Pierre Lissy
email: lissy@ceremade.jussieu.fr

Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory

Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory

Pierre Lissy

Introduction

Presentation

The main motivation of this paper is to continue the study of [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] concerning the estimation for the cost of fast "boundary" controls for a class of linear equations of parabolic or dispersive type. The precise scope of the paper will be made more precise later.

Let us introduce some usual notations. Let H be some Hilbert space and U be another Hilbert space. Let A : D(A) → H be a self-adjoint operator with compact resolvent. The eigenvalues of A (which are here supposed to be different from 0 without loss of generality) are called (λ k ) k 1 and supposed to be with multiplicity 1 in all what follows. To each eigenvalue we associate a normalized eigenvector corresponding to the eigenvalue λ k , which is called e k . We assume that -A generates on H a strongly continuous semigroup S : t → S(t) = e -tA . The Hilbert space D(A * ) (= D(A) ) will be equipped with the norm

||x|| 2 D(A) = < x, e k > 2 H λ 2 k .
We call B ∈ L c (U, D(A) ) an admissible control operator for this semigroup, i.e. such that for every time T > 0, there exists some constant C(T ) > 0 such that for every z ∈ D(A), one has

T 0 ||B * S(t) * z|| 2 U C(T )||z|| 2 H .
We consider the following class of controlled semigroups:

y t + Ay = Bu (parabolic case) (1)
and

y t + iAy = Bu (dispersive case). (2)
In the case of Equation ( 1), A will moreover be supposed to be a positive operator to ensure the existence of solutions. Then, it is well-known (see for example [2, Chapter 2, Section 2.3], the operators -A or -iA generates a strongly continuous semigroup under the hypothesis given before thanks to the Lummer-Phillips or Stone theorems) that if u ∈ L 2 ((0, T ), U ), System (1) or [START_REF] Coron | Control and nonlinearity[END_REF] with initial condition y 0 ∈ H has a unique solution satisfying y ∈ C 0 ([0, T ], H).

From now on and until the end of the paper, we will assume that B is a scalar control, that is to say of the form Bu = bu, where b ∈ D(A) and u ∈ L 2 ((0, T ), K) where here U = K with K = R or C. From now on, we will call

b k =< b, e k > (D(A) ,D(A)) ,
where <, > (D(A) ,D(A)) is here the duality product between D(A) and D(A) with pivot space H. It is well-known (see [START_REF] Ho | Admissible input elements for systems in Hilbert space and a Carleson measure criterion[END_REF]

) that if ||(b k ) k∈N * || ∞ < +∞ and if (λ k ) k 1 is regular in the sense that inf m =n |λ m -λ n | > 0,
then B is an admissible control operator.

Let us now introduce the notion of cost of the control. Assume that that system (1) or ( 2) is null controllable at some time T 0 > 0 (i.e. for every y 0 ∈ H, there exists some control u ∈ L 2 ((0, T 0 ), U ) such that y(T 0 , •) ≡ 0). One can then verify easily that there exists a unique control u opt ∈ L 2 ((0, T 0 ), U ) with minimal L 2 ((0, T 0 ), U )-norm. Moreover, the map y 0 → u opt is linear continuous (see for example [2, Chapter 2, Section 2.3]). The norm of this operator will be from now on called the cost of the control at time T 0 and denoted C T0 in this section. Thanks to the definition of C T0 , this constant is also exactly the smallest constant C > 0 such that for every y 0 ∈ H, there exists some control u driving y 0 to 0 at time T 0 with ||u|| L 2 ((0,T0),U ) C||y 0 || H .

Here we attend to give some precise upper bounds on C T when T → 0 for some families of operators A for which equations ( 1) and ( 2) are null controllable in arbitrary small time. Some applications to fractional heat and Schrödinger equations will be provided later.

Let us explain now precisely the scope of the paper:

1. We give a new family of possible multipliers for the moment method that depend on the asymptotic growth of (λ k ) k 1 and seem well-adapted (this is the main result of the paper).

The author believes that it would be quite difficult to improve the construction given here, notably because the construction approximates quite well some optimal problem (see Section 3.2), but there is still some small possibility of doing better (see Remark 4) that the author did not manage to exploit suitably.

2. We manage to extend the results of [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] to the wider possible range of exponents for the asymptotic polynomial behavior of the eigenvalues (λ k ) k∈N * . In section 2.3, we will explain into more details why the multiplier used in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] was not really adapted to the case of eigenvalues λ k behaving roughly like k α with α = 2 and how we managed to improve it.

3. As we will see later (see notably Figures 1 and3), we will not improve the upper bounds in the case of the classical heat equation and Schrödinger equation given in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]. However, we will also see that there are a wide range of families of eigenvalues for which our estimates are better than in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], and notably when α → ∞ we dramatically improve the results of [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] and we are quite close to the lower bounds of [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF].

State of the art

As far as the author know, the study of the behavior of fast control for partial differential equations started with the paper [START_REF] Seidman | Two results on exact boundary control of parabolic equations[END_REF], where an upper bound for the cost of fast boundary control for the one-dimensional heat equation was given. A lower bound were then given in [START_REF] Güichal | A lower bound of the norm of the control operator for the heat equation[END_REF], proving that the cost of fast controls had to be necessarily roughly (up to fractional terms in T ) of the form exp(K/T ) for some K > 0 as T → 0. The same result for the boundary control of onedimensional Schrödinger equations was proved later in [START_REF] Miller | How Violent are Fast Controls for Schrödinger and Plate Vibrations?[END_REF]. The next natural question is then to try to estimate precisely K and notably its dependence with respect to the geometry of the problem (i.e. to the behavior on the eigenvalues λ k ). In the case of the heat equation, if we call L the length of the interval on which we control our one-dimensional equation, we know that

L 2 /2 K 3L 2 /4.
The best upper bound was obtained in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF] whereas the lower bound was obtained in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF]. For the Schrödinger equation, we have

L 2 /4 K 3L 2 /2.
The upper bound was obtained in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF] and the lower bound in [START_REF] Miller | How Violent are Fast Controls for Schrödinger and Plate Vibrations?[END_REF].

