Optimal shape and location of sensors for parabolic equations with random initial data
Abstract
In this article, we consider parabolic equations on a bounded open connected subset of . We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer?
We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable subsets of of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semianalytic and thus has a finite number of connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schrödinger) studied in [57] for which relaxation does occur.
We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the DirichletLaplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power.
The results are illustrated with several numerical simulations.
Keywords: parabolic equations, optimal design, observability, minimax theorem.
AMS classification: 93B07, 35L05, 49K20, 42B37.
Contents
1 Introduction
Given a bounded domain of , in this paper we model and solve the problem of finding an optimal observation domain for general parabolic equations settled on . We want to optimize not only the placement but also the shape of , over all possible measurable subsets of having a certain prescribed measure. Such questions are frequently encountered in engineering applications but have been little treated from the mathematical point of view. Our objective is here to provide a rigorous mathematical model and setting in which these questions can be addressed. Our results will be established in a general parabolic framework and cover the cases of heat equations, anomalous diffusion equations or Stokes equations. For instance for the heat equation we will answer to the following question (that we will make more precise later on):
What is the optimal shape and location of a thermometer?
Brief state of the art.
Due to their relevance in engineering applications, optimal design problems for the placement of sensors for processes modeled by partial differential equations have been investigated in a large number of papers. Let us mention for instance the importance of the shape and placement of sensors for transportreaction processes (see [5, 17]). Several difficulties overlap for such problems. On the one hand, the parabolic partial differential equations under consideration constitute infinitedimensional dynamical systems, and, consequently, solutions live in infinitedimensional spaces. On the other hand, the class of admissible designs is not closed for the standard and natural topology. Few works take into consideration both aspects. Indeed, in many contributions, numerical tools are developed to solve a simplified version of the optimal design problem where either the partial differential equation has been replaced with a discrete approximation, or the class of optimal designs is replaced with a compact finite dimensional set (see for example [7, 24, 63] and [46] where such problems are investigated in a more general setting). In other words, in most of these applications the method consists in approximating appropriately the problem by selecting a finite number of possible optimal candidates and of recasting the problem as a finitedimensional combinatorial optimization problem. In many studies the sensors have a prescribed shape (for instance, balls with a prescribed radius) and then the problem consists of placing optimally a finite number of points (the centers of the balls) and thus it is finitedimensional, since the class of optimal designs is replaced with a compact finitedimensional set. Of course, the resulting optimization problem is already challenging. We stress however that, in the present paper, we want to optimize also the shape of the observation set, and we do not make any a priori restrictive assumption to compactly the class of shapes ( to be of bounded variation, for instance) and the search is made over all possible measurable subsets.
From the mathematical point of view, the issue of studying a relaxed version of optimal design problems for the shape and position of sensors or actuators has been investigated in a series of articles. In [48], the authors study a homogenized version of the optimal location of controllers for the heat equation problem (for fixed initial data), noticing that such problems are often illposed. In [3], the authors consider a similar problem and study the asymptotic behavior as the final time goes to infinity of the solutions of the relaxed problem; they prove that optimal designs converge to an optimal relaxed design of the corresponding twophase optimization problem for the stationary heat equation. We also mention [47] where, for fixed initial data, numerical investigations are used to provide evidence that the optimal location of nullcontrollers of the heat equation problem is an illposed problem. In [56] we proved that, for fixed initial data as well, the problem of optimal shape and location of sensors is always well posed for heat, wave or Schrödinger equations (in the sense that no relaxation phenomenon occurs); we showed that the complexity of the optimal set depends on the regularity of the initial data, and in particular we proved that, even for smooth initial data, the optimal set may be of fractal type (and there is no relaxation).
A huge difference between these works and the problem addressed in this paper is that all criteria introduced in the sequel take into consideration all possible initial data. Moreover, the optimization will range over all possible measurable subsets having a given measure. This the idea developed in [54, 55, 57], where the problem of the optimal location of an observation subset among all possible subsets of a given measure or volume fraction of was addressed and solved for conservative wave and Schrödinger equations. A relevant spectral criterion was introduced, viewed as a measure of eigenfunction concentration, in order to design an optimal observation or control set in an uniform way, independent of the data and solutions under consideration. Such a kind of uniform criterion was earlier introduced for the onedimensional wave equation in [27, 28] to investigate optimal stabilization issues.
