Yannick Privat
email: yannick.privat@upmc.fr

Emmanuel Trélat
email: emmanuel.trelat@upmc.fr

Enrique Zuazua
email: enrique.zuazua@uam.es

Actuator design for parabolic distributed parameter systems with the moment method

Keywords: heat equation, parabolic systems, shape optimization, null controllability, moment method, lumped control AMS classification: 93B07, 35L05, 49K20, 42B37

In this paper, we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset Ω of IR n . We optimize not only the location but also the shape of actuators, by finding what is the optimal distribution of actuators in Ω, over all possible such distributions of a given measure. Using the moment method, we formulate a spectral optimal design problem, which consists of maximizing a criterion corresponding to an average over random initial data of the largest L 2 -energy of controllers. Since we choose the moment method to control the PDE, our study mainly covers one-dimensional parabolic operators, but we also provide several examples in higher dimensions.

We consider two types of controllers: either internal controls, modeled by characteristic functions, or lumped controls, that are tensorized functions in time and space. Under appropriate spectral assumptions, we prove existence and uniqueness of an optimal actuator distribution, and we provide a simple computation procedure. Numerical simulations illustrate our results.

Introduction and modeling of the problem

In this article, we model and solve the problem of finding the optimal shape and location of internal controllers for parabolic equations with (mainly) Dirichlet boundary conditions and (mainly) in the one-dimensional case Ω = (0, π). Such questions are frequently encountered in engineering applications. We provide a possible mathematical model for investigating such issues.

For mathematical reasons that will be clarified in the sequel, we will focus in the whole article on controls obtained by using the so-called moment method. As it will be underlined, it requires in general some spectral gap assumptions on the operators involved that essentially reduce the applications of our results to one-dimensional partial differential equations, but our results also cover several particular situations in larger dimension.

To avoid technicalities and highlight the main ideas, we first present the results in the simplified framework of the controlled one-dimensional heat equation with Dirichlet boundary conditions, without introducing (at this step) the more general parabolic framework in which our results are actually valid.

Generalizations to a more general framework will be described in Section 3. Unlike the simplified case of the one-dimensional heat equation, it requires a discussion on the Müntz-Szász theorem as well as specific spectral considerations.

Notice that the general control framework in which this problem could be addressed is much more intricate and will be evoked as a possible perspective at the end of this article.

Reminders on the controllability of the 1D heat equation

Consider the internally controlled one-dimensional heat equation

∂ t y(t, x) -∂ xx y(t, x) = χ ω (x)u(t, x), (t, x) ∈ (0, T ) × (0, π), (1)
with Dirichlet boundary conditions y(t, 0) = y(t, π) = 0, t ∈ (0, T ),

where u ∈ L 2 ((0, T ) × (0, π)) is a control function, and ω is a measurable subset of (0, π) standing for the support of the controller. Here, χ ω is the characteristic function of ω, defined by χ ω (x) = 1 if x ∈ ω and χ ω (x) = 0 otherwise. For a given subset ω, the equation ( 1) is said to be exactly null controllable in time T whenever every initial datum y(0, •) ∈ L 2 (0, π) can be steered to 0 in time T by means of an appropriate control function u ∈ L 2 ((0, T ) × (0, π)). It is well known that, for a given subset ω, the system (1) is exactly null controllable if and only if there exist a positive constant C (only depending on T and ω) such that

C π 0 z(T, x) 2 dx T 0 ω z(t, x) 2 dx dt, (3)
(observability inequality) for every solution of ∂ t z(t, x) -∂ xx z(t, x) = 0, (t, x) ∈ (0, T ) × (0, π), z(t, 0) = z(t, π) = 0, t ∈ (0, T ), [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF] with z(0, •) ∈ L 2 (0, π).

Exact null controllability by the moment method. The observability inequality [START_REF] Ammar-Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF] has been shown to hold true for any subset ω of [0, π] of positive Lebesgue measure in [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF] by the moment method, that we will use as well in the present paper and that we recall hereafterin.

The eigenfunctions of the Dirichlet-Laplacian, given by φ j (x) = 2 π sin(jx) for every j ∈ IN * , associated with the eigenvalues λ j = j 2 , make up an orthonormal basis of L 2 (0, π). From the Müntz-Szász theorem, there exists a sequence (θ T j ) j∈IN * of L 2 (0, T ), biorthogonal to the sequence of functions t → e -j 2 t . The following lemma provides an exact null controllability result for (1)- [START_REF] Allaire | Long time behavior of a two-phase optimal design for the heat equation[END_REF]. Lemma 1. [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF] Let T > 0 and let ω be a measurable subset of (0, π) of positive measure. Then every initial datum y(0, x) = y 0 (x) = +∞ j=1 a j sin(jx), in L 2 (0, π), can be steered to zero in time T with the control u ∈ L 2 ((0, T ) × (0, π)) defined by u(t, x) = -+∞ j=1 a j e -j 2 T ω sin 2 (jy) dy θ T j (T -t) sin(jx).

A proof of this lemma is given in a more general setting in Section A.1. We set Γ ω (y 0 ) = χ ω u. The operator Γ ω : L 2 (0, π) → L 2 ((0, T ) × (0, π)) is linear and continuous, and is called the moment control operator. The norm of this operator, given by Γ ω = sup{ Γ ω (y 0 ) L 2 ((0,T )×(0,π)) | y 0 L 2 (0,π) = 1}, provides an account for the worst possible initial datum to be controlled to zero, in terms of the effort (L 2 energy) required to steer this initial datum to zero. Minimizing Γ ω over a class of admissible domains (that we will denote by U L in the sequel) is then an interesting problem, that will be discussed in the next section.

State of the art

When realizing exact null controllability in practice, an important question is to know where to place and how to shape optimally the actuators (modeled here by the subset ω), in order to minimize the efforts done to steer any possible initial data to zero. In this paper, we want to optimize not only the location, but also the shape of actuators, without any specific restriction on the regularity of ω.

The literature on optimal sensor or actuator location problems is abundant in engineering applications (see, e.g., [START_REF] Armaoua | Optimal actuator/sensor placement for linear parabolic PDEs using spatial H 2 norm[END_REF][START_REF] Harris | Optimal sensor location with an application to a packed bed tubular reactor[END_REF][START_REF] Kumar | Optimal location of measurements for distributed parameter estimation[END_REF][START_REF] Morris | Linear-quadratic optimal actuator location[END_REF][START_REF] Sigmund | Systematic design of phononic band-gap materials and structures by topology optimization[END_REF][START_REF] Ucinski | Sensor network design for the estimation of spatially distributed processes[END_REF][START_REF] Van De Wal | A review of methods for input/output selection[END_REF][START_REF] Vande Wouwer | An approach to the selection of optimal sensor locations in distributed parameter systems[END_REF] and references therein), where the aim is often to optimize the number, the place and the type of sensors or actuators in order to improve the estimation of the state of the system. Fields of applications are very numerous and concern for example active structural acoustics, piezoelectric actuators, vibration control in mechanical structures, damage detection and chemical reactions, just to name a few of them. In most of these applications the method consists of approximating appropriately the problem by selecting a finite number of possible optimal candidates and of recasting the problem as a finite-dimensional combinatorial optimization problem. In many of these contributions the sensors or actuators 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 is finite-dimensional, since the class of optimal designs is replaced with a compact finite-dimensional set. We stress that, in the present paper, the shape of the control domain is an unknown of the optimization procedure.

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 [START_REF] Münch | Optimal distribution of the internal null control for the 1D heat equation[END_REF], 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 ill-posed. In [START_REF] Allaire | Long time behavior of a two-phase optimal design for the heat equation[END_REF], the authors consider a similar problem and study the asymptotic behavior as the final time T goes to infinity of the solutions of the relaxed problem; they prove that optimal designs converge to an optimal relaxed design of the corresponding two-phase optimization problem for the stationary heat equation. We also mention [START_REF] Münch | Numerical null controllability of semi-linear 1D heat equations: fixed point, least squares and Newton methods[END_REF] where, for fixed initial data, numerical investigations are used to provide evidence that the optimal location of null-controllers of the heat equation problem is an ill-posed problem.

Concerning the problem of optimal shape and location of sensors for fixed initial data (instead of controllers in [START_REF] Münch | Numerical null controllability of semi-linear 1D heat equations: fixed point, least squares and Newton methods[END_REF]) we proved in [START_REF] Privat | Complexity and regularity of maximal energy domains for the wave equation with fixed initial data[END_REF] that it 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). In [START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF], we modeled and solved the problem of optimal shape and location of the observation domain having a prescribed measure. This problem was motivated by the question of shaping and placing sensors in some domain in such a way to optimize the quality of the observation.

Here, we rather investigate the dual question of the best shape and location of actuators. In [START_REF] Münch | Numerical approximation of null controls for the heat equation : ill-posedness and remedies[END_REF], the authors investigate numerical approximations of exact or trajectory controls for the heat equation, by developing a numerical version of the so-called transmutation method.

1.3 Modeling of the optimal design problems: a randomization procedure

In the present paper, our objective is to search the internal control domain over all possible subsets of (0, π), without assuming any a priori regularity. We optimize not only the placement but also the shape of the actuators.

Note that, for any problem consisting of optimizing the quality of the control, certainly the best strategy consists of controlling the solutions over the whole domain (0, π). This is however obviously not reasonable and in practice the domain covered by actuators is limited, due for instance to cost considerations. From the mathematical point of view, we model this basic limitation by considering as the set of unknowns, the set of all possible measurable subsets ω of (0, π) that are of Lebesgue measure |ω| = Lπ, where L ∈ (0, 1) is some fixed real number. Any such subset ω represents the actuators put in (0, π). Finally, for mathematical reasons, it is more convenient to assimilate a measurable subdomain ω of (0, π) to its characteristic function χ ω , vanishing outside ω and equal to 1 else. Hence, let us introduce the class of admissible control domains

U L = {χ ω ∈ L ∞ (0, π, {0, 1}) | ω ⊂ Ω measurable, |ω| = Lπ}. (5)
In view of modeling the optimal design of actuators, a first approach consists of minimizing the functional χ ω → Γ ω over the set U L . However, even for simple choices of control domains ω, the quantity Γ ω is not explicitly computable and therefore the cost functional is hard to handle. Besides, note that the moment control operator norm Γ ω is deterministic and thus provides an account for the worst possible case; in this sense, it is a pessimistic constant. One can argue that, in practice, when running a large number of experiments, it is expected that the worst possible case does not occur so often. For these reasons, we are next going to consider an average criterion which, in some sense, does not take into account rare events. Nevertheless, we stress that the issue of minimizing Γ ω with respect to the domain ω has not only a mathematical interest, but appears also naturally in some practical situations, where it is imperative that the worst possible case be avoided, even if it is a rare event. We refer to the conclusion section 3.4 for some comments about such a problem. The same kind of difficulty arises when modeling optimal design problems for sensors, as discussed in [START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF][START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF].