Let us now explain with more details what was precisely done [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]. In this article, the author proved precise upper bounds concerning the cost of the control for some large classes of linear parabolic or dispersive equations when the time T goes to 0, where the underlying "elliptic" operator was chosen to be self-adjoint or skew-adjoint with eigenvalues roughly as Rk α or ±Rk α (only for (2)) for some R > 0 and α 2 when k → +∞. The cost of the control is proved to be bounded from above by exp K/(RT ) 1/(α-1) where K is some explicit constant depending on α. This does not cover all possible cases, because it is well-known that equations like (1) and ( 2) are controllable in arbitrary small time if and only if α > 1.

Explicit lower bounds have later been derived in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] for all controllable in small time fractional heat or Schrödinger equations. Let us mention that in the case of the Schrödinger equation (which corresponds to λ k = k 2 and equation ( 2)), the lower bound proved in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] is exactly the same as the one given by Miller in [START_REF] Miller | How Violent are Fast Controls for Schrödinger and Plate Vibrations?[END_REF], whereas the lower bound for the heat equation controlled on the boundary (which corresponds to the case λ k = k 2 and equation [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]) is twice the one obtained by Miller in [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] and was conjectured to be to exact behavior for the cost of the control until now.

To finish, let us mention that the study of the behavior of fast controls may also be applied in some cases to study the uniform controllability of convection-diffusion equations in the vanishing viscosity limit as explained in [START_REF] Lissy | A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation[END_REF] and [START_REF] Lissy | An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit[END_REF], however the results given in the present article does not enable us to deduce directly new results for this problem.

Main results and comments

In this section, we are going to give the main results of this paper and some additional comments. Let us first set some notations. In all what follows, f g (with f and g some complex valued functions depending on some variable x in some set S) means that there exists some constant C > 0 such that for all x ∈ S, one has |f (x)| C|g(x)|, (such a C is called an implicit constant in the inequality f g), and f g means that we have both f g and g f . Sometimes, when it is needed, we might detail the dependence of the implicit constant with respect to some parameters. Theorem 2.1 Assume that (λ n ) n 1 is a regular increasing sequence of positive numbers verifying moreover that there exist some α > 1 and some R > 0 such that

λ n = Rn α + O n→∞ (n α-1 ) (3)
holds (as n → ∞), and assume that b n 1 (in the sense that the sequence (|b n |) n∈N is bounded from below and above by positive constants). Then system (2) is null controllable and the cost of the control C T verifies for T small enough: 1) , for every K > C S (α) := 4

1. If α 2, then C T e K (RT ) 1/(α-1) , for every K > C S (α) := 2 1 α-1 π α α-1 (α -1) 1 α-1 sin( π α ) α α-1 . 2. If α ∈ (1, 2), then C T e K (RT ) 1/(α-
1 α-1 π α α-1 (α -1) 1 α-1 sin( π α ) α α-1
.

(the implicit constant in the previous inequalities might depend on α or R but not on T )

We also have a theorem in the dispersive case when the eigenvalues are not supposed anymore to be positive. Theorem 2.2 Assume that the sequence of increasing eigenvalues (λ n ) n∈Z * of A is a regular sequence of non-zero numbers verifying moreover that there exist some α > 1 and some constant

R > 0 such that        λ n = Rn α + O n→∞ (n α-1 ), n > 0, λ -n = -Rn α + O n→∞ (n α-1 ), n < 0, sgn(λ n ) = sgn(n), (4)
and assume that b n 1. Then system (2) is null controllable and the cost of the control C T verifies for T small enough:

1. If α 2, then C T e K (RT ) 1/(α-1) , for every K > C S+-(α) := 2 α+1 α-1 π α α-1 (α -1) 1 α-1 sin( π α ) α α-1 . 2. If α ∈ (1, 2), then C T e K (RT ) 1/(α-1) , for every K > C S+-(α) := 2 1 α-1 (2 1 α + 1) α α-1 π α α-1 (α -1) 1 α-1 sin( π α ) α α-1
.

(the implicit constant in the previous inequalities might depend on α or R but not on T )

Concerning the parabolic case, we obtain the following result:

Theorem 2.3 Assume that (λ n ) n 1
is a regular increasing sequence of positive numbers verifying moreover that there exists some α > 1 and some constant R > 0 such that (3) holds. Assume that b n 1. Then system (1) is null controllable. Moreover, the control can be chosen in the space C 0 ([0, T ], U ) and the cost of the control C T (in norm L ∞ (0, T ), so this is also true in L 2 (0, T )) verifies for T small enough

C T e K (RT ) 1/(α-1) , for every K > π α α-1 2(α -1) 1 α-1 sin( π 2α ) α α-1
.

(the implicit constant in the previous inequalities might depend on α or R but not on T ) Remark 1

1. Concerning the asymptotic behavior of C S (α), we obtain by easy Taylor expansions that

C S (α) exp(1) 4 (α -1) α+1 1 α-1 as α → 1 + , C S (α) α as α → +∞.
2. We observe that for α = 2, the cost of the control is bounded by e Kπ 2 RT for every K > C S (2) = 2π 2 , which is worse than [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF] which stated that we have the same result for every K > 3π 2 /2. However, we see on figure 1 (which compares the upper bound C S (α) and the one found in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] in the case α 2) that our new bound becomes better for α 2.76 and is linearly better as α → ∞. [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] with respect to α.

Another interesting comparison can be done between the upper bound C S (α) and the lower

bound given in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF]. This is what is done in figure 2. We can see that our upper bound becomes close to the lower bound given in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] when α → ∞, more precisely the difference between the two quantities converges to 1 + ln(2). However, the upper bound is very far from the lower bound for α → 1 + .

Figure 2: Difference between C S (α) and the lower bound of [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] with respect to α.