The main difference of the previous analyses of conservative wavelike problems with respect to the present one is that, here, due to strong dissipativity of the heat equation (or of more general parabolic equations), highfrequency components are penalized in the spectral criterion, thus making optimal shapes to be determined by the low frequencies only, which, in particular, avoids spillover phenomena to occur.
Overview of the results of this paper.
Let us now provide a short overview of the results of the present paper, without introducing (at this step) the whole general parabolic framework in which our results are actually valid.
Let be an open bounded connected subset of . Let be a fixed (arbitrary) positive real number. To start with a simple model, let us consider the heat equation
(1) 
with Dirichlet boundary conditions. For any measurable subset of , we observe the solutions of (1) restricted to over the horizon of time , that is, we consider the observable , where denotes the characteristic function of . The subset models sensors, and a natural question is to determine what is the best possible shape and placement of the sensors in order to maximize the observability in some appropriate sense, for instance in order to maximize the quality of the reconstruction of solutions. In other words, we ask the question of determining what is the best shape and placement of a thermometer in .
At this stage, a first challenge is to settle the problem properly, to make it both mathematically meaningful and relevant in view of practical issues.
Throughout the paper, we fix a real number , and we will work in a class of domains such that . In other words the set of unknowns is
This is done to model the fact that the quantity of sensors to be employed is limited and, hence, that we cannot measure the solution over in its whole.
We stress again that we do not make any restriction on the regularity or shape of the subsets . We are trying to determine whether or not there exists an ”absolute” optimal observation domain. We will see that such a domain exists in the parabolic case under slight assumptions on the operator and on the domain (in contrast to the case of hyperbolic equations studied in [57]).
Let us now define the observability problem under consideration.
Recall that, for a given measurable subset of , the heat equation (1) is said to be observable on in time whenever there exists such that
(2) 
for every solution of (1) such that (the set of functions defined on , that are smooth and of compact support). It is well known that, if is , then this observability inequality holds true (see [18, 21, 40, 62]). Note that this result has been recently extended in [6] to the case where is bounded Lipschitz and locally starshaped.
The observability constant is defined as the largest possible constant such that (2) holds. This constant gives an account for the wellposedness of the inverse problem of reconstructing the solutions from measurements over (see, e.g., the textbook [15] for such inverse problems). Of course, the larger the constant is, the more stable the inverse problem will be.
Hence it is natural to model the problem of best observation for the heat equation (1) as the problem of maximizing the functional over the set , that is,
(3) 
Such a problem is however very difficult due to the presence of crossed terms at the righthand side of (2) when considering spectral expansions (see Section 2.1 for details). On the other hand, actually, the observability constant is (by nature) pessimistic in the sense that it corresponds to a worst possible case, and in practice it is expected that the worst case will not occur very often. In practice, to reconstruct solutions one is often led to achieve a large number of measurements, and in the problem of finding a best observation domain it is reasonable to design a set that will optimize the observability only in average.
In view of that, we define an averaged version of the observability inequality, where the average runs over random initial data. This procedure, described in detail in Section 2.1, consists of randomizing the Fourier coefficients of the initial data. To explain it with few words, let us fix an orthonormal Hilbert basis of consisting of eigenfunctions of the (negative of) DirichletLaplacian associated with the positive eigenvalues , with . Every solution of (1) can be expanded as
We randomize the solutions (actually, their initial data) by considering
for every event , where is a sequence of independent real random variables on a probability space having mean equal to , variance equal to , and a super exponential decay (for instance, Bernoulli laws). The randomized version of the observability inequality (2) is then defined as
where the expectation ranges over the space with respect to the probability measure . Here, is defined as the largest possible constant such that this randomized observability inequality holds, and is called randomized observability constant. It is easy to establish that
(4) 
for every measurable subset of . Moreover, note that (and the second inequality may be strict, as we will see further).
Following the previous discussion, instead of considering as a criterion the deterministic observability constant (and then, the problem (3)), we find more relevant to model the problem of best observation domain as the problem of maximizing the functional over the set , that is the problem
(5) 
This spectral model is discussed and settled in a more general parabolic framework in Section 2.1. As a particular case of our main results established in Section 2.2, we have the following result for the heat equation (1) with homogeneous Dirichlet boundary conditions.
Theorem.
Let arbitrary. Assume that is piecewise . There exists a unique^{5}^{5}5Here, it is understood that the optimal set is unique within the class of all measurable subsets of quotiented by the set of all measurable subsets of of zero measure. optimal observation measurable set , solution of (5). Moreover:

.

The optimal set is open and semianalytic. In particular, it has a finite number of connected components and .

The optimal set is completely characterized from a finitedimensional spectral approximation, by keeping only a finite number of modes. More precisely, for every , there exists a unique measurable set such that maximizes the functional
over . Moreover is open and semianalytic. Furthermore, the sequence of optimal sets is stationary, and there exists such that for every . The stationarity integer decreases as increases and whenever is large enough. In that case, the optimal shape is completely determined by the first eigenfunction.
A more general result (Theorem 1) will be established in a general parabolic framework. In the case of the heat equation, one of the important ingredients of the proof is a fine lower bound estimate (stated in [6]) of the spectral quantities , which is uniform over measurable subsets of a given measure.
Note that this existence and uniqueness result holds for every orthonormal basis of eigenfunctions of the DirichletLaplacian, but the optimal set depends, in principle, on the specific choice of the basis. Of course, for large enough, the optimal set is independent of the basis since it is completely determined by the first eigenfunction.
These properties, stated here for the heat equation (1) (and proved more generally for parabolic equations under an appropriate spectral assumption, see further) are in strong contrast with the results of [55, 56, 57] established for conservative wave and Schrödinger equations. In that context of wavelike equations it was proved that:

when considering the problem with fixed initial data, the optimal set could be of Cantor type (hence, ) even for smooth initial data;

the corresponding randomized observability constant is equal to , and, with respect to (4), the evident difference is that all weights are equal to . This is not surprising in view of the conservative properties of the wave or Schrödinger equation, however the fact that all frequencies have the same weight causes a strong instability of the optimal sets (maximizers of the corresponding spectral approximation). It was proved in [28, 55] that the best possible set for modes is actually the worst possible one when considering modes (spillover phenomenon).
In contrast, for the parabolic problems under consideration, we prove that this instability phenomenon does not occur, and that the sequence of maximizers is constant for large enough, equal to the optimal set . This stationarity property is of particular interest in view of designing the best observation set in practice.
In Section 2.2 we provide more details on these results, and state them in a far more general setting, involving in particular the Stokes equation and anomalous diffusion equations (with fractional Laplacian). For the Stokes equation
(6) 
considered on the unit disk with Dirichlet boundary conditions, we establish that there exists a unique optimal observation set in , sharing nice regularity properties as above.
Let us mention a striking feature occuring for the anomalous diffusion equation
(7) 
considered on some domain , where is some positive power of the DirichletLaplacian. Note that such equations are well recognized as being relevant models in many problems encountered in physics (plasma with slow or fast diffusion, aperiodic crystals, spins, etc), in biomathematics, in economy, also in imaging sciences (see for instance [43, 45, 61]). Hence they provide an important class of parabolic equations entering into the general framework developed in the paper.
Given arbitrary, we prove that if is piecewise and if (or if and is large enough) then there exists a unique optimal observation domain, independently on the Hilbert basis of eigenfunctions under consideration. Furthermore, we prove the unexpected facts that:

in the Euclidean square , when considering the usual Hilbert basis of eigenfunctions consisting of products of sine functions, for every there exists a unique optimal set in (as in the theorem), which is moreover open and semianalytic and thus has a finite number of connected components (and this, whatever the value of may be);

in the Euclidean disk , when considering the usual Hilbert basis of eigenfunctions parametrized in terms of Bessel functions, for every there exists a unique optimal set (as in the theorem), which is moreover open, radial, with the following additional property:

if then consists of a finite number of concentric rings that are at a positive distance from the boundary;

if (or if and is small enough) then consists of an infinite number of concentric rings accumulating at the boundary!

This surprising result shows that the complexity of the optimal shape does not only depend on the operator but also on the geometry of the domain .
It must be underlined that the proof of these properties (done in Section 3.5) is lengthy and particularly difficult in the case . It requires the development of very fine estimates for Bessel functions, combined with the use of quantum limits (semiclassical measures) in the disk, nontrivial minimax arguments and analyticity considerations.
Several numerical simulations based on the spectral approximation described previously are provided in Section 2.4. They show in particular what is the optimal shape and location of a thermometer in a square or in a disk.
The paper is structured as follows.
Section 2 is devoted to model and solve the problem of finding a best observation domain for parabolic equations. The model is discussed and defined in Section 2.1, based on the introduction of the randomized observability inequality. The problem is solved in a general parabolic setting in Section 2.2, where it is shown that, under an appropriate spectral assumption, there exists a unique optimal observation set, which can moreover be recovered from a finite dimensional spectral approximation problem. Section 2.3 is devoted to the application to the Stokes equation on the unit disk. In Section 2.4, we study the case of anomalous diffusion equations and then we provide several numerical simulations illustrating our results and in particular the stationarity feature of the sequence of optimal sets. Further comments on the spectral assumption are presented in Section 2.5, from a semiclassical analysis viewpoint.
All results are proved in Section 3. It must be underlined that the proof concerning the anomalous diffusion equations, in particular in the case , is long and very technical. It is actually unexpectedly difficult. The proof concerning the Stokes equation is as well for a large part based on facts derived in the previous proof.
Section 4 provides a conclusion and several further comments and open problems.
2 Optimal sensor shape and location / optimal observability
Let be an open bounded connected subset of . Throughout the paper we consider the problem of determining the optimal observation domain for the abstract parabolic model
(8) 
where be a densely defined operator. Precise assumptions on will be done further. As the main reference, we can keep in mind the typical example of the heat equation with Dirichlet boundary conditions overviewed in the introduction. But our analysis and results will be established for a large class of parabolic operators.
At this stage all what we need to assume, in order to establish the model that we will study, is that there exists a normalized Hilbert basis of consisting of (complexvalued) eigenfunctions of , associated with the (complex) eigenvalues .
2.1 The model
The aim of this section is to introduce and define a relevant mathematical model of the problem of best observation. The first ingredient is the notion of observability inequality.
Observability inequality.
For every , there exists a unique solution of (8) such that . For every measurable subset of , the equation (8) is said to be observable on in time if there exists such that
(9) 
for every solution of (8) such that . This inequality is called observability inequality, and the constant defined by
(10) 
is called the observability constant. It is the largest possible nonnegative constant for which (9) holds. In other words, the equation (8) is observable on in time if and only if .
Remark 1.
It is well known that, if is the negative of the Dirichlet, or Neumann, or Robin Laplacian, then the equation (8) is observable (see [18, 21, 40, 62]), for every open subset of . The observability property holds as well, e.g., for the linearized CahnHilliard operator corresponding to , , with the boundary conditions (see [62]). For the Stokes operator, the observability property follows from [20, Lemma 1].^{6}^{6}6More precisely, in order to derive the usual observability inequality from the Carleman estimate proved in this reference, it suffices to estimate from below the lefthand side weight on , to estimate from above the righthand weight, and to use the fact that the function is nonincreasing.
As explained in the introduction, throughout the paper we fix a real number and we will search an optimal domain in the set
(11) 
This gives an account for the fact that we can measure the solutions only over a part of the whole domain .
Having in mind the observability inequality (9), it is a priori natural to model the question of the optimal location of sensors in terms of maximizing the observability constant over the set defined by (11), where is fixed. Actually, when implementing a reconstruction method, the observability constant gives an account for the wellposedness of the corresponding inverse problem. More precisely, the larger the observability constant is, and the better conditioned the inverse problem is.
However at this stage two remarks are in order.