In this paper, we propose another approach based on the controllability result stated in Lemma 1, and on a randomization argument reflecting what happens when a large number of experiments is expected to be done. We are going to use a probabilistic argument, by considering random initial data. We follow the approach developed in [START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF][START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF]. Let us fix an arbitrary initial datum y(0, •) = y 0 (•) ∈ L 2 (0, π) of the controlled system (1), with Fourier coefficients defined by

a j = π 0 y 0 (x) sin(jx) dx, (6)
These coefficients are now randomized according to a ν j = β ν j a j for every j ∈ IN * , where (β ν j ) j∈IN * is a sequence of independent real random variables on a probability space (X , F, P), having mean equal to 0, variance equal to 1, and with a super exponential decay1 (for instance, independent Bernoulli random variables, see [START_REF] Burq | Large-time dynamics for the one-dimensional Schrödinger equation[END_REF][START_REF] Burq | Random data Cauchy theory for supercritical wave equations. I. Local theory[END_REF] for details and properties of randomization). For every event ν ∈ X , the control steering the initial datum

y 0 ν (x) = +∞ j=1 β ν j a j sin(jx)
to zero by the moment method is, according to Lemma 1,

u ν (t, x) = - +∞ j=1 β ν j a j e -j 2 T ω sin 2 (jy) dy θ T j (T -t) sin(jx).
Using the previous notations, one has Γ ω (y 0 ν ) = χ ω u ν . We propose, then, to model the problem of best actuator shape and location as the problem of minimizing the averaged functional

K(χ ω ) = sup y 0 L 2 (0,π) =1 E Γ ω (y 0 ν ) 2 L 2 ((0,T )×(0,π))
over U L , where E is the expectation over the probability space (X , F, P). This is the randomized counterpart to the deterministic quantity Γ ω . One of the advantages is that K(χ ω ) can be explicitly computed, as follows.

Lemma 2. One has

K(χ ω ) = inf j∈IN * e 2j 2 T T 0 θ T j (t) 2 dt ω sin 2 (jx) dx -1
, for every measurable subset ω ⊂ (0, π).

Lemma 2 is proved in Section A.2. Therefore, the problem of best shape and location of actuators is finally written as

sup χω∈U L inf j∈IN * e 2j 2 T T 0 θ T j (t) 2 dt ω sin 2 (jx) dx. ( 7
)
The article is organized as follows: Section 3.1 is devoted to comments on the control of parabolic equations by the moment method, the use of biorthogonal sequences and modeling of the optimal design problem issues. In Section 3.2, we solve the problem and provide a numerical illustration as well as a series of examples, mainly in the 1D case due to the restrictions imposed by the choice of the control method. Finally, in Section 3.3, we investigate a variant of the previously studied optimal design problem, where the control acts on the system by means of tuning the time-intensity.

2 Solving the problem (7)

Main results, comments and illustration

We first provide an existence result.

Theorem 1. The shape optimization problem (7) has a unique2 solution χ ω * . This theorem is proved in Section 2.3. In addition to this result, what is remarkable is that we have a simple and numerically efficient procedure to compute the optimal control domain ω * .

Algorithmic computation procedure. The optimal set ω * of Theorem 1 can actually be built from a finite-dimensional spectral approximation, by keeping only a finite number of modes. Let us provide the details of the procedure. For every integer N ∈ IN * , we define the functional J N by

J N (χ ω ) = inf 1 j N e 2j 2 T T 0 θ T j (t) 2 dt ω sin 2 (jx) dx,
for every measurable subset ω of (0, π). The functional J N is a spectral truncation to the N first terms. We consider the shape optimization problem

sup χω∈U L J N (χ ω ), (8)
called truncated problem, which is a spectral approximation of the problem [START_REF] Bucur | Variational methods in shape optimization problems[END_REF].

We have then the following results, proved in Sections 2.2 and 2.3.

Proposition 1. For every N ∈ IN * , the truncated problem (8) has a unique solution χ ω N ∈ U L .
Moreover, ω N has a finite number of connected components, and there exists η N > 0 such that

ω N ⊂ [η N , π -η N ].
Proposition 2 further (see Section 3.2) will provide an extension of this result to higher dimensions. We will however provide two different proofs. Indeed, in the one-dimensional case investigated here, we will show in the proof that the problem (8) can be expressed in an equivalent way as a classical optimal control problem. This point of view (already used in [START_REF] Privat | Optimal location of controllers for the one-dimensional wave equation[END_REF]) is interesting not only for the proof but also in order to derive efficient numerical methods for the numerical computation of the optimal domains.

Let us now give the main result that is at the base of the algorithmic procedure.

Theorem 2. There exists N 0 ∈ IN * such that ω * = ω N for every N N 0 . Furthermore, we have N 0 N 0 , where N 0 is the first integer (which exists and is finite) such that

∀j N 0 , θ T j 2 L 2 (0,T ) e 2 (πL -sin(πL)) 128 e 2T (j 2 -1) .
As a result, N 0 is equal to 1 if T is large enough.

In other words, Theorem 2 says that the sequence (ω N ) N ∈IN * of optimal sets, whose existence is stated in Proposition 1, is stationary. The numerical procedure consists of computing these sets, and once it has become stationary, then we have found the optimal set ω * , solution of the shape optimization problem [START_REF] Bucur | Variational methods in shape optimization problems[END_REF].

A natural issue concerns the characterization of the minimal integer N 0 such that the sequence of optimal domains (ω N ) N N0 remains constant. Even if a partial answer is provided in Theorem 2, it is likely that the determination of N 0 is in general intricate.

As a numerical illustration of this computation procedure, we provide on Figure 1 several numerical simulations of the optimal control domain, solution of the truncated problem (8) in the 1D case, for the Dirichlet-Laplacian. We observe the expected stationarity property of the sequence of optimal domains ω N from N = 5 on. In the forthcoming section devoted to providing the proofs of the results above, it will be required to consider a convexified version of the problem [START_REF] Bucur | Variational methods in shape optimization problems[END_REF], which may fail to have some solutions because of the hard constraint3 χ ω ∈ U L (which is a binary constraint almost everywhere). This is usually referred to as relaxation (see, e.g., [START_REF] Bucur | Variational methods in shape optimization problems[END_REF]). Since the set U L (defined by ( 5)) does not share nice compactness properties, we consider the convex closure of U L for the weak star topology of L ∞ , which is

U L = a ∈ L ∞ (0, π; [0, 1]) | Ω a(x) dx = Lπ . (9)
Such a relaxation was used as well in [START_REF] Münch | Optimal distribution of the internal null control for the 1D heat equation[END_REF][START_REF] Privat | Optimal observability of the one-dimensional wave equation[END_REF][START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF]. Replacing χ ω ∈ U L with a ∈ U L , we consider the relaxed (or convexified) formulation of the problem [START_REF] Bucur | Variational methods in shape optimization problems[END_REF] given by sup

a∈U L J (a), (10)
where the functional J is naturally extended to U L by

J (a) = inf j∈IN * e 2j 2 T T 0 θ T j (t) 2 dt Ω a(x) sin 2 (jx) dx, (11)
for every a ∈ U L . We consider as well a relaxed formulation of the truncated optimal design problem (8) by sup

a∈U L J N (a), (12)
where the functional J N is naturally extended to U L by

J N (a) = inf 1 j N e 2j 2 T T 0 θ T j (t) 2 dt Ω a(x) sin 2 (jx) dx, (13)
for every a ∈ U L .

Being defined as the infimum of linear functions, continuous for the L ∞ weak star topology, the functional J is upper semi continuous for the L ∞ weak star topology. The set U L being compact for this topology, we then have the following result. Lemma 3. For every L ∈ (0, 1), the relaxed problem (10) (respectively [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF], for any N ∈ IN * ) has at least one solution a * ∈ U L (respectively a N ∈ U L ).

Proof of Proposition 1

Considering the functions a(•) of U L as controls, and interpreting the problem (8) as an optimal control problem, leads to consider the control system

y (x) = a(x), y j (x) = e 2j 2 T T 0 θ T j (t) 2 dt a(x) sin 2 (jx), j ∈ {1, . . . , N }, z (x) = 0, (14)
for almost every x ∈ [0, π], with initial conditions

y(0) = 0, y j (0) = 0, j ∈ {1, . . . , N }. (15)
The additional function z above stands for the cost functional J N (a) and will be defined with the help of inequality constraints below since it is written as the minimum of the quantities y j (π) over j ∈ {1, . . . , N }.

The relaxed problem [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] is then equivalent to the optimal control problem of determining a control a ∈ U L steering the control system (14) from the initial conditions [START_REF] Hironaka | Subanalytic sets, in: Number Theory, Algebraic Geometry and Commutative Algebra[END_REF] to the final condition

y(π) = Lπ, (16)
and maximizing the quantity z(π) (or similarly z(0), since z in constant on [0, π]), with the additional final conditions z(π) y j (π), for all j ∈ {1, . . . , N }.