Remark 2 1. Concerning the asymptotic behavior of C H (α), we obtain by easy Taylor expan-sions that

C H (α) π α α-1 2(α -1) 1 α-1 as α → 1 + , C H (α) α as α → +∞.
2. We observe that for α = 2, the cost of the control is bounded by e Kπ 2 RT for every K > C H (2) = π 2 , which is worse than [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF] which stated that we have the same result for every K > 3π 2 /4. However, we see on figure 3 (which compares the upper bound C H (α) and the one found in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] in the case α 2) that our new bound becomes better for α 2.221 and is linearly better as α → ∞. [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] with respect to α.

3. To finish, let us compare this upper bound and the lower bound given in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF]. This is what is done in figure 4. We can see that our upper bound becomes close to the lower bound given in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] when α → ∞, more precisely the difference between the two quantities tends to 1. However, the upper bound is very far from the lower bound for α → 1 + .

Figure 4: Difference between C H (α) and the lower bound of [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] with respect to α.

Application to fractional heat or Schrödinger equations with "boundary control"

Let us give the two main applications of this result, which have already been mentioned widely. Let us consider the 1-D Laplace operator ∆ with domain D(∆) := H 1 0 (0, L) and state space H := H -1 (0, L). It is well-known that -∆ : D(∆) → H -1 (0, L) is a positive definite operator with compact resolvent, the k-th eigenvalue is

λ k = k 2 π 2 L 2 , with associated eigenvector e k (x) := sin kπx L /|| sin kπx L || H -1 (0,L) .
Thanks to the continuous functional calculus for positive self-adjoint operators, one can define any positive power of -∆. Let us define our "boundary" control (for more explanations, see notably

[11, Sections 3.3 & 3.4] b ∈ (D(-(-∆) α/2
)) as follows:

b := (∂ x δ 0 ) • ∆ -1 .
We set K := R (for ( 5)) or C (for ( 6)). Let us consider here some γ > 1 and two different equations, one particular case of (1), which are the fractional heat equations

y t = -(-∆) γ/2 y + bu in (0, T ) × (0, L), y(0, •) = y 0 in (0, L), (5)
and one particular case of (2), which are the fractional Schrödinger equations

y t = -i(-∆) γ/2 y + bu in (0, T ) × (0, L), y(0, •) = y 0 in (0, L). (6)
Equation ( 5) is often used to model anomaly fast or slow diffusion (see for example [START_REF] Metzler | The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[END_REF]), whereas [START_REF] Ho | Admissible input elements for systems in Hilbert space and a Carleson measure criterion[END_REF] was introduced to study the energy spectrum of a 1-D fractional oscillator or for some fractional Bohr atoms (see for example [START_REF] Guo | Some physical applications of fractional Schrödinger equation[END_REF]).

As explained in details in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], the above equations exactly fit our abstract setting and we directly deduce from our main Theorems the following results: Theorem 2.4 System (6) is null controllable and the cost of the control C T verifies for T small enough:

1. If γ 2, then C T e KL 2γ/(2γ-1) T 1/(2γ-1) , for every K > 2 1 2γ-1 (2γ -1) 1 2γ-1 sin( π 2γ ) 2γ 2γ-1 . 2. If γ ∈ (1, 2), then C T e KL 2γ/(2γ-1) T 1/(2γ-1) , for every K > 4 1 2γ-1 (2γ -1) 1 2γ-1 sin( π 2γ ) 2γ 2γ-1 .
System (5) is null controllable. Moreover, the control can be chosen in the space C 0 ([0, T ], U ) and the cost of the control C T verifies for T small enough

C T e KL 2γ/(2γ-1) T 1/(2γ-1) , for every K > 1 2(2γ -1) 1 2γ-1 sin( π 4γ ) 2γ 2γ-1 .
(the implicit constant in the previous inequalities might depend on α but not on T ).

Gevrey functions, the moment method and the Bray-Mandelbrojt construction

In this "heuristic" Section, our goal is to give some informal explanations on the strong link between Gevrey functions and the moment method of Fattorini and Russell [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] that will be used here to prove our theorems. Let us consider here the parabolic case (1), the dispersive case being quite similar. Let us decompose our initial condition on our Hilbert basis of eigenfunctions:

y 0 (x) = ∞ k=1 a k e k (x),
with (a k ) k∈N * ∈ l 2 (N * ). Then, using the notations of the previous section, it is well-known that we have for all k ∈ N * and t ∈ (0, T ] that

y(t), e k H = a k e -λ k t + b k t 0 e -λ k (t-s) u(s)ds.
Imposing that y(T, •) = 0 is then equivalent to imposing that for every k ∈ N * , one has

a k e -λ k T + b k T 0 e -λ k (T -s) u(s)dt = 0, i.e. T 0 e -λ k (T -s) u(s)dt = -a k e -λ k T b k ,
the right-hand side being in l 2 (N * ) as soon as b k 1.

Hence, if we assume that we are able to exhibit a bi-orthogonal family to {t → e -λn(T -t) } in L 2 (0, T ), i.e. a family of functions {ψ m } m∈N * such that for every (k, l) ∈ (N * ) 2 one has

e λ k t , ψ l L 2 (0,T ) = δ kl , (7)
then one can use as a control function

u(t) := - k∈N * a k e -λ k T b k ψ k (t). (8)
A usual way (but not the only one, see notably [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]) to construct the family{ψ k } is to use the Paley-Wiener Theorem (see for example [START_REF] Rudin | Real and complex analysis[END_REF]Theorem 19.3, Page 370]), which says that ψ k can be constructed (after some adequate translation) as the inverse Fourier Transform of a L 2 (R)function J k of exponential type T /2 verifying moreover J k (iλ l ) = δ kl . As we will see during the proof, finding such a function J k can be decomposed into two main steps:

• First exhibit some family of Weiertrass product Ψ k involving the eigenvalues and verifying Ψ k (iλ l ) = δ kl (see notably Lemma 3.1).