Firstly, settled as such, the problem is difficult to handle, due to the presence of crossed terms at the rightand side of (9) when considering spectral expansions. This problem, which has been discussed thoroughly in [55, 57], is quite similar to the open problem of determining the best constants in Ingham’s inequalities (see [30, 31]). Here, one is faced with the problem of determining the infimum of eigenvalues of an infinite dimensional symmetric nonnegative matrix (namely, the Gramian, see below). Although this criterion has a clear sense, it leads to an optimal design problem which does not seem to be easily tractable.
Secondly, even though the problem of maximizing the observability constant seems natural at the first glance, it is actually not so relevant with respect to the practical issues that we have in mind. Indeed in practice one is led to deal with a large number of solutions: when implementing a reconstruction process, one has to carry out in general a very large number of measures; likewise, when implementing a control procedure, the control strategy is expected to be efficient in general, but maybe not exactly for all cases. The issue that we raise here is the fact that the above observability inequality (9) is deterministic, and thus the observability constant is pessimistic since it corresponds to a worst possible case. It is likely that in practice this worst case will not occur very often, and hence the deterministic observability constant is not a relevant criterion when realizing a large number of experiments. Instead of that, we are going to propose an averaged version of the observability constant, better suited to our purposes, and defined in terms of probabilistic arguments.
Randomized observability inequality.
Every solution of (8) such that can be expanded as
(12) 
where
(13) 
for every . Using this spectral decomposition, the change of variable and an easy density argument, we get
(14) 
As briefly explained previously, appears as the infimum of the eigenvalues of a Gramian operator, which is the infinitedimensional Hermitian nonnegative matrix
(15) 
Due to the crossed terms appearing when expanding the square in (14), the resulting optimal design problem, consisting of maximizing over the set , is not easily tractable, at least in view of deriving theoretical results. Moreover, from the practical point of view the problem of modeling the best observation has to be done, having in mind that the best observation domain should be designed to be the best possible in average, that is, over a large number of experiments. The observability constant above is by definition deterministic, and thus pessimistic in the sense that is gives an account for the worst possible case. In practice, when carrying out a large number of experiments, it can however be expected that the worst possible case does not occur very often. Having this remark in mind, we next define a new notion of observability inequality by considering an average over random initial data. We then define below a notion of randomized observability constant, which is in our view better suited to the model of best observation. We follow [57], accordingly to early ideas developed in [51] for harmonic analysis issues and recently in [11, 12] in view of ensuring the probabilistic wellposedness of classically illposed supercritical wave or Schrödinger equations.
For any given , the Fourier coefficients of , defined by (13), are randomized by defining for every , where is a sequence of independent real random variables on a probability space having mean equal to , variance equal to , and a super exponential decay (for instance, independent Bernoulli random variables, see [11, 12] for more details on randomization possibilities and properties). For every , the solution corresponding to the initial data is then . Instead of considering the deterministic observability inequality (9), we define the randomized observability inequality by
(16) 
for every solution of (8) such that , where is the expectation over the space with respect to the probability measure . The nonnegative constant is called randomized observability constant and is defined (by density) by
(17) 
It is the randomized counterpart of the deterministic constant defined by (14). Note that
(18) 
for every measurable subset of . The inequalities can be strict (see Theorem 1 further).
Proposition 1.
Let arbitrary. For every measurable subset of , we have
with
(19) 
Proof.
Using the Fubini theorem and the independence of the random laws, one has
and the conclusion follows easily. ∎
This result clearly shows how the randomization procedure rules out the offdiagonal terms in the Gramian (15).
Conclusion: the optimal shape design problem.
For every measurable subset of , we set
(20) 
Throughout the paper, we will consider the problem of maximizing the functional over the set defined by (11), where the coefficients are defined by (19). In other words, we consider the problem
(21) 
According to the previous discussion, this optimal shape design problem models the best sensor shape and location problem for the parabolic equation (8).
The functional defined by (20) corresponds to an energy concentration measure. As we will see, solving this problem requires spectral assumptions.
2.2 The main result
In our main result below, it will be useful to consider the functional defined by
(22) 
for every measurable subset of , for every . The functional is the spectral truncation of the functional to the first terms. We consider as well the shape optimization problem
(23) 
which is a spectral approximation of the problem (21). We call it the truncated problem.
Let us now provide the general parabolic framework and the required spectral assumptions.
Framework and assumptions.
Let be an open bounded connected subset of , and let and be arbitrary. Let be a densely defined operator, generating a strongly continuous semigroup on . We assume that there exists a Hilbert basis of consisting of (complexvalued) eigenfunctions of , associated with (complex) eigenvalues such that , and such that the following assumptions are satisfied:

(Strong Conic Independence Property) If there exists a subset of of positive Lebesgue measure, an integer , a tuple , and such that almost everywhere on , then there must hold and for every .

For every such that , one has

The eigenfunctions are analytic in .
We start with a simple preliminary result for the truncated problem.
Proposition 2.
Under , for every , the truncated problem (23) has a unique^{7}^{7}7Here and in the sequel, it is understood that the optimal set is unique within the class of all measurable subsets of quotiented by the set of all measurable subsets of of zero measure. solution . Moreover, under , is an open semianalytic^{8}^{8}8A subset of a real analytic finite dimensional manifold is said to be semianalytic if it can be written in terms of equalities and inequalities of analytic functions. We recall that such semianalytic subsets are stratifiable in the sense of Whitney (see [23, 29]), and enjoy local finitetess properties, such that: local finite perimter, local finite number of connected components, etc. set, and thus, in particular, it has a finite number of connected components.
Remark 2.
If is defined on a domain such that the eigenfunctions vanish on (Dirichlet boundary conditions), then moreover there exists such that the (Euclidean) distance between and is larger than .
Our main result is the following.
Theorem 1.
Under and , the optimal shape design problem (21) has a unique solution .
Moreover, there exists a smallest integer such that
for every . In other words, the sequence of maximizers of is stationary, that is, for .
The function is nonincreasing, and if as then whenever is large enough.
Under the additional assumption , we have moreover that:

;