Indeed, this follows directly from the observation that the unique solution of

max{z | z y j (π), j ∈ {1, . . . , N }} is z = min 1 j N y j (π).
Therefore, a * is a solution of the optimal control problem above. The existence of an optimal control is standard. According to the Pontryagin Maximum Principle (see [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]), if a is optimal then there exist real numbers4 (p y , p 1 , . . . , p N ) ∈ IR -× IR N + \(0, . . . , 0), such that

a(x) = 1 if ϕ N (x) > 0, 0 if ϕ N (x) < 0, (18)
for almost every x ∈ [0, π], where the so-called switching function ϕ N is defined by

ϕ N (x) = p y + N j=1 e 2j 2 T T 0 θ T j (t) 2 dt p j sin 2 (jx). (19)
Moreover, the control a(•) is nonsingular (see [START_REF] Trélat | Contrôle optimal, théorie & applications (French) [Optimal control, theory and applications[END_REF]) since ϕ N is a finite trigonometric sum and thus cannot be constant on any subset of positive measure. In particular, this implies that the optimal control a N is the characteristic function of a measurable subset ω N (L) of [0, π] of measure Lπ.

Note that the minimum of ϕ N on [0, π] is reached at 0 and π, hence from (18) the optimal set ω N does not contain 0 and π.

To prove uniqueness, according to the previous discussion where it is stated that every maximizer of J over U L is the characteristic function of some subset of [0, π], assume that there exist two distinct minimizers χ ω1 and χ ω2 in U L . As a maximum of linear functionals, the functional a → J (a) is convex on U L , and it follows that for every t ∈ (0, 1) the function tχ ω1 + (1 -t)χ ω2 is also a solution of the problem [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF], which is in contradiction with the fact that any solution of this problem is extremal.

Finally, the fact that ω N (L) has at most N connected components follows from the facts that the elements of ∂ω N (L) are the solutions of ϕ N (x) = 0 and that ϕ N can be written as

ϕ N (x) = p y + 1 2 N j=1 e 2j 2 T T 0 θ T j (t) 2 dt p j - 1 2 N j=1 e 2j 2 T T 0 θ T j (t) 2 dt p j T 2j (cos x),
where T 2j denotes the 2j-th Chebychev polynomial of the first kind. The degree of the polynomial ϕ N (arccos X) (in the argument X) is at most 2N , whence the result.

Proofs of Theorems 1 and 2

The main idea of this proof is close to the one of [START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF]Theorem 1]. According to Lemma 3, the relaxed optimal design problem (10) has at least one solution a * ∈ U L . We will prove simultaneously Theorems 1 and 2, by showing that a * coincides with the solution a N of the truncated problem [START_REF] Burq | Large-time dynamics for the one-dimensional Schrödinger equation[END_REF] for N large enough. First of all, as a consequence of [27, Lemma 6], we have π 0 a * (x) sin 2 (jx) dx Lπ-sin(Lπ)

2

, and therefore,

e 2j 2 T T 0 θ T j (t) 2 dt π 0 a * (x) sin 2 (jx) dx e 2j 2 T (Lπ -sin(Lπ)) 2 T 0 θ T j (t) 2 dt (20)
and moreover, there exists

N 0 ∈ IN * such that inf j>N0 e 2j 2 T T 0 θ T j (t) 2 dt Ω a * (x) sin 2 (jx) dx > e 2T T 0 θ T 1 (t) 2 dt . ( 22
)
Since there holds in particular

J N0 (a * ) e 2j 2 T T 0 θ T j (t) 2 dt Ω a * (x) sin 2 (x) dx e 2j 2 T T 0 θ T j (t) 2 dt
, we infer from [START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF] that

J (a * ) = min J N0 (a * ), inf j>N0 e 2j 2 T T 0 θ T j (t) 2 dt Ω a * (x) sin 2 (jx) dx = J N0 (a * ).
Let us actually prove that J (a * ) = J N0 (a N0 ), where a N0 ∈ U L denotes the unique maximizer of J N0 , as stated in Lemma 3. Since a N0 maximizes J N0 over U L , one has J (a * ) = J N0 (a * ) J N0 (a N0 ). Let us argue by contradiction and assume that J N0 (a * ) < J N0 (a N0 ). For every t ∈ [0, 1], we set a t = a * + t(a N0 -a * ). Since J N0 is concave (as an infimum of linear functionals), we get

J N0 (a t ) (1 -t)J N0 (a * ) + tJ N0 (a N0 ) > J N0 (a * ) = J (a * ),
for every t ∈ (0, 1], which means that inf

1 j N0 e 2j 2 T T 0 θ T j (t) 2 dt Ω a t (x) sin 2 (jx) dx > inf 1 j N0 e 2j 2 T T 0 θ T j (t) 2 dt Ω a * (x) sin 2 (jx) dx J (a * ), (23)
for every t ∈ (0, 1]. Besides, for every ε > 0 there exists t > 0 small enough such that

e 2j 2 T T 0 θ T j (t) 2 dt Ω a t (x) sin 2 (jx) dx (1 -t) e 2j 2 T T 0 θ T j (t) 2 dt Ω a * (x) sin 2 (jx) dx e 2T T 0 θ T 1 (t) 2 dt + ε, for every j > N 0 . Therefore, inf j>N0 e 2j 2 T T 0 θ T j (t) 2 dt Ω a t (x) sin 2 (jx) dx > e 2T T 0 θ T 1 (t) 2 dt . ( 24
)
Since there holds in particular J N0 (a t )

e 2T T 0 θ T 1 (t) 2 dt
, we infer from ( 23) and ( 24) that J (a t ) = J N0 (a t ) > J (a * ), which contradicts the optimality of a * .

Therefore J N0 (a * ) = J (a * ) = J N0 (a N0 ), whence the result.

Estimate of the integer N 0 . It remains to provide an estimate for N 0 . We claim that any nonzero integer N 0 such that the inequality inf

j> N0 e 2j 2 T T 0 θ T j (t) 2 dt Ω χ ω N (x) sin(jx) 2 dx > e 2T T 0 θ T 1 (t) 2 dt
, holds true satisfies N 0 N 0 (in the sequel, we denote by N 0 any integer such that the sequence (ω N ) N N0 remains constant).

To prove this claim, let us consider the simple case where T 1. Notice that, in the next explanations, the lower bound on the time T is not a restriction of our approach and can be chosen as small as desired with a slight adaptation of the following arguments. It is possible to perform more precise computations since in this case, we know at the same time several properties on the involved biorthogonal sequences (θ T j ) j∈IN * as well as the useful spectral property: for all j ∈ IN * , one has

π 0 χ ω N (x) sin 2 (jx) dx Lπ-sin(Lπ) 2
according to [27, Lemma 6]. As a consequence, following the proof of Theorem 1 and by using in particular [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF], N 0 can be chosen to be any integer such that

∀j N 0 , e 2j 2 T θ T j 2 L 2 (0,T ) πL -sin(πL) 2 e 2T θ T 1 2 L 2 (0,T )
.

According to [22, Theorem 3.2], there holds θ T

1 2 L 2 (0,T ) e 2
64 , and we infer that N 0 can also be chosen such that 1) .

∀j N 0 , θ T j 2 L 2 (0,T ) e 2 (πL -sin(πL)) 128 e 2T (j 2 -
It remains to provide an upper bound of the quantity θ T j for any j ∈ IN * . To this aim, we will use that for a given j ∈ IN * , the mapping v : [0, T ] t → (-1) j 2j e -j 2 T θ T j (t) is the control of minimal L 2 (0, T )-norm for the boundary control problem of steering the system

∂ t ϕ(t, x) -∂ xx ϕ(t, x) = 0, (t, x) ∈ (0, T ) × (0, π), ϕ(t, 0) = 0, ϕ(t, π) = v(t), t ∈ (0, T ), (25)
with initial datum ϕ(0, x) = sin(jx) to zero in time T , as highlighted in [22, Proposition 2.2].

Consider the particular control function w j vanishing on the time interval [0, T -1] and equal to the control constructed with the help of the Hilbert Uniqueness Method (HUM) for the control problem of steering the system (25) with initial datum ϕ(0, x) = e -j 2 (T -1) sin(jx) to zero in time 1. The existence of w j is well-known and we refer for instance to [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF][START_REF] Zuazua | Controllability and Observability of Partial Differential Equations: Some results and open problems[END_REF]. More generally, the controllability property of (25) also implies the existence C > 0 that does not depend on j nor T such that w j L 2 (0,T ) Ce -(T -1)j 2 for all j ∈ IN * since the sequence of functions (x → sin(jx)) j∈IN * is uniformly bounded in L 2 (0, π). We thus infer that

θ T j L 2 (0,T ) 2Cje j 2
for every j ∈ IN * , and therefore, it suffices to choose N 0 such that j N 0 ⇒ 2Cje j 2 e 2 (πL -sin(πL)) 128 e 2T (j 2 -1) .

This estimate shows in particular that N 0 and thus N 0 are equal to 1 if T is large enough.

Generalization to parabolic distributed parameter systems, and lumped control

In this section, we generalize the results obtained previously for the one-dimensional heat equation, to a large family of parabolic systems. In a second step, we consider an alternative way of acting on the system, by means of lumped controls.

Problem setting

Let n ∈ IN * be an integer, and let Ω be a bounded open connected subset of IR n . We consider the internally controlled parabolic distributed parameter system

∂ t y + A 0 y = χ ω u, t ∈ (0, T ), (26)
where

A 0 : D(A 0 ) → L 2 (Ω, C
) is a densely defined operator that generates a strongly continuous semigroup on L 2 (Ω, C), u ∈ L 2 ((0, T ) × Ω, C) is the control function, and ω ⊂ Ω is a measurable subset standing for the control domain. We assume that there exists an orthonormal basis (φ j ) j∈IN * of L 2 (Ω, C) consisting of eigenfunctions of A 0 , associated with (complex) eigenvalues (λ j ) j∈IN * such that Re(λ 1 )

• • • Re(λ j ) • • • .
The one-dimensional heat equation investigated previously enters into this frame, but now the setting is much more general.

The objective of this section is to give a precise sense to the question of optimizing the control domain ω. As a first remark, let us note that, since the equation is parabolic and thus has smoothing properties, we focus on the exact null controllability problem, that is the problem of steering the system from any initial condition (in an appropriate functional space) to zero, within a time T > 0.