• Then multiply Ψ k with some adequate function M k (called multiplier ) so that the product J k := Ψ k M k will then be of exponential type T /2 and belongs to L 2 (R). Moreover M k has to verify the normalization condition

M k (iλ k ) = 1. (9)
The crucial point is that the behavior for x → ∞ of the Weierstrass product is very bad and can be proved to be exactly of the form exp(K|x| 1/α ) for some constant K > 0 if we assume that λ k k α for some α > 1. Hence, we observe that if we want to obtain a function J k which is in L 2 (R), it is necessary that the multiplier M k behaves at least as exp(-K|x| 1/α ) at infinity. This exactly means that M k has to be the Fourier transform of some function G k which has to be Gevrey of order α, i.e. verifying

||G (j) k || ∞ C Gev (α, T, R)R j j! α , (10)
for some constant R > 0 > 0 (see for example [19, Section 1.6, Page 30]). Moreover, as it will be observed during the proof of our theorems, an important point is that the constant R determines in an optimal way the exact behavior of the exponential decreasing of M k (i.e. the constant K). Hence, imposing here that M k compensates quite "exactly" the growth of Ψ n means that this constant ν in [START_REF] Lissy | An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit[END_REF] is totally imposed by the data of our problem. To finish, since we want M k to be of exponential type T /2, applying one more time the Paley-Wiener Theorem, it is equivalent to saying that G k has to be of compact support [-T /2, T /2]. Using some estimations made more precise later, the function ψ k will also be bounded by C Gev (α, T, R) and then using [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF] we deduce that C H (α, T ) is also bounded by C Gev (α, T, R). Hence, if we want to obtain a precise bound on C H (α, T ), it is necessary to make all our possible to obtain some constant C Gev (α, T, R) which is as small as possible. Let us also mention that the normalization condition ( 9) can be easily replaced by the condition |G k | = 1 (and some suitable inequality on G k (iλ k ) that we do not detail here for the sake of clarity, see notably (60)).

Taking into account all these remarks, the rules of the game can be gathered as the following.

Optimizing the cost of the control is closely related to the following problem: given some T > 0, construct a Gevrey function G of order α, with support equal to [-T /2, T /2], verifying |G| = 1, with imposed coefficient R appearing in the growth of the derivative and minimizing the quantity C Gev (α, T, R) (which has of course to explode as T → 0) that is appearing in [START_REF] Lissy | An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit[END_REF].

Let us now explain a possible way to construct Gevrey function (or more generally C ∞ functions) with compact support. This construction was first described by Szolem Mandelbrojt in [13, Section 13], but in this book it is mentioned that the construction comes from previous unpublished works of Hubert Bray, whence the name "Bray-Mandelbrojt construction" adopted here. The idea is to use repeated mean-values of functions, i.e. to make an infinite convolution product of rectangle functions on [-a k , a k ], (a k ) k 0 being supposed to be a convergent series of positive numbers (whose sum will exactly be the support of the final function). This construction can also be found in [START_REF] Hörmander | The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis[END_REF]. Let us mention that since the Fourier transform of a rectangle function is exactly some cardinal sine function, and that the Fourier transform changes convolution into products, this construction is totally equivalent to the ones we can find notably in [16, Proof of Lemma 4.4], which is also used in [START_REF] Miller | How Violent are Fast Controls for Schrödinger and Plate Vibrations?[END_REF] and that goes back to the work of Ingham in [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF] (we refer to [START_REF] Redheffer | Completeness of sets of complex exponentials[END_REF] for extra explanations on the usefulness of this construction).

To finish, let us mention that in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF], the authors used to following Gevrey function to construct the multiplier:

G(x) := exp(-ν/(1 -x 2 )),
where ν > 0 is some large parameter to be chosen. It can be proved that G is a Gevrey function of order exactly 2. Notably, this function is not Gevrey of order α for α ∈ (1, 2) (but it is a Gevrey function of order α for every α > 2), and this explains why in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] (where the same multiplier was used) we were not able to treat the case α ∈ (1, 2) and why our estimations were quite bad for large α.

3 Proofs of Theorems 2.1, 2.2 and 2.3

Estimates on the Weierstrass product involving the eigenvalues

As usual when the moment method is concerned, we first study the asymptotic behavior for large z ∈ C of the Weierstrass product constructed from the family of eigenvalues (λ n ) n∈N . Lemma 3.1 Let (λ n ) n 1 be a regular increasing sequence of positive numbers verifying moreover that there exists some α > 1 and some constant R > 0 such that (3) holds.

Let Φ n be defined as follows:

Φ n (z) := k =n (1 - z λ k -λ n ). (11)
Then,

1. If α ∈ (1, 2), then for every z ∈ C, Φ n (z) e 2 1/α π R 1/(α-1) sin(π/α) |z| 1 α P (λ n , |z|), ( 12
)
where P is a polynomial.

2. If α 2, then for every z ∈ C,

Φ n (z) e π R 1/(α-1) sin(π/α) |z| 1 α P (λ n , |z|), ( 13
)
where P is a polynomial.

3. If x ∈ R, Φ n (-ix -λ n ) e π 2R 1/(α-1) sin(π/2α) |x| 1 α P (λ n , |x|), ( 14
)
where P is a polynomial.

(In the previous inequalities, the implicit constant may depend on α and R but not on z, x or n)

Remark 3 One can see numerically by taking the particular case λ k = k α that inequalities [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF] and ( 14) are optimal. However, as explained in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]Remark 3] we are unable to extend inequality (13) to the case α ∈ (1, 2), where is seems numerically not to be true anymore. However, it is likely that estimate (12) is far from being optimal (because notably of the gap when α → 2 -with what we have in (13) for α = 2).

Proof of Lemma 3.1. Without loss of generality, we can assume that R = 1 (one can go back to the general case by an easy scaling argument).