the optimal observation set is an open semianalytic set and thus it has a finite number of connected components.
Remark 3.
In the next sections we will comment in detail on the assumptions done in the theorem, and provide classes of examples where they are satisfied (note however that proving their validity is far from obvious): heat and anomalous diffusion equations, Stokes equation.
We can however note, at this stage, that these assumptions are of different natures.
The assumption will be treated essentially with analyticity considerations. Indeed note that holds true as soon as the eigenfunctions are analytic in (that is, under the assumption ) and vanish along . This is often the case, for instance, for elliptic operators with analytic coefficients. It can be noted that a generalization of the property has been studied for the DirichletLaplacian in [53], where the are arbitrary real numbers, and is proved to hold generically with respect to the domain . The validity of in general (for instance, in for Neumann boundary conditions) is an open problem.
The assumption , which can as well be seen from a semiclassical point of view (see comments in Section 2.5 further) is related with nonconcentration properties of eigenfunctions. For instance proving it for heatlike equations will require the use of fine recent results providing lower bound estimates that are uniform with respect to the observation domain .
Before coming to these applications, several remarks are in order.
Remark 4.
The fact that the sequence of optimal sets of the truncated problem (35) is stationary is in strong contrast with the results of [27, 28, 54, 55, 57] in which such optimal design problems have been investigated for conservative wave or Schrödinger equations. In these references it was observed and proved that the corresponding maximizing sequence of subsets does not converge in general, except in very particular cases. Moreover, in dimension one, this sequence of sets has an instability property known as spillover phenomenon. Namely, the best possible set for modes is actually the worst possible one when considering modes. This instability property has negative consequences in view of practical issues for designing a relevant notion of optimal set.
In contrast, Theorem 1 shows that, for the parabolic equation (8), the maximizing sequence of subsets is stationary, and hence only a finite number of modes is enough in order to capture all the information necessary to design the true optimal set. In other words, higher modes play no role. Although this result can appear as intuitive because we are dealing with a parabolic equation, deriving such a property however requires the spectral property , which is commented and analyzed further.
Remark 5.
The fact that the optimal set is semianalytic is a strong (and desirable) regularity property. In addition to the fact that has a finite number of connected components, this implies also that is Jordan measurable, that is, . This is in contrast with the already mentioned fact that, for wavelike equations, when maximizing the energy for fixed data, the optimal set may be a Cantor set of positive measure, even for smooth initial data (see [56]).
Remark 6 (A convexified formulation of (21)).
It is standard in shape optimization to introduce a convexified version of a maximization problem, since it may fail to have some solutions because of hard constraints. This is what is usually referred to as relaxation (see, e.g., [10]).
Since the set (defined by (11)) does not share nice compactness properties, we consider the convex closure of for the weak star topology of , which is
(24) 
Such a relaxation was used as well in [48, 55, 57]. Replacing with , we define a relaxed formulation of the optimal shape design problem (21) by
(25) 
where the functional is naturally extended to by
(26) 
for every . Moreover, one has the following existence result.
Lemma 1.
For every , the relaxed problem (25) has at least one solution .
Proof of Lemma 1.
For every , the functional is linear and continuous for the weak star topology of . Hence is upper semicontinuous as the infimum of continuous linear functionals. Since is compact for the weak star topology of , the lemma follows. ∎
Note that, obviously,
But, in fact, from Theorem 1 (and from its proof) we deduce that the two suprema coincide, and that the problem (21) and the relaxed problem (25) have the same (unique) solution. This means two things. First, there is no gap between the optimal values of the problem (21) and its relaxed formulation (25). A similar result was established in [57] for wave and Schrödinger like equations under spectral assumptions on the domain . But, in contrast to these hyperbolic equations where relaxation occurs except for some very distinguished discrete values of , here, in the parabolic setting, relaxation does not occur, at least under the assumption , which is fulfilled for the DirichletLaplacian for piecewise domains (see Theorem 3 further).
In particular, in the parabolic setting, contrarily to what happens in wavelike equations, the constant function is not an optimal solution. Note that this constant function corresponds intuitively (at the weak limit) to equidistribute the sensors over the domain . This strategy is however not optimal for parabolic problems.
Remark 7.
The assumption can be actually weakened (as can be easily seen from the proof of the theorem). To ensure that the conclusion of Theorem 1 holds, it is sufficient to assume that
(27) 