We use the moment method in order to derive a relevant model of optimal sensor shape and location with results valuable for almost every initial data. This method provides a way of constructing a control achieving exact null controllability, for some given initial data y 0 ∈ L 2 (Ω). As explained below, this approach suffers however from restrictions related to the Müntz-Szász theorem, and then cannot be applied to any parabolic system.

We address this control problem in the framework developed in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equation in one space dimension[END_REF] (see also the survey [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF]) where the controllability problem is reduced to a moment problem which is solved explicitly with the help of a biorthogonal sequence to the family of exponential functions Λ = (e -λj t ) j 1 .

Consider the control system [START_REF] Privat | Complexity and regularity of maximal energy domains for the wave equation with fixed initial data[END_REF] with the initial data

y(0) = y 0 = j∈IN * a j φ j ∈ L 2 (Ω). (27)
The moment method provides a control steering the parabolic system (26) to zero, as stated in the following result.

Lemma 5. We define formally the function u by

u(t, x) = - j∈IN * a j e -λj T ω |φ j (y)| 2 dy θ T j (T -t)φ j (x), (28)
for almost every t ∈ (0, T ) and every x ∈ Ω. If this series defines a function of L 2 ((0, T ) × Ω), then this control is a solution of the problem of steering the system (26) from y 0 to 0 in time T .

The proof of this lemma is done in Section A.1.

Remark 1. Recall that such a biorthogonal sequence exists if and only if the family Λ is minimal, that is, every element t → e -λj t lies outside of the closure in L 2 (0, T ) of the vector space spanned by all other elements t → e -λ k t , with k = j. If this condition is fulfilled, then this biorthogonal sequence is uniquely determined if and only if the family Λ is complete in L 2 (0, T ). It is well known, by the Müntz-Szász theorem, that the family Λ is complete in L 2 (0, T ) (but not independent) if and only if

j∈IN * 1 Re(λ j ) + λ = +∞,
for some real number λ such that Re(λ j ) + λ > 0 for every j ∈ IN * (for instance, λ = -Re(λ 1 ) + 1 is suitable). On the contrary, if this series is convergent then the closure of the span of Λ is a proper subspace of L 2 (0, T ), moreover Λ is minimal and thus a biorthogonal sequence exists.

Then, here, we are led to assume that the series is convergent, which is a quite strong restriction on the parabolic system under consideration.

For every y 0 ∈ L 2 (Ω), we set Γ ω (y 0 ) = χ ω u, where u is the control defined by ( 28), steering the system (26) from y 0 to 0 in time T . This defines an operator Γ ω : L 2 (Ω) → L 2 ((0, T ) × Ω), called the moment control operator, which is linear and continuous. Its norm is Γ ω = sup{ Γ ω (y 0 ) L 2 ((0,T )×Ω) | y 0 L 2 (Ω) = 1}. As in the previous section, we randomize the Fourier coefficients of a given y 0 ∈ D(A 0 ), with y 0 = +∞ j=1 a j φ j , by defining a ν j = β ν j a j for every j ∈ IN * , where (β ν j ) j∈IN * is a sequence of independent real-valued random variables on a probability space (X , A, P) having mean equal to 0, variance equal to 1, and a super exponential decay (for instance, independent Bernoulli random variables). Then we define

K(χ ω ) = sup y 0 L 2 (Ω) =1 E Γ ω (y 0 ν ) 2 L 2 ((0,T )×Ω) ,
where y 0 ν = +∞ j=1 β ν j a j φ j , and E is the expectation over the space X with respect to the probability measure P. Lemma 6. There holds

K(χ ω ) = inf j∈IN * γ j (T ) ω |φ j (x)| 2 dx -1
, where the coefficients γ j (T ) are defined by

γ j (T ) = e 2Re(λj )T T 0 θ T j (t) 2 dt , ( 29
)
for every j ∈ IN * .
This lemma is proved in Section A.2. As discussed previously, we model the best actuator shape and placement problem as the problem of minimizing K over the set U L defined by

U L = {χ ω ∈ L ∞ (Ω, {0, 1}) | ω ⊂ Ω measurable, |ω| = L|Ω|}. ( 30
)
According to Lemma 6, the problem of optimal actuator placement is equivalent to the problem sup

χω∈U L inf j∈IN * γ j (T ) ω |φ j (x)| 2 dx, (31)
where the coefficients γ j (T ) are defined by [START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF]. In what follows, we define

J (χ ω ) = inf j∈IN * γ j (T ) ω |φ j (x)| 2 dx,
for every measurable subset ω ⊂ Ω.

Main result and examples

We consider the following assumptions.

(H 2 ) For every a ∈ L ∞ (Ω; [0, 1]) such that Ω a(x) dx = L|Ω|, one has lim inf j→+∞ γ j (T ) Ω a(x)|φ j (x)| 2 dx > γ 1 (T );
(H 3 ) The eigenfunctions φ j are analytic in Ω.

These assumptions have been considered as well in [START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF] and are commented in that reference. For instance, they are satisfied for A 0 = (-) α with α > 1 2 and is the Dirichlet-Laplacian on a piecewise C 1 domain Ω (see [START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF]Section 2.4]).

The problem ( 31) is similar to the optimal design problem ( 7), except that now the weights γ j (T ) are defined by ( 29). It appears then important to estimate the asymptotics of γ j (T ) as j tends to +∞. But this has been done in [START_REF] Avdonin | Families of exponentials. The method of moments in controllability problems for distributed parameter systems[END_REF][START_REF] Fattorini | Exact controllability theorems for linear parabolic equation in one space dimension[END_REF][START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF][START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF]. Those estimates will impose further restrictions on the problem under consideration.

For every N ∈ IN * , we define the truncated criterion

J N (χ ω ) = inf 1 j N γ j (T ) ω |φ j (x)| 2 dx,
for every measurable subset ω ⊂ Ω. We have the following result.

Proposition 2. Let N ∈ IN * . Under (H 1 ), the problem sup χω∈U L J N (χ ω ) ( 32
)
has a unique solution χ ω N in U L . Moreover, under the additional assumption

(H 3 ), ω N is an open semi-analytic 5 set.
This proposition is proved in Section A.3. The main result is then the following theorem, proved in Section A.4. Theorem 3. Assume that there exist m 1 > 0, m 2 ∈ (0, 2T ), and a sequence (θ T j ) j∈IN * biorthogonal to the family Λ = (t → e -λj t ) j 1 , such that

θ T j 2 L 2 (0,T ) m 1 e m2Re(λj ) , (33)
for every j ∈ IN * . Then, under (H 1 ) and (H 2 ), the problem (31) has a unique solution χ ω * ∈ U L . Moreover there exists N 0 ∈ IN * such that ω * = ω N , for every N N 0 . In particular, if (H 3 ) is moreover satisfied, then ω * is an open semi-analytic subset of Ω, and thus, it has a finite number of connected components.

The same considerations as in Section 2 on the algorithmic computation procedure still hold in this general framework.

To finish, we provide hereafter some classes of examples for which the existence of a biorthogonal sequence satisfying ( 33) is known.

• Assume that there exist δ > 0, β > 1, ε > 0, A 0 and B δ such that

|λ j -λ k | δ|j β -k β | and ε(A + Bj β ) |λ j | < A + Bj β , (34)
for all (j, k) ∈ (IN * ) 2 , where the elements of the sequence (λ k ) k∈IN * are assumed to lie in {λ ∈ C | | arg λ| θ} for some given θ ∈ (0, π/2). As argued in Remark 1, under the condition [START_REF] Trélat | Contrôle optimal, théorie & applications (French) [Optimal control, theory and applications[END_REF] there exists a sequence (θ T j ) j∈IN * biorthogonal to Λ, and it is proved in [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF] that there exist two positive constants à and B such that

θ T j 2 L 2 (0,T ) Be Ãj ,

and since

Re(λ j ) |λ j | cos θ ε(A + Bj β ) cos θ for every j ∈ IN * , we infer the existence of m 1 and m 2 such that the estimate [START_REF] Sion | On general minimax theorems[END_REF] holds.

We also refer to [START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF]Theorem 3.2] for an elementary proof of [START_REF] Sion | On general minimax theorems[END_REF] for the eigenvalues the one-dimensional Dirichlet Laplacian operator.

For example, assume that A 0 = (-) α is a positive power of the one-dimensional Dirichlet-Laplacian on Ω = (0, π); then (34) is satisfied if and only if α > 1/2.

In [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF] other examples are provided where [START_REF] Trélat | Contrôle optimal, théorie & applications (French) [Optimal control, theory and applications[END_REF] is satisfied, such as the damped Euler-Bernoulli plate in dimension two.

• Assume that (λ n ) n∈IN * is a sequence of positive real numbers and that there exist K > 0, α > 0 and β > 1 such that

λ n = K(n + α) β + o(n β-1 ),
as n tends to +∞. It is proved in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equation in one space dimension[END_REF]Formula (3.25)] that there exists two constants à and B such that

θ T j 2 L 2 (0,T )
Ãe Bλ 1/β j for every j ∈ IN * and the estimate (33) then holds true. Note that the authors of [START_REF] Fattorini | Exact controllability theorems for linear parabolic equation in one space dimension[END_REF] use it to derive exact controllability results for a Sturm-Liouville one-dimensional equation. We also mention the article [START_REF] Ammar-Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF] where the authors extend the above approach and estimate to the framework of systems of one-dimensional parabolic equations, in view of establishing exact boundary controllability properties.

• Assume that A 0 is the Dirichlet-Laplacian on the unit ball Ω = {x ∈ IR n | x < 1}, with n arbitrary. Using a refined study of the sequences of eigenfunctions and eigenvalues, it is proved in [10, Section 6, (6.27)] that (33) holds true with a constant m 2 not depending on T , and the authors use it to investigate boundary controllability issues for the heat equation in Ω. Then Theorem 3 can be applied, provided that T is large enough (since it is required that m 2 ∈ (0, 2T )).

Optimal lumped controls

In this section, we investigate a variant of the previously studied optimal design problem, based on another kind of controls referred to in the literature as the lumped controls (see [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF]Chapter 4] or [START_REF] Khapalov | Controllability of partial differential equations governed by multiplicative controls[END_REF]Chapter 1.4]). This wording designates tensorized controls that are the product of separated variables functions in time and space, the space profile of the control term being given. Then one only acts on the system by means of tuning the time-intensity of the control.