Let us prove inequality (12) (inequality ( 13) was already proved in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]). We go back to the computations done in [11, Proof of Lemma 2.1]. One notably has

|Φ n (z)| (1 + |z|/γ) 2C e |z| 0 ∞ γ (λn +s) 1 α -((λn-s) + ) 1 α (t+s) 2 dsdt . ( 15
)
We also have

|z| 0 ∞ γ (λ n + s) 1 α -((λ n -s) + ) 1 α (t + s) 2 dsdt = |z| ∞ γ (λ n + s) 1 α -((λ n -s) + ) 1 α s(s + |z|) ds |z| λ 1-1 α n (U ( |z| λ n ) + V ( |z| λ n )), (16)
where

U (x) := 1 0 (1 + v) 1 α -(1 -v) 1 α v(v + x) dv and V (x) := ∞ 1 (v + 1) 1 α v(v + x) dv.
Using the change of variables t = x/v, we easily obtain

x 1-1/α V (x) = x 0 (1/t + 1/x) 1 α 1 + t dt (17)
and

x 1-1/α U (x) = ∞ x (1/t + 1/x) 1 α -(1/x -1/t) 1 α 1 + t dt. ( 18
)
Using expression [START_REF] Miller | On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian[END_REF], one has

x 1-1/α V (x) 2 1 α x 0 1 t 1 α (1 + t) dt.
Using expression [START_REF] Redheffer | Completeness of sets of complex exponentials[END_REF], we also have

x 1-1/α U (x) 2 1 α ∞ x 1 t 1 α (1 + t) dt.
Hence we deduce [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] taking into account that (see for example [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]

, Lemma 2.2]) ∞ 0 dt t 1 α (1 + t) = sin( π α ). (19)
Concerning ( 14), one verifies that the proof provided in [11, Page 2661] for α 2 is also valid for α ∈ (1, 2) because we can use [START_REF] Robino | Linear partial differential operators in Gevrey spaces[END_REF], replacing α by 2α (> 1).

In the dispersive case, we also need some estimate in the case where the eigenvalues are not supposed to be positive anymore. Lemma 3.2 Let (λ n ) n 1 be a regular increasing sequence of positive numbers verifying moreover that there exists some α > 1 and some constant R > 0 such that (4) holds.

Let Φ n be defined as follows:

Φ n (z) := k =n (1 - z λ k -λ n ). (20)
Then,

1. If α ∈ (1, 2), then for every z ∈ C, Φ n (-z -λ n ) e (2 1/α +1)π R 1/(α-1) sin(π/α) |z| 1 α P (λ n , |z|), (21)
where P is a polynomial.

2. If α 2, then for every z ∈ C,

Φ n (-z -λ n ) e 2π R 1/(α-1) sin(π/α) |z| 1 α P (λ n , |z|), (22)
(In the previous inequalities, the implicit constant may depend on α and R but not on z, x or n)

This Proposition was already proved in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] for the case α 2. For α ∈ (1, 2), the proof is exactly the same as the corresponding case in Lemma 3.1 that we combine with the computations made in the proof of [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]Lemma 2.4 ] (see Pages 2666-2668 in this reference) and will be omitted.

Construction of adequate multipliers

As explained before, we now have to construct an adequate multiplier, which has to be the Fourier transform of a Gevrey function with compact support. Let us emphasize that the main contribution of this paper is the construction given in (43).

To begin, let us give some useful estimates concerning the Bray-Mendelbrojt construction of functions with compact support (see [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF]Section 13]). Proposition 3.1 Let (a k ) k 0 be a decreasing sequence of positive numbers such that

a := k a k < ∞.
Then, there exists a nonnegative function u with compact support included in [-a, a] verifying

a -a u = 1, (23)
u(0) 1 2a 0 , (24)
u is even,

u is increasing on [-a, 0], (25)
such that for every j ∈ N, one has

||u (j) || ∞ 1 j i=0 a i , (27)
Proof of Proposition 3.1 We follow step by step the proof given in [START_REF] Hörmander | The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis[END_REF]. For every b > 0 we call

H b := 1 [-b,b] 2b . ( 28
) Let us remark that R H b = b -b H b = 1 (29)
and H b is even.

We then consider

u n := H a0 * H a1 * • • • * H an ,
where * represents the convolution product. u n is of class C n-1 and

Supp(u n )[- n 1 a i , n 1 a i ](⊂ [-a, a]). (31)
Moreover, one easily verifies that, for j n -1 we have

u (j) n (x) = j-1 0 1 2a i (τ -ai -τ ai ) H aj * • • • * H an (x), (32)
where

τ b : u → (x → u(x -b)).
Using (29), the fact that for u and v some regular enough functions one has u * v = u v, and |u * v| ||u|| ∞ ||v|| 1 , we deduce, taking into account (32), that

||u (j) n || ∞ 1 j i=0 a i .
Let us also remark that u n is even (33) and verifies

a -a u n (x)dx = H a1 • • • H an = 1, (34)
moreover

u n (0) = a -a H a0 (s)H a1 * • • • * H an (-s)ds = 1 2a 0 a0 -a0 H a1 * • • • * H an (-s)ds 1 2a 0 a -a H a1 • • • a -a H an = 1 2a 0 . (35)
Let us now prove that u n is increasing on [-a, 0] for every n ∈ N. Let us prove this by induction. it is true for n = 0 and n = 1. Let us assume that u n is increasing on [-a, 0]. Then we compute the derivative of u n+1 , we obtain

u n+1 (x) = 1 2a n+1 (τ -an+1 -τ an+1 )(H a0 * • • • * H an (x)) = 1 2a n+1 (u n (x+a n+1 )-u n (x-a n+1 )). ( 36
)
Let x ∈ [-a, 0]. Then, we distinguish two cases:

1. If x -a n+1 , then x -a n+1 x + a n+1 0, hence, by using the fact that u n is increasing on [-a, 0], it is clear that u n (x -a n+1 ) u n (x + a n+1 ), and then by (36), u n+1 (x) 0.

If

x -a n+1 , then we have -a n+1 -x 0. But we also have a n+1 -x -x -a n+1 0.

Using the fact that u n is even, we know by induction that u n is decreasing on [0, a], hence we deduce that

u n (x + a n+1 ) = u n (-x -a n+1 ) u n (a n+1 -x) = u n (x -a n+1 ),
and then by (36), u n+1 (x) 0.

