instance, when considering fractional Laplacians of the form with 1 2, see Remark 4.2 (a) below. The result itself is probably well known and can be proved, for instance, with a standard reasoning using Stone's formula, which relates the spectral family of a self-adjoint operator with its resolvents. We give a brief variant of this reasoning below for the convenience of the reader.

Lemma 3.3 Let and be lower semibounded self-adjoint operators on Hilbert spaces
and , respectively, and let be a bounded linear operator such that . Then, the spectral families E and E for and , respectively, satisfy We close this section with a remark on a general lower bound on the control cost in the Hilbert space setting. Up to a relation between and , this allows to interpret the lower bound for in terms of the lower bound for . This gets particularly interesting when , , and , a situation which we encounter in Theorem 1.1, see Section 4 below. In this case, the factor agrees with the factor obtained in the bound from Theorem 3.1. .

Proof of the Main Theorem
The extension relation is then obtained by approximation since is an operator core for . Theorem 3.1 can then be applied with and .

Remark 4.3
In the above considerations, the operator models the extension of a function in 2 to a function in 2 by reflection with respect to one hyperplane. However, in view of the general form of Theorem 3.1, also other forms of extensions are feasible. For instance, one could consider to prolong functions on a sector of angle in the plane to functions on the whole plane by successive reflections with respect to different lines. In a similar way, in the work [22] the author considers the prolongation of functions on the hemiequilateral triangle with corners 0 0 , 0 1 , 1 3 0 to functions on the rectangle 0 3 0 1 . One then has to prove results analogous to Lemma 4.1 and relation (4.9), allowing to infer null-controllability results on such sectors and the (hemi-) equilateral triangle from those on the whole plane and the rectangle, respectively. In fact, in case of the hemiequilateral triangle, these analogous results follow to some extend from the considerations in [22] already.
One may also consider the rescaling that maps a given sector of arbitrary angle 0 2 to the sector with angle 4. This transforms the pure Laplacian on the given sector to a divergence-type operator on the sector of angle 4 with constant diagonal matrix and, therefore, fits into the framework of our abstract Theorem 3.1. However, there are currently no results on null-controllability of the form discussed in Section 2 available for such operators on this specific sector, so that Theorem 3.1 can not be applied yet.
corresponds to the face of exactly one other cube , , but then with the opposite direction of the corresponding outward unit normal, or the face belongs to the boundary of . In the latter case, the integral over the face vanishes unless the face belongs to the hyperplane 0 1 , since supp 0 1 lies in the interior of . This means that after summing over all cubes , , in (A.1) only boundary integrals for faces in the hyperplane 0 1 can survive, and these faces have an outward unit normal in the negative direction in the first coordinate ( 1). Thus, after summing in (A.1) over all cubes , , in both cases (a) and (b)  . This proves (c) and, hence, completes the proof of the lemma.