Let Ω be an open connected subset of IR n and A 0 : D(A 0 ) → L 2 (Ω, C) be a densely defined operator that generates a strongly continuous semigroup on L 2 (Ω, C). We adopt the same framework as in Section 3.1, assuming the existence of an orthonormal basis (φ j ) j∈IN * of L 2 (Ω, C) consisting of eigenfunctions of A 0 , associated with (complex) eigenvalues (λ j ) j∈IN * such that Re(λ 1 )

• • • Re(λ j ) • • • .
Consider the internally controlled parabolic system

∂ t y(t, x) + A 0 y(t, x) + g(x)u(t) = 0, (t, x) ∈ (0, T ) × Ω, (35)
with Dirichlet boundary conditions, where g ∈ L 2 (Ω, C) is the control profile and u ∈ L 2 (0, T ) is the control function. The controlled system ( 35) is a particular version of (1).

In some sense, the function g plays the role of χ ω in ( 26), but here, the control function u depends only on t. The function g is usually fixed and the control is u. Here, we propose to optimize the control profile g.

Performing the same analysis as in Section 1.1 and using the same notations, one proves easily that every initial datum y 0 = +∞ j=1 a j φ j ∈ L 2 (Ω) can be steered to zero in time T with the control u ∈ L 2 (0, T ) given by

u(t) = - +∞ j=1 a j e -λj T Ω g(y)φ j (y) dy θ T j (T -t),
provided that the Fourier coefficients Ω g(y)φ j (y) dy of g do not vanish.

As previously, we define the moment control operator Γg : L 2 (Ω) → L 2 ((0, T ) × Ω) by Γg (y 0 ) = f , with f (t, x) = g(x)u(t). Its norm is given by Γg = sup

y 0 L 2 (Ω) =1 Γg (y 0 ) L 2 ((0,T )×Ω) = g L 2 (Ω) sup y 0 L 2 (Ω) =1 u L 2 (0,T )
Following the framework developed in Sections 1.1 and 3.1 leads to define a randomized criterion by defining a ν j = β ν j a j for every j ∈ IN * . Then we define Kg (χ ω ) = sup

y 0 L 2 (Ω) =1 E( Γg (y 0 ν ) 2 L 2 ((0,T )×Ω) ),
where y 0 ν denotes the function of L 2 (Ω) whose Fourier coefficients are the a ν j defined above. Lemma 7. There holds

Kg (χ ω ) = sup j∈IN * e -2Re(λj )T T 0 θ T j (t) 2 dt Ω g(x)φ j (x) dx 2 Ω |g(x)| 2 dx.
The proof is similar to the proofs of Lemmas 1 and 5, and thus is skipped. We model the "best design of lumped controller" as the problem of minimizing Kg (χ ω ) over the set of all possible profiles g ∈ L 2 (Ω). The functional g → Kg (χ ω ) being homogeneous according to the previous lemma, the problem of optimal lumped control placement is then equivalent to the problem sup

g L 2 (Ω) =1 inf j∈IN * γ j (T ) Ω g(x)φ j (x) dx 2 , (36)
with

γ j (T ) = e 2Re(λj )T T 0 θ T j (t) 2 dt
, for every j ∈ IN * . Let us now solve this optimal design problem.

Theorem 4. We assume that

+∞ j=1 1 γ j (T ) < +∞. ( 37
)
Then, the problem (36) has at least one solution, and we have

sup g L 2 (Ω) =1 inf j∈IN * γ j (T ) Ω g(x)φ j (x) dx 2 =   +∞ j=1 1 γ j (T )   -1 .
Moreover, the set of solutions consists of all functions g in L 2 (Ω) that can be expanded as

g = +∞ j=1 g j φ j with |g j | 2 =   +∞ j=1 1 γ j (T )   -1 1 γ j (T )
.

This theorem is proved in Section A.5.

Remark 2. Consider the case where A 0 = -∂ xx is defined on H 2 (0, π)∩H 1 0 (0, π) (one-dimensional Dirichlet-Laplacian). Then φ j (x) = 2 π sin(jx) and λ j = j 2 for every j ∈ IN * . Denote by g any solution of the problem [START_REF] Ucinski | Sensor network design for the estimation of spatially distributed processes[END_REF]. According to [START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF]Theorem 3.2], there exists a positive constant C T such that

e 2j 2 T T 0 θ T j (t) 2 dt C T e 2j 2 T -2πj ,
for every j ∈ IN * . According to Theorem 4, it follows that the Fourier coefficients g j decrease exponentially with respect to j, and as a consequence, the optimal functions g are analytic (see e.g. [1, Chapter 11, §63]).

Remark 3. It might seem natural and of physical interest to investigate what happens if we restrict our search of the control profile g to a set of characteristic functions of a measurable subset ω, with the measure of ω possibly fixed. Doing this, we get a kind of instability: indeed, assuming that ω is the finite union of rational intervals (in other words, intervals whose extremities are rational multiples of π), one can easily check that inf j∈IN * γ j (T )

π 0 χ ω (x) sin(jx) dx 2 = 0.
Therefore, this problem appears to be ill-posed in some sense, and is probably not so much relevant with respect to practical issues.

Conclusion

To conclude, let us provide several further comments and open problems.

Generalization to other methods of control and higher dimensions. As underlined in the previous sections, the use of controllers obtained by the moment method reduces mainly the perimeter of our study to one-dimensional operators.

In view of generalizing our approach to other control operators, let us use the framework described in Section 3.1, considering the controlled system

∂ t y + A 0 y = χ ω u, t ∈ (0, T ), (38)
where y(0, •) = y 0 ∈ L 2 (Ω) and u is a control steering this system from y 0 to 0 in time T , whenever it is possible. Let us assume that for every T > 0 and every Lebesgue measurable subset ω of positive measure, the system ( 38) is null-controllable in time T . In this case, let us write Γ ω = χ ω u.

For instance, the Hilbert Uniqueness Method (see [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]) is a well-known method used to design a null control for (1)-( 2), with the additional property that this control has a minimal L 2 norm over all possible null controls. The null-controllability property in time T of this system is equivalent to an observability property on the pair (ω, T ). Note that in the case where A 0 is the Dirichlet-Laplacian operator -, it has been showed in [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF] that the observability inequality holds true for every T > 0 and every Lebesgue measurable subset ω of positive measure.

Following the approach described in Section 3.1, we define

K(χ ω ) = sup y 0 L 2 (Ω) =1 E Γ ω (y 0 ν ) 2 L 2 ((0,T )×Ω) ,
where y 0 ν = +∞ j=1 β ν j a j φ j , and E is the expectation over the space X with respect to the probability measure P. Similar computations as those of Section A.2 enable to show that

K(χ ω ) = sup j∈IN * Γ ω (φ j ) 2 L 2 ((0,T )×Ω)
As discussed previously, we model the best actuator shape and placement problem as the problem of minimizing K over the set U L . Analyzing this optimal design problem does not seem easy since it requires to know fine regularity properties of each control function u j defined by Γ ω (φ j ) = χ ω u j .

Analysis of the full control operator. One of the main issues that remains to be developed is whether one can attack the problem of the optimal design of the controllers and actuators without the diagonalization procedure by randomization. The issue is then much harder to handle, as it occurs at the level of the observability problem. Note also that in that case, because of possible interactions of all modes, it is unclear how complex the optimal sets are.

Actually, if one defines the Gramian operator G T as the infinite dimensional symmetric nonnegative matrix whose coefficient at row j and column k is given by T 0 e (j 2 +k 2 )t dt ω sin(jx) sin(kx) dx, the operator norm Γ ω is the inverse of the smallest eigenvalue of G T . The randomization procedure consists in dropping the non-diagonal terms in G T , by considering the inverse of smallest eigenvalue of diag(G T ).

Concerning the particular case of controllers given by the Hilbert Uniqueness Method (see [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]) to design a null control for (1)-( 2), minimizing the control efforts in a deterministic way is actually equivalent to maximizing

C T (χ ω ) = inf T 0 ω |y(t, x)| 2 dx dt y(T, •) 2 L 2 (0,π) y(0, •) ∈ L 2 (0, π) \ {0} ,
which is the largest possible observability constant C in the inequality (3), over U L , because of the duality between controllability and observability. An interesting problem then consists of maximizing the functional C T (χ ω ) over the set U L . This problem has been discussed in [START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF][START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF], and for the same reasons as above it has appeared more relevant to introduce the concept of randomized observability constant C T,rand (χ ω ). At this step, one may think of coming back, by duality, to the controllability problem. Unfortunately, the problem of maximizing C T,rand (χ ω ) does not admit any nice interpretation in terms of controlling, say, almost every initial data to 0 in time T . This is due to the fact that the randomization procedure does not commute with the duality operator realizing the duality between observability and controllability.

More precisely, the Gramian G T defined above does not commute with the randomization procedure. To describe which kind of initial data can be steered to 0 in a random way, it would be required to compute the image under G T of the random laws used in the randomization procedure, and then show that these random laws share appropriate probability properties, as in [START_REF] Burq | Random data Cauchy theory for supercritical wave equations. I. Local theory[END_REF].

Hence, here, we have found more relevant to combine the randomization procedure with the moment method, in which case the problem of the lack of commutation arising in the HUM procedure disappears.

Use of other biorthogonal families. We have here used the moment problem approach but in a very special way, taking advantage of the fact that eigenvalues grow sufficiently fast so to ensure the existence of a family of time-biorthogonals that allow to build by separation of variables biorthogonal families for all possible supports of the control ω. Of course the issue can be formulated without that restrictive assumption taking advantage of the existence of biorthogonal families in the (x, t) variables, in other words of families Λ

= (θ ω,T k ) k∈IN * such that T 0 ω θ ω,T k (t, x)e -λj t φ j (x) dxdt = δ jk .
But their dependence with respect to ω seems to be hard to analyze.