We conclude as in [START_REF] Hörmander | The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis[END_REF] by letting n → ∞, u being the limit (that exists) of the u n .

According to Section 2.3 and to inequality (27) (that has to be verified for every n), we see that the problem of minimizing what we called C Gev (α, T, R) in the case of the Bray-Mandelbrojt construction can be reformulated as maximizing the following quantity with respect to the sequence (a n ) n∈N :

max j∈N j i=0 (i + 1) α a i ν α-1 ,
for some large parameter ν > 0 to be chosen later. It is clear that this problem is equivalent to maximizing

∞ i=0 (i + 1) α a i ν α-1 .
Taking into account all these considerations, it is quite natural to investigate the following optimization Problem.

Problem (1) Let ν > 0 and a > 0. Find the sequence (a k ) k∈N that maximizes the quantity

∞ k=0 ln (k + 1) α a k ν α-1
with the constraints:

1. a k > 0 for every k ∈ N.

2. (a k ) k∈N is non-increasing.

3.

∞ k=0 a k = a. (37)
Unfortunately, we have the following proposition: Proposition 3.2 There are no solutions to Problem [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].

Proof of Proposition 3.2. Let f ((a k )) := ∞ k=0 ln (k + 1) α a k v α-1 . (38)
One has

∂f ∂a i ((a k )) = 1 a k . ( 39
) Let g((a k )) := ∞ k=0 a k . (40)
One has

∂g ∂a i ((a k )) = 1. (41)
We want to maximize f under the constraint g((a k )) for every j ∈ N (the constraint a k > 0 can be forgotten here). We do not need here to use that we want (a k ) k∈N to be decreasing.

Let λ ∈ R and (µ j ) j∈N , we consider the Lagrangian

L((a k )) := f ((a k )) -λg((a k )). (42)
We remark that for every j ∈ N, we have ∂L ∂aj = 0 if and only if a j = λ according to (39) and (41), hence the sequence (a j ) has to be constant. If we take into account the fact that we want the constraint g((a k )) = a to be verified, we see that Problem (1) cannot have any solution.

We observe thanks to the previous study that if we want to "mimic" in some sense an optimal sequence, a good idea would be to consider some sequence (a k ) k∈N which is constant at least for small k and then decreases in some suitable way such that it becomes summable. Hence, taking into account this remark, we set ν > 0 some parameter (that is destined to compensate the bad growth of the Weierstrass product [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], see (59), and to be very large so we will always assume ν C > 0), and we consider the following sequence (a k ) k∈N defined by

a k = 1 ν if k ν -1, a k = ν α-1 (1+k) α if k ν , (43)
The main advantage of this construction is that for k ν , we have

(k + 1) α a k v α-1 = 1,
which will simplify a lot the forthcoming computations.

Remark 4

The second part of the construction of (43) may seem to be chosen quite arbitrarily, however the author did not find a more suitable construction. It might be possible that another appropriate decreasing that would enable to improve the bounds given in this article.

We consider the corresponding function σ ν constructed as in the proof of Lemma 3.1 from the sequence (a k ) k∈N . We call a ν its support, that is to say

a ν := 1 + ∞ k= ν ν α-1 (1 + k) α . (44)
It may seem quite strange that we do not impose the support to be [-T /2, T /2] here, however for the sake of clarity we prefer to adjust the support thanks to the parameter β to be introduced later (see (58)).

Taking into account the definition of a ν , we may observe that a ν does not depend too much on a ν in the sense that

a ν ∼ α α -1 as ν → ∞. (45)
Let us remark that

aν aν σ ν = 1, (46)
σ ν (0) ν 2 , ( 47
)
σ ν is even, ( 48
)
σ ν is increasing on [-a v , 0]. (49)
Then, using (27), we obtain

||u (j) || ∞ 1 j i=0 a i . (50)
We can then deduce the following crucial estimate :

Lemma 3.3 One has for every j ∈ N ||u (j) || ∞ e αν (j!) α (ν α-1 ) j . (51)
Now we choose ν such that α α (βν α-1 ) > π sin( π α ) and close to π/ sin(π/α) (see estimates [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF] and (59)), for example ν := (π + δ) α/(α-1) (α sin(π/α)) α/(α-1) β 1/(α-1) .

(69)

Then, using (67), ( 13), ( 68), ( 69) and (59), we obtain

|g n (x)| e αν+π/ sin(π/α)|x+λn| 1 α -(π+δ/2)/ sin(π/α)|x+λn| 1 α P (|x + λ n |).
It is clear that

e π/ sin(π/α)|x+λn| 1 α -(π+δ/2)/ sin(π/α)|x+λn| 1 α P (|x + λ n |) 1 1 + (x + λ n ) 2 ,
hence we deduce using (69) and (68) that

|g n (x)| e αν 1 + (x + λ n ) 2 e α(2α) 1/(α-1) (π+δ) α/(α-1) (α sin(π/α)) α/(α-1) (T (α-1)(1-δ/2)) 1/(α-1) 1 + (x + λ n ) 2 e 2 1/(α-1) (π+δ) α/(α-1) (sin(π/α)) α/(α-1) (T (α-1)(1-δ/2)) 1/(α-1) 1 + (x + λ n ) 2 .
Let us fix some K > 2 1/(α-1) π α/(α-1) (α -1) 1/(α-1) sin(π/α) α/(α-1) T 1/(α-1) . Considering δ as close as 0 as needed, we deduce that

|g n (x)| e K T 1/(α-1) 1 + (x + λ n ) 2 (70)
This notably proves that g n ∈ L 2 (R). Moreover, using ( 13), ( 67 The case α ∈ (1, 2) follows by doing exactly the same proof, the only difference being that we use [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] instead of [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF].

Proof of Theorem 2.2. The proof is exactly the same as the one of Theorem 2.1, using (21) instead of [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] for α ∈ (1, 2) and ( 22) instead of (13) for α 2, and will hence be omitted.