where is any optimal solution of the relaxed problem (25). In other words, it is sufficient to restrict the assumption to the sole . Note that such an assumption is impossible to check since is not known a priori, but this remark will however be useful in Section 3.5.
Note that, since , it follows that , and hence in particular there always holds
Remark 8.
The existence and uniqueness of an optimal set, stated in Theorem 1, holds true for any Hilbert basis of eigenfunctions of as soon as this basis satisfies the assumptions , and . However the optimal set may depend on the specific choice of the basis.
Remark 9.
As noted before, the issue of solving the optimal design problem
where is the observability constant of the parabolic equation (8) defined by (10), is natural and interesting, although this problem is very difficult to handle from the theoretical point of view, even for the truncated criterion, and not as much relevant as the one we consider here, from the practical point of view (as already discussed).
Note that the truncated version of the criterion is the lowest eigenvalue of the diagonal matrix , whereas the truncated version of the criterion is the lowest eigenvalue of the Gramian matrix
(28) 
which is the truncation of the Gramian defined by (15). Under the conditions of Theorem 1, the sequence of the minimizers over of the truncated version of the randomized constant is stationary. An interesting problem consists of investigating theoretically or numerically whether this stationarity property holds true or not for the truncated version of the observability constant .
Notice that, extending the definition of to the functions by
one gets easily that the optimal design problem of maximizing over has at least one solution. Furthermore, it is interesting to note that, by adapting the proof of [55, Proposition 2], we get the following partial result.
Lemma 2.
For every and every , the constant function is not a maximizer of the functional over .
Remark 10.
Finally, let us comment on the role of the time . Recall that has been arbitrarily fixed at the beginning of the analysis. Its role is in the weights coming into play in the definition of the functional (defined by (20)). If the eigenvalues are such that , then the larger is, and the quicker the weights tend to . As a consequence, as stated in Theorem 1, the integer decreases as increases, and if is large enough then . This says that, if one can observe the solutions of the equation over a large enough horizon of time, then the optimal observation domain can be designed from the first mode only. This fact is in accordance with the strong damping properties of a parabolic equation, at least, under the assumption . In large time the energy of the solutions is essentially carried by the first mode.
2.3 Application to the Stokes equation in the unit disk
In this section, we assume that is the Euclidean unit disk of , and we consider the Stokes equation (6) in the unit disk of , with Dirichlet boundary conditions.
Note that the Stokes system does not exactly enter in the framework defined in Section 2.2, but it suffices to make the following very slight modification. The Stokes operator is defined by , with , , , and is the Leray projection. Then is an unbounded operator in the Hilbert space (and not on ), endowed with the norm (see [9]).
We consider here the Hilbert basis of of eigenfunctions, indexed by , and , defined by
(29) 
and
(30) 
whenever , where are the usual polar coordinates (see [35, 41]). The functions are defined by and , with the agreement that , and is the Bessel function of the first kind of order . Denoting by is the positive zero of , the eigenvalues of are the doubly indexed sequence , where is of multiplicity if , and if .
Note that, in , , and in the definition (20) of the functional , we replace with the Euclidean norm of .
Theorem 2.
This result is proved in Section 3.6. The proof is technically based on the explicit form of the basis of eigenfunctions under consideration, and we did not investigate what can happen in higher dimension. Also, what can happen for more general domains is not known.
2.4 Application to anomalous diffusion equations
In this section, we assume that is Lipschitz, and we consider the DirichletLaplacian defined on its domain . Note that if is then .
We set (where is the DirichletLaplacian), with arbitrary, defined spectrally, based on the spectral decomposition of the DirichletLaplacian. This case corresponds to the anomalous diffusion equation (7).
To be more precise with the functional framework, the domain of the operator , as an unbounded operator in , is defined as follows. If , then ; if then (LionsMagenes space), and if then (see [42] or [8, Appendix]).
For the operator is defined by composing integer powers of with the fractional powers above. For instance one can take with the boundary conditions : in that case (8) corresponds to a linearized model of CahnHilliard type.
In the general case , the equation (8) models a physical process exhibiting anomalous diffusion (see for instance [43, 45, 61]). Of course if then (8) is the heat equation with Dirichlet boundary conditions, as overviewed in the introduction.
Note that the eigenfunctions of are those of the DirichletLaplacian, and therefore the assumptions and are satisfied. Only the assumption has to be discussed in the sequel. We have the following three results.
2.4.1 A general result
Theorem 3.
Assume that is piecewise^{9}^{9}9Actually a more general assumption can be done: is Lipschitz and locally starshaped (see [6] for the definition and details). . If , then the assumption is satisfied for any Hilbert basis of eigenfunctions of . For , the conclusion holds true as well provided that