A Proofs

In what follows and similarly to what has been done in Section 2, we will consider a convexified version of the problem [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF] to overcome the difficulty related to the non-compactness of the set U L defined by [START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF] for the L ∞ weak-star topology. We refer to this section for more comments on this procedure. The convex closure of U L for the weak star topology of L ∞ is

U L = a ∈ L ∞ (Ω; [0, 1]) | Ω a(x) dx = L|Ω| . (39)
Replacing χ ω ∈ U L with a ∈ U L , we consider the convexified formulation of the problem [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF] given by sup

a∈U L J (a), (40)
where the functional J is naturally extended to U L by

J (a) = inf j∈IN * γ j (T ) Ω a(x)|φ j (x)| 2 dx, (41)
for every a ∈ U L . We consider as well a relaxed formulation of the truncated optimal design problem (32) by sup

a∈U L J N (a), (42)
where the functional J N is naturally extended to U L by

J N (a) = inf 1 j N γ j (T ) Ω a(x)|φ j (x)| 2 dx, (43)
for every a ∈ U L .

We have the following result.

Lemma 8. For every L ∈ (0, 1), the relaxed problem (40) (respectively (42), for any N ∈ IN * ) has at least one solution a * ∈ U L (respectively a N ∈ U L ).

We refer to Section 2.1 for a proof of this result.

A.1 Proofs of Lemmas 1 and 5

We prove Lemma 5, which is a generalization of Lemma 1.

We seek a control function u = u(t, x) in L 2 ((0, T ) × Ω) achieving the null controllability for the system [START_REF] Privat | Complexity and regularity of maximal energy domains for the wave equation with fixed initial data[END_REF] with the initial condition y(0, x) = y 0 (x) = +∞ j=1 a j φ j (x), that is, such that y(T, •) = 0. Setting y(t, x) = +∞ j=1 y j (t)φ j (x), we get

y j (T ) = e -λj T a j + T 0 ω e -λj (T -t) u(t, x)φ j (x) dx dt,
for every j ∈ IN * . In order to realize the null controllability in time T , the control u must be such that

T 0 ω e -λj (T -t) u(t, x)φ j (x) dx dt = -a j e -λj T , (44)
for every j ∈ IN * . In order to solve these equations, assume that there exists a sequence (θ T j ) j∈IN * of functions biorthogonal to the family Λ, that is,

T 0 e -λj t θ T k (t) dt = δ jk , for all (j, k) ∈ (IN * ) 2
, where δ jk = 1 whenever j = k, and δ jk = 0 otherwise. Then the function u defined by ( 28) is a formal solution of the moment problem (44).

A.2 Proof of Lemmas 2 and 6

We randomize the initial datum y 0 (x) = +∞ j=1 a j φ j (x) according to y ν 0 (x) = +∞ j=1 β ν j a j φ j (x). Then, the corresponding control u = Γ ω (y ν 0 ) coming from the moment method, steering y ν 0 to 0 in time T , is The result follows.

A.3 Proof of Proposition 2

This proof is similar to the one of [START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF]Proposition 2]. We include it in the present paper, for the sake of completeness and readability. For every N ∈ IN * , we consider the relaxed truncated problem (43), where the functional J N is defined by (42). Using the same arguments as in the proof of Lemma 8, it is clear that the problem (43) has at least one solution a N ∈ U L . Let us prove that a N is the characteristic function of a set ω N such that χ ω N ∈ U L . Defining the simplex set S N = α = (α j ) 1 j N ∈ [0, 1] N | N j=1 α j = 1 , it follows from the Sion minimax theorem (see [START_REF] Sion | On general minimax theorems[END_REF]) that We set ϕ N (x) = N j=1 α N j γ j (T )|φ j (x)| 2 , for every x ∈ Ω. It follows from (H 1 ) that ϕ N is never constant on any subset of Ω of positive measure. Therefore, there exists λ N such that a N (x) = 1 whenever ϕ N (x)

λ N , and a N (x) = 0 otherwise. In other words, a N = χ ω N ∈ U L , with

ω N = {x ∈ Ω | ϕ N (x) > λ N }.
The uniqueness of a N follows from the fact that, as proved above, any optimal solution is a characteristic function. Indeed if there were two optimal sets, then any convex combination would also be an optimal solution because J N is concave. This raises a contradiction since any maximizer has to be a characteristic function.

Under the additional assumption (H 3 ), the function ϕ N is analytic in Ω and therefore ω N is an open semi-analytic set.

A.4 Proof of Theorem 3

This proof is a immediate adaptation of the one of Theorem 1.

Indeed, notice that according to the assumption (H 2 ), the estimate [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF], constituting the starting point of the proof of Theorem 1, can be replaced by lim inf j→+∞ γ j (T ) Ω a * (x)|φ j (x)| 2 dx > γ 1 (T ), where a * stands for a solution of the relaxed problem (40).

The rest of the proof is then unchanged.

A.5 Proof of Theorem 4

We expand g = +∞ j=1 g j φ j , with g j = Ω g(x)φ j (x) dx for every j ∈ IN * . Note that, since g L 2 (Ω) = 1, we have Therefore, the optimal value for the problem [START_REF] Ucinski | Sensor network design for the estimation of spatially distributed processes[END_REF] coincides with the optimal value of a convexified problem, as follows. Writing α = (α j ) j∈IN * = (g 2 j ) j∈IN * , there holds

2 0 2 0 2 0 2 0 2 0

 22222 the Heat equation (Dirichlet case) with N=1, T=0.05 and L=0.the Heat equation (Dirichlet case) with N=2, T=0.05 and L=0.the Heat equation (Dirichlet case) with N=3, T=0.05 and L=0.the Heat equation (Dirichlet case) with N=4, T=0.05 and L=0.domain for the Heat equation (Dirichlet case) with N=5, T=0.05 and L=0.domain for the Heat equation (Dirichlet case) with N=6, T=0.05 and L=0.domain for the Heat equation (Dirichlet case) with N=7, T=0.05 and L=0.domain for the Heat equation (Dirichlet case) with N=8, T=0.05 and L=0.2

Figure 1 :

 1 Figure 1: Ω = (0, π), L = 0.2, T = 0.05. From left to right, and top to down: optimal solution χ ω N for N = 1, . . . , 8.

a

  for every j ∈ IN * . Besides, we have the following result on the growth of the biorthogonal sequence (θ T j ) j∈IN * , following from [22, Theorem 3.2]. Lemma 4. Let T > 0. There exists C T > 0 such that C T T 0 θ T j (t) 2 dt e 2πj , for every j ∈ IN * . It follows from this result that e 2j 2 T T 0 θ T j (t) 2 dt C T e 2j 2 T -2πj , for every j ∈ IN * . Combining these two facts, we infer that lim j→+∞ * (x) sin 2 (jx) dx = +∞,

(H 1 )

 1 (Strong Conic Independence Property) If there exist a subset E of Ω of positive Lebesgue measure, an integer N ∈ IN * , a N -tuple (α j ) 1 j N ∈ (IR + ) N , and C 0 such that N j=1 α j |φ j (x)| 2 = C almost everywhere on E, then there must hold C = 0 and α j = 0 for every j ∈ {1, • • • , N }.

and hence u ν 2 L 2 (

 22 u ν (t, x) = -+∞ j=1 β ν j a j e -λj T θ T j (T -t) φ j (x) ω |φ j (y)| 2 dy , (0,T )×Ω) = +∞ j,k=1 β ν j β ν k a j āk e -(λj + λk )T T 0 θ T j (t)θ T k (t) dt ω φ j (x)φ k (x) dx ω |φ j (x)| 2 dx ω |φ k (x)| 2 dx , and therefore, E u ν 2 L 2 ((0,T )×Ω) = +∞ j=1 |a j | 2 e -2Re(λj )T ω |φ j (x)| 2 dx T 0 θ T j (t) 2 dt.

  γ j (T ) Ω a(x)|φ j (x)| 2 dx = max a∈U L min α∈S N Ω a(x) N j=1 α j γ j (T )|φ j (x)| 2 dx = min α∈S N max a∈U L Ω a(x) N j=1 α j γ j (T )|φ j (x)| 2 dx,and that there existsα N ∈ S N such that (a N , α N ) is a saddle point of the functional (a, α) ∈ U L × S N -→ N j=1 α j γ j (T ) Ω a(x)|φ j (x)| 2 dx.Therefore, a N is solution of the optimal design problem max a∈U L Ω a(x) N j=1 α N j γ j (T )|φ j (x)| 2 dx.

+∞ j=1 |g j | 2 = 1 .

 21 We define the convex set S byS = β = (β j ) j∈IN×IN * ∈ 1 (IR + ) | j∈IN * β j = 1 , Note that, if (η j ) j∈IN * is a sequence of positive real numbers, then inf j∈IN * η j = inf β∈S +∞ j=1 β j η j .

sup g L 2 F 1 γj 1 ,1

 211 (0,π) =1 inf j∈IN * γ j (T ) Ω g(x)φ j (x) dx 2 (α, β),where the functional F is defined by F (α, β) = +∞ j=1 γ j (T )α j β j . In accordance with[START_REF] Van De Wal | A review of methods for input/output selection[END_REF], define the sequence λ * = (λ * j ) j∈IN * by λ * j = (T ) for every j ∈ IN * . Clearly, λ * ∈ S and we have sup α∈S inf β∈S F (α, β) sup α∈S F (α, λ * ) = and the supremum is reached if, and only if |g j | 2 = γ j (T ) for every j ∈ IN * . The conclusion follows.

Recall that the sequence (β ν j ) j∈IN * is said to have a super-exponential decay whenever∃(C, δ) ∈ (IR * + )

| ∀α ∈ IR, E(e α|β ν j | ) Ce δα 2 .

Here and in the sequel, it is understood that the optimal set is unique within the class of all measurable subsets of (0, π) quotiented by the set of all measurable subsets of Ω of zero measure.

Indeed, equality constraints in L ∞ are in general not preserved by the natural topologies such as the L ∞ weak-star topology.

Note that, since the dynamics of (14) do not depend on the state, it follows that the adjoint states of the Pontryagin Maximum Principle are constant.