Proof of Theorem 2.3. We now consider the parabolic case (Equation ( 1)). Let δ > 0 a small enough parameter. We call

h n (z) := Φ n (-iz -λ n ) H β (z) H β (iλ n ) . ( 72
)
One has, using [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], that h n (iλ k ) = δ kn . We want to apply at the end the Paley-Wiener Theorem (see estimate (58)) in an optimal way, so we want a ν β to be close to T /2. Let us consider ν large enough such that (see ( 45))

|a ν - α α -1 | δ/4.
Assume that a ν β < T /2 and close to T /2, for example

β = T (α -1)(1 -δ/2) 2α . ( 73
)
Now we choose ν such that α α (βν α-1 ) > π/(2 sin(π/2α))

and close to π/(2 sin(π/(2α))) (see estimates [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF] and (59)), for example ν := (π + δ) α/(α-1) (2α sin(π/2α)) α/(α-1) β 1/(α-1) .

(74)

Moreover, thanks to (72), ( 14), (59), (60), ( 74) and (73), one has |h n (x)| νe αν+π/(2 sin(π/2α))|x| (2 sin(π/2α)) α/(α-1) (T (α-1)(1-δ/2)) 1/(α-1)

(1 + (x + λ n ) 2 ) .
Let us fix some K > π α/(α-1) 2(sin(π/2α)) α/(α-1) (α -1) 1/(α-1) .

Considering δ as close as 0 as needed, we deduce that

|h n (x)| e K T 1/(α-1) (1 + (x + λ n ) 2 ) , (75)
This notably implies that h n (x) ∈ L 1 (R) ∩ L 2 (R) and

Figure 1 :

 1 Figure 1: Difference between C S (α) and the upper bound of [11] with respect to α.

Figure 3 :

 3 Figure 3: Difference between C H (α) and the upper bound of [11] with respect to α.

  ), (68) and (58), we obtain|g n (z)| e T |z|/2 .Hence, using the Paley-Wiener Theorem, g n is the Fourier transform of a function f n ∈ L 2 (R) with compact support [-T /2, T /2]. Moreover, by construction {f n } is biorthogonal to the family {e iλnt }. Then, one can create the control thanks to the family {f n }. Let us consider y 0 = a k e k the initial condition, we callu(t) := -k∈N (a k /b k )e -iT λ k /2 f k (t -T /2). (71)This expression is meaningful since b k 1, moreover the corresponding solution y of (2) verifies y(T, •) ≡ 0. We obtain by proceeding as in [22, Page 87-88] that ||u(t)|| L 2 (0,T ) α-1) ||y 0 || H .

1 α

 1 -((π+δ/2)/(2 sin(π/2α)))|x| 1 α -βλn 4ν P (|x|, |λ n |) νe αν-δ/(2 sin(π/2α))|x| 1 α -βλn 4ν P (|x|, λ n |) ν e α (π+δ) α/(α-1)(2α sin(π/2α)) α/(α-1) β 1/(α-1) (1 + (x + λ n ) 2 ) ν e α (2α) 1/(α-1) (π+δ) α/(α-1) (2α sin(π/2α)) α/(α-1) (T (α-1)(1-δ/2)) 1/(α-1) (1 + (x + λ n ) 2 )ν e 2 1/(α-1) (π+δ) α/(α-1)

  ||h n || L 1 (R) e K T 1/(α-1) .(76)Moreover, using[START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF], (72), (58) and (73)|h n (z)| e T |z|/2 , so using the Paley-Wiener Theorem, h n is the Fourier transform of a function w n ∈ L 2 (R) with compact support [-T /2, T /2]. Moreover, by construction {w n } is biorthogonal to the family {e λnt }. Then, one can create the control thanks to the family {h n }. Let us consider y 0 = a k e k the initial condition, we callu(t) := -(a k /b k )e -T λ k /2 w k (t -T /2),(77)This expression is meaningful since b k 1, moreover the corresponding solution y of (1) verifies y(T, •) ≡ 0. One easily verifies that u ∈ C 0 ([0, T ], R). Using (77), |b k | 1 and inequality (76), we obtain ||u(t)|| L ∞ (0,T ) e K T 1/(α-1) |a k |e -T λ k /2 . Using the Cauchy-Schwarz inequality, one deduces ||u(t)|| L ∞ (0,T ) e K T 1/(α-1) ||y 0 || H .

Proof of Lemma 3.3. Using the expression of a k given in (43) and (50), we deduce that for j ν we have

ν α-1 /(i + 1) α (i + 1) α a i /ν α-1 (53)

Now, using an usual lower bound on the factorial, we obtain

from which we deduce (51).

The same estimation is also clearly true for j ν since in this case we have

which will be our multiplier (up to some homothety). We have the following properties:

Lemma 3.4 For every δ > 0 small enough, one has

Proof of Lemma 3.4. Equality (57) comes from (46). Inequality ( 58) is an immediate consequence of the definition of H β given in (56). Let us now prove (59). Since all derivatives of σ ν vanish at t = -a ν and t = a ν , we have

for every x > 0 and j ∈ N. Combining (61) and (51), we deduce that

We set

Then we have

Using ( 64) and ( 62) we obtain

Combining ( 65), (63), and inequality

we deduce |H β (x)| e αν e -αj j α/2 e αν e -α(βν α-1 x) 1/α +δ/(2 sin(π/(α)))x 1/α , which proves the desired estimate.

Let us know prove (60). We have, since σ ν 0 and using (49) together with (48),

Using ( 46), ( 47), ( 48) and ( 49), we have

We deduce that for ν large enough (which is equivalent to T small enough, see (69))

which gives the desired result thanks to (66).