A subset ω of a real analytic finite dimensional manifold M is said to be semi-analytic if it can be written in terms of equalities and inequalities of analytic functions. We recall that such semi-analytic subsets are stratifiable in the sense of Whitney (see[START_REF] Hardt | Stratification of real analytic mappings and images[END_REF][START_REF] Hironaka | Subanalytic sets, in: Number Theory, Algebraic Geometry and Commutative Algebra[END_REF]), and enjoy local finitetess properties, such that: local finite perimeter, local finite number of connected components, etc.

Acknowledgment. The first author was partially supported by the ANR project OPTIFORM. This work was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency -GA 694126, ICON of the French ANR-2016-ACHN-0014-01, FA9550-15-1-0027 of AFOSR, A9550-14-1-0214 of the EOARD-AFOSR, and the MTM2014-52347 Grant of the MINECO (Spain).


Issah W Wabuyabo

Dr Peter W Bukhala

Dr Gaudin Benoit

Issah Kweyu Wabuyabo

Socio-Cultural Determinants of Athletics Abilities Among Kenyan Elite & Sub-Elite Middle and Long Distance Runners

published or not. The documents may come

Introduction

Kenya has participated in the Commonwealth games since 1954, the Olympics since 1964, the All African Games since 1965 and the World championship since 1983. They have won a total of 86 medals at Olympics out of which 56 are in middle and long distance track event and 100 medals (38 Gold, 33 Silver and 29 Bronze) at World Championships. 2 The running tradition began with Kenya winning their first Olympic bronze medal in the800mat the 1964 Tokyo Olympic Games. A comparison of performance between Kenyan athletes and their counterparts from other nations in the Top 25 all time lists of track events (800m, 1500m, 3000m flat, 3000m steeplechase, 5000m and 10,000m) is shown in table 1 The majority of Kenyan middle and long distance runners come from the Kalenjin tribe, which has seven sub-tribes namely Kipsigis, Nandi, Tugen, Marakwet, Keiyo, Pokot, Sabaot and Terik. For instance, both Kipchoge Keino and Mike Boit are from Nandi sub-tribe, Wilson Kipketer and Moses Tanui and Tegla Loroupe are from the Pokot tribe, five-time world cross-country champion and world 10,000m record holder Paul Tergat is from the Tugen tribe while three-time world steeplechase champion Moses Kiptanui is from Marakwet. 3 The table below shows a distribution of medals tallies from various Kenyan communities at theOlympics, world championships and in the marathon. Because of this level of performance, myths have been used to explain this dominance of Kenyan middle and long distance runners. Most of the scientific studies conducted to explain the same phenomenon focused on the biological characteristics and ethnic traits of the athletes. However 3 Chesaina C.,Oral Literature of the Kalenjin, Nairobi: Heinemann Kenya, 1991. none of the studies has been able to conclusively explain what underlies the excellent performance of Kenyan middle and long distance athletes. This has prompted a social science approach in the search for answers to this phenomenon. This paper seeks to highlight the sociocultural determinants of athletic abilities among elite and sub elite Kenya middle and long distance runners.

Theoretical framework

This study will be guided by the theory of social constructionism as developed by Pierre Bourdieu's concept of the habitus. The concept of habitus refers to the durable and generalized disposition that defines a person's actions throughout an entire domain of life or mould of the personality. 4 Habits are substantially formed during early life of an individual and remain open to development as the interdependent networks of people get involved making it more complex. This is a useful framework for understanding culture according to Bourdieu. 5 Habitus can be understood as those aspects of culture that are anchored in the body or daily practices of individuals, groups, societies, and nations. It includes the totality of learned habits, bodily skills, styles, tastes, and other non-discursive knowledge that might go without being noticed. This relates to Smith's 6 view of an ethnic group whose members will share a common set of understandings, values and attitudes that reflect their common culture. A socially structured system of traits and dispositions determine people's attitudes towards the way they use their body in everyday life as well as in sport. These ideals mirror and generate gender differences. Bourdieu's 7 considerations regarding cultural capital, habitus, taste and processes of social distinction can contribute to our understanding of the challenges and difficulties of participating in sport for people of different ethnic origins. One is socialized into running early in life, which includes learning habits, bodily skills, styles, tastes, and knowledge passed on from earlier generations. As junior athletes they join the hierarchies by learning from their senior counterparts (mentors) at school or camps.

As one joins this athletics family he or she learns the skills, techniques and tactics in specific events while sharing common values and attitudes that reflect their common athletics culture and identity. This includes the myths and cultural belief about the community and the athletics culture. During training in camps one learns from the seniors as well as tapping into their social networks to enable them to access competitions locally and abroad.

During competition the runners (individually and collectively) feel under obligation to defend the identity of winning that is historical. After winning during any event the athlete returns at home 4 Van Krieken Robert, Norbert Elias, Routledge, 1998.

5 Bourdieu P., Distinction: A Social Critique of the Judgment of Taste, London, Routledge, 1984; Bourdieu P., The to a welcome and recognition as a cultural capital for the community. Consistent outcomes in athletic events especially if it reflects ethnic or gender patterns, it creates a stereotype that later influences how people perceive and use their bodies in the same community. This may explain the performance discrepancies between different ethnicities and gender. Alternatively the stereotype could also influence gender and ethnic specializations in some sports.

Myths about Kenyan Middle and Long Distance Running Superiority

There are many myths that have been used to explain Kenyan middle and long distance running superiority. Long distance running athletic ability has been attributed to biological and ethnic traits. For instance, the myth of racial superiority as fronted by Entine 8 has used genetic, racial, and environmental determinism to illustrate a genetic basis for the predominance of black athletes in sports. Saltin et al 9 describe Kenyan elite athletes as having run or walked an average of 8-12 km a day, five days a week from age 7-8 years, increasing to 90 km a week as adolescents, contrary to Bale and Sang's report 10 that 14 of 20 elite Kenyan athletes interviewed had never been required to run to school. Malik 11 argued that populations with roots in equatorial Africa are more likely to have lower natural fat levels, which is likely to be a key factor in running. Bale and Sang 12 argue that if altitude was the only factor involved in athletic performance, then all countries with high altitude populations such as Nepal, Peru, and Mexico would be producing many world class athletes in middle and long distance running like Kenya.

Onywera et al 13 studied the traditional Kenyan diet for Kenyan distant runners. They reported that the diet was composed of 10% protein, 13% fat and 77% carbohydrate which come from a traditional Kenyan maize meal (Ugali) and tea (Chai). He concluded that while the diets seem favorable for middle and long distance training and performance, the Kenyans' diets do not appear uniquely different from the training diets of most other international competitors. These myths have not explained why foreign athletes who come to train with Kenyan athletes at the same altitude and possibly eat the same foods never perform equally well during international events. High altitude is not only found in Kenya to the extent that it can be a determinant of Kenyan athletic prowess. Likewise, all runners have genes and eat food with nutrients. What the studies based on biological characteristics and ethnic traits have been unable to do is to specify which gene or food nutrient (if any) makes Kenyans good runners in middle and long distance events.

Socio-cultural factors

Traditional culture

Before 1895 cattle raiding was common among the pastoral communities like the Kalenjin tribe in Kenya. This was an activity exclusively for young men. The most fierce and dreaded of all the Kalenjin cattle raiders were the Nandi, which is the second largest of the Kalenjin sub-groups. Unlike other cattle raiders, they would often cover long distances such as 50 to 60 kilometers before striking at dawn. Apparently running abilities were a prerequisite for being a successful cattle raider. 14 Coincidentally they produce the highest number of middle and long distance runners in Kenya. Skills of running acquired from this cultural practice were very compatible with modern athletics introduced during colonial period. This quickly translated into good performances in the initial competitions and as an established culture it has been able to sustain production of elite runners from the 1920s onwards.

Colonial influence

Cattle raiding as a cultural practice posed a challenge to British colonizers in the early 19 th century. They deemed the tradition uncivilized "stock-theft". They introduced running as a safe alternative to such tribal traditions and as a means of social control. In the 1920s, the colonial administration organized local track meets. The winners were offered blankets and cooking pots as prizes. 15 These activities later improved and were performed on precisely measured 440 yard tracks in the 1940s and 1950s.

Colonial British settlers and Asian contractors formed many professional teams and clubs. This happened as early as 1922 before the establishment of formal schools. 16 The colonial government institutionalized sports in schools; Fraser 17 and the Phelps-Stokes Commission of 1924 emphasized that drills and Physical Training were some of the mandatory subjects at the elementary school level. Sports were introduced in schools in 1925. The syllabus for teaching sports in schools as an extracurricular activity was introduced in 1935. Football (soccer) and athletics (track and field) respectively were the first sports to be professionally organized in schools. 18 In 1924 the Arab, Africa Sports Association (AASA) was formed and organized the first AASA athletics meet at the racecourse in Nairobi, where teams from various districts would compete.

Athletic competitions

Kenyan athletes had early exposure to many standard competitions as early as the 1920s. The colonial police commissioner (RGB Spicer) in 1925 organized running competitions for police recruits [START_REF] Spicer | Annual reports[END_REF] in Murang'a. In 1933 Keino's father won a 6 mile race and was awarded 4 gallons of oil 20 while in 1934 the first inter-territorial athletics competition was organized between Kenya and Uganda. The championship was hosted in Kampala, Uganda. [START_REF]L'Athletisme Africain '94[END_REF] 1936 witnessed a new record set by Kamunya in a 1 mile race, while Luka Njeru set a record in the half mile race in the school championships. [START_REF] Greaves | Carey Francis of Kenya[END_REF] Kenya participated in Malayan athletic championships of 1952 where Kipsang from the King African rifles won the 3 mile race. The following year (1953), the Kenya team also participated in an athletic championship in Zambia (Northern Rhodesia) and dominated the event setting a record of 61metres in the javelin. In 1954 just before the Vancouver games, the Kenyan athletics team participated in the Indian Ocean Games in Madagascar where Nyandika Maiyoro emerged victorious in the 3000m. [START_REF] Bale | [END_REF] These activities accelerated Kenya's affiliation to the IAAF in 1954 and the creation of the Kenya Olympic Association in 1955. Kenya was able to participate in the 5 th edition of the British Empire Commonwealth Games in 1954. This prepared the country for the 1956 Melbourne Olympics where Kenya participated in athletics, hockey, sailing and shooting. Similarly in the 1960 Olympics Kenyan competitors participated in athletics, hockey sailing and shooting. During this Olympics Nyandika Mayoro competed in the men's 5000m and finished in 6 th place. 24 The Kenyan athletics team also participated in the 1962 Commonwealth Games in Perth where Antao won in 110 yard and 220 yard sprints while Kimaru Songok won the 440 yards hurdles. [START_REF] Watman | Keino 'the world's most exciting athlete[END_REF]

Sports institutions

To build on earlier activities of the Arab, African Sports Association (AASA), the Kenya Amateur Athletics Association (KAAA) was founded in 1951 to govern athletics (track &field) in Kenya. Arthur Evans is credited for starting organized athletics and forming the Kenya Amateur Athletics Association (KAAA) in 1951.Track and field activities were intensified from local village sports activities to serious competition around the Provinces. This transformation paved the way for the Kalenjin runners who also turned out to be exceptionally gifted athletes. 26 In 1952 KAAA organized an event between Uganda Protectorate and Tanganyika Territory, the East African Territorial Athletics Championships. The Kenyan men performed well compared with the women and dominated the championships. The championships became annual forming the cornerstone of the East Africa Athletics programme. Later other nations like Somalia, Malawi, Ethiopia, and Zambia joined the "East African Amateur Athletic Board". 27 The National Olympic Committee of Kenya (NOCK) was established in 1955 to facilitate Kenya's participation in the 1956 Olympic Games and in preparation for the 1960 Olympics, 1964 Olympics in Melbourne and the All African Games of 1965.The Kenya National Sports Council (KNSC) was established in 1966 to coordinate the activities of the National Sports Federations and also assist in the preparation of national teams for the 1968 Olympic Games.

The Presidential Decree of 1980 made Physical Education (PE) a mandatory subject in all schools and Teachers Training Colleges was a landmark decision that saw improved athletics training. 28 Physical Education classes constitute the foundation of teaching sports skills in schools. Learners who are good in a particular sport will practice during games time in specific school teams to be presented during school competitions. During competitions regional dominance in differences sports will always indicate preferences. Schools from the Rift valley dominate running events to date.

Education and learning institutions

Athletics alongside other sports is organized from grass roots institutions (primary schools) up to university level. For instance, Kenya Universities Sports Association (KUSA) is the governing body for Universities Sports in Kenya. Each university has a department in charge of running sports in the university. This is mainly to prepare university teams for the inter-university championships among other open championships in the country. Out of the KUSA championships the best teams (which meet the qualifying times) represent the country in other championships.

The Kenya Teacher Colleges Sports Association (KTCSA) has brought together teacher trainees from all colleges around the country to compete in different Sports and Games for the last 27 years. The association documents the best times recorded in each event, which guides training as a target in subsequent competitions. Kenya Secondary Schools Sports Association (KSSSA) is an association that deals exclusively with secondary school sports in Kenya. KSSSA's main tasks include, nurturingand developing sports talent when it is young, encouraging sports in schools as a co-curricular activity and developing an all rounded student both in academics and sports. Likewise Kenya Primary Schools Sports Association (KPSSA) is the governing body for sports in primary schools. Many youth in primary and secondary schools with upcoming athletic careers especially in the rift valley (where many athletes come from) are motivated with the presence of world-class athletes in their own villages and in several training camps in villages nearby. The champions are close relatives; they are fathers, brothers, cousins, and neighbors. 29 Children bear witness to the fortunes their heroes make out of winning an athletics event. This includes the homes and cars those fortunes can buy. For instance,Tegat's one-year deal of US$300,000, with the Flora London Marathon is the most lucrative in the history of road running. 30 Kenyan athletes see running as a means of escaping poverty and providing money for their families given that statistics show 40% of Kenyans are unemployed. 31 Pitsiladis et al. observe that 31% of Kenyan world-class athletes became runners for economic empowerment. 32

Role models

Achievement orientation

According to Ames 33 goal orientation with achievement motivation results to increased physical activity participation and success. Members from the Kalenjin community displayed the highest achievement orientation of all the Kenyan tribes according to Berg-Schlosser (cited by Bale and Sang 34 ). Achievement orientation refers to an internalized tendency to strive for excellence. High levels of achievement orientation can contribute to success by impacting on behaviour whose indicators include performance, effort and persistence. 35 This is more applicable in individual sport like running where the athletes are competing against time or set record.

Ascetic experience

Entine found out that Kalenjin students perceive the ascetic experience of sport as more significant than others. 36 The ascetic experience refers to the postponement of gratification where the middle and long distance runners train hard on a daily basis for success in the distant future. Both factors (achievement orientation and ascetic experience) develop over time and are important for successful performance in individual sports like distance running. According to Festinger cognitive dissonance theory suggests that we have an inner drive to hold all our attitudes and beliefs in harmony and avoid disharmony. 37 Gill further argues that attitudes may predispose the individual to certain behaviour which tend to be directed towards specific objects,

Establishment of training camps

The establishment of athletic training Camps in Eldoret, Ngong and Kapsabet among others also increased Kenya's fortunes. These camps provide basic services like accommodation, meals and coaching for the runners. Establishing training camps closer to the homes of runners has helped to increase the numbers who can access training. This also increases the chances of training under a coach unlike training privately. International funding and endorsement of some of these athletic camps has created an avenue for the recruitment and training more athletes.

Conclusion

The development of Kenyan athletics culture has taken many years. The cumulative effect of what started in 1895 when Kenya became a British protectorate is the foundation of the current athletic prowess being witnessed today. Different generations have been able to consolidate gains made by their predecessors to improve Kenyan performance internationally. This dominance in athletics is not unique to Kenya; different parts of the world dominate indifferent sports. For instance, martial arts in Japan, rugby in New Zealand and boxing in Cuba are cumulative products of their cultures and do not necessarily mean that participants have genes for dancing or boxing. Even in running, Scandinavian countries once dominated the world of athletics and specifically distance running in the 1920s. Therefore looking at this phenomenon as an outcome of human choices would pay attention to social and cultural factors that have specificity to known cultures in the regional dominance of various sports. For Kenya's case in particular they are Kenya's traditional culture, colonial influence, the existence of role models, sports institutions, the availability of training camps, the achievement orientation of the runners and their ascetic experience, and early exposure to standard facilities and competitions.

  27 http://www.athleticskenya.or.ke 28 Rintaugu E, G., Mwisukha A. & Munayi S., Sports: On the right track, in Njonjo S.K, Kitonga A. & Awour P. (eds), Youth research compendium, Institute of Economic Affairs, Nairobi, 2011.

38

  Gill DL. & Williams L., Psychological dynamics of sport, Champaign, IL: Human Kinetics Publishers, Inc., 1986. 39 Mazrui Ali, The Africans: A Triple Heritage, Little Brown & Co, 1987.

  below. Table1: Showing medals tallies in Top 25 all time list in Olympic Games, Commonwealth Games, All African Games track events

	ETHIOP	M	1	0	7	0	6	6	20	52
	IA	W	0	1	6	4	11	10	32
	Morocco M	0	3	1	2	4	1	11	13
		W	0	1	0	0	1	0	2
	USA	M	2	1	1	9	0	1	14	17
		W	0	0	0	2	0	1	3
	UK	M	1	3	0	0	0	1	5	7
		W	0	0	0	0	1	1	2
	Source: compiled from http/www.iaaf.org.
	Country Gend	800m	1500m 3000m 3000m	5000	10,000	Tota	Over
		er				St.	m	m	l	all
						Chase				Total
	KENYA M	9	11	8	17	12	13	70	85
		W	1	0	4	7	0	3	15

Table 2 :

 2 Distribution of medals amongst various Kenyan communities

Runner origin Kenyan tribes Medal tallies compared Olympics World Championships Marathon -Top 25 list

	Kalenjin	54	77	206
	Kikuyu	6	3	22
	Kisii	3	2	16
	Kamba	1	0	10
	Maasai	1	7	2
	Turkana	0	1	0
	Runner

origin Kalenjin sub tribes medal tallies compared

	Nandi	33	39	86
	Keiyo	5	10	38
	Kipsigis	7	6	45
	Marakwet	5	15	5
	Pokot	1	3	2
	Tugen	3	3	22
	Sabaot	0	1	8
	Source: Tucker et al 2015.

  8Entine Jon, Taboo: Why Black Athletes Dominate Sports and Why We Are Afraid to Talk About It,New York: B, Terrados N, Larsen H. et al., Morphology, enzyme activities and buffer capacity in leg muscles of Kenyan and Scandinavian runners, Scand J Med Sci Sports, 5, 1995, pp. 222-30. 10 Bale J. and Sang J., op. cit., 1996. 11Malik K., Why Black Will Beat White in the Olympics: The role of race in athletic success, New Statesman,

	Times Company, 2000.
	4504:129, 2000.

9Saltin 12 Bale J. and Sang J., op. cit., 1996. 13Onywera V.O., Kiplamai F.K., Tuitoek P.J., Boit M. and Pitsiladis Y., Food and macronutrient intake of elite Kenyan distance runners, International Journal of Sport Nutrition and Exercise Metabolism, 14 (6), 2004.

29

  Bale J. and Sang J., op. cit., 1996. Ames C., Achievement goals, motivational climate and motivativational processes, in Roberts G. (ed), , or ideas.38 This athletics culture has developed over time from the utilitarian from before 1895 to current modern elite athletics.39

	30 http/www.iaaf.org
	31Pitsiladis et al, op. cit., 2007.
	32 Ibid.
	Motivation in sport and exercise, Champaign, I.L: Human Kinetics, 1992.
	34 Bale J. and Sang J., op. cit., 1996.
	35 Nicholls J.G., The competitive ethos and democratic education, Cambridge, M.A: Harvard University Press,
	1989.
	36 Entine, op. cit., 2008.

33

people

1Head, Dept of Health Promotion and Sport Science, Maside Muliro University of Science and Technology, Kenya.