Proof of Theorem 2.1. The proof follows the one of [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]Theorem 3.1 and 3.4]. We still assume without loss of generality that R = 1. Let us first consider the dispersive case (Equation ( 2)) and the case α 2. Let δ > 0 a small enough parameter. We call

so that one has g n (-λ k ) = δ kn by ( 57) and [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]. We want to apply at the end the Paley-Wiener Theorem (see estimate (58)) in an optimal way, so we want a ν β to be close to T /2. From now on we will always consider ν large enough such that (see (45))

Assume that a ν β < T /2 and close to T /2, for example


Laurent Baratchart
email: laurent.baratchart@sophia.inria.fr

Matthias Caenepeel
email: matthias.caenepeel@vub.ac.be

Yves Rolain
email: yves.rolain@vub.ac.be

Harmonic-like Identification of Nonlinear Systems around an Equilibrium

Keywords:

.

Introduction

Several classes of models have been proposed for black-box nonlinear system identification. They range from Volterra and multiple Fourier series [START_REF] Schetzen | The Volterra and Wiener Theories of Nonlinear Systems[END_REF][START_REF] Rugh | Nonlinear System Theory âĂŞ The Volterra Wiener Approach[END_REF] to Hammerstein systems (i.e. a static nonlinear map followed by a linear dynamical system) or Wiener systems (i.e. a linear dynamical system followed by a static nonlinear map) and combinations thereof like the Wiener-Hammerstein model (a linear system sandwiched between two static nonlinearities) [START_REF] Haber | Nonlinear System Identification-Input Output Modeling Approach[END_REF]. A variant of the Wiener model, where the dynamics is nilpotent, has become especially popular in recent years under the name of Nonlinear Autoregressive Moving Averages with Exogenous inputs (NARMAX) models [START_REF] Billings | Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains[END_REF]. Still, it is natural to ask how general these classes of models with respect to the wild variety of all nonlinear models.

In the present work, we approach a local, but otherwise fairly general version of nonlinear identification for finite-dimensional continuous time systems around an equilibrium point, provided that the latter is hyperbolic meaning that the linearized dynamics has no pure imaginary eigenvalue. We show in passing that Wiener-Hammerstein systems are very special, and highly non-generic in the class of all such systems, even locally. More importantly, we also show that stable nonlinear models lend themselve locally to a generalization of classical harmonic identification of linear systems, at least when the state is observed. To be specific, we will see that one can in principle determine the poles of the linearized approximation from the response to periodic inputs.

For this, we dwell on local linearization results from [START_REF] Baratchart | A Grobman-Hartman theorem for control systems[END_REF] saying that when a nonlinear system with observed state, around a hyperbolic equilibrium point, is feeded forward by another finite-dimensional nonlinear system, then it is locally conjugate to a linear system feeded forward in the same way. As will be established below, the local character in time of this result can be eliminated if the initial system is stable (however, the result remains local with respect to initial states). Now, in this construction, the conjugacy map generally depends on the feeding system that was used, and observable Wiener-Hammerstein systems are those for which such a dependency does not hold; this is why they are so special. The results we just mentioned appear in sections 2 and 3.

Pursuing this line of investigation, we set up in section 4 a framework for harmonic identification of stable nonlinear system with observed state around an equilibrium point, and we establish equations for identification to be solved in this context.

Local linearization of control systems

We consider a control system with observed state of the form:

ẋ = f (x, u) , x ∈ R n , u ∈ R m , (1)
and we suppose that f (0, 0) = 0, i.e. we work around an equilibrium point that we choose to be the origin without loss of generality. By a solution of (1) that remains in an open set Ω ⊂ R n+m , we mean a mapping γ defined on a real interval I, say

γ : I → Ω t → γ(t) = ( γ I (t) , γ II (t) ) (2)
with γ I (t) ∈ R n and γ II (t) ∈ R m , such that :

• γ is measurable, locally bounded, and γ I is absolutely continuous,

• whenever [T 1 , T 2 ] ⊂ I, we have :

γ I (T 2 ) -γ I (T 1 ) = T2 T1 f ( γ I (t) , γ II (t) ) dt . Given another system ż = g(z, v) , z ∈ R n , v ∈ R m , (3)
we say that system (1) is conjugate to (3) if there is a local homeomorphism

χ : Ω → Ω (x, u) → χ(x, u) = ( χ I (x, u) , χ II (x, u) ) (4)
between two open subsets of R n+m such that, for any real interval I, a map γ : I → Ω is a solution of (1) that remains in Ω if, and only if, χ • γ is a solution of (3) that remains in Ω . In other words, trajectories are mapped to trajectories in a pointwise bijective, bi-continuous, and time-preserving manner. When the control variables u, v are not present, so that the component χ II (x, u) does not appear, we recover the definition of local conjugacy for ordinary differential equations (i.e. autonomous systems). We say that ( 1) is locally linearizable if it is topologically conjugate to a linear control system at (0, 0). Now, let us assume that f is continuously differentiable with respect to x and u. Subsequently, we single out the linear part of f by setting A = ∂f ∂x (0, 0) and B = ∂f ∂u (0, 0) so that (1) can be rewritten as

ẋ = Ax + Bu + F (x, u) with F (0, 0) = ∂F ∂x (0, 0) = ∂F ∂u (0, 0) = 0 . ( 5
)
If ( 1) is locally linearizable, then it is easy to see that it must be topologically conjugate to its linear approximation ż = Az + Bu. However, it is shown in [START_REF] Baratchart | On the local linearization of control systems[END_REF] that this property is seldom satisfied, that is, it cannot hold unless some rather stringent conditions hold on f . In fact, it is proved in [START_REF] Baratchart | On the local linearization of control systems[END_REF] that linearizability of system (1) implies that it is linearizable under feedback and that the linearizing feedback has quite a bit of smoothness; then, classical necessary conditions for smooth feddback local linearization [START_REF] Jakubczyk | On linearization of control systems[END_REF] are to the effect that the derivatives of f must satisfy some integrability conditions which are highly non-generic. As a next best thing, however, the following theorem was proved in [START_REF] Baratchart | A Grobman-Hartman theorem for control systems[END_REF]: Theorem 1.[6, thm.3.1] With f as in ( 5), consider the system

ẋ = Ax + Bh(ζ) + F (x, h(ζ)), ζ = g(ζ), ( 6
)
where g : R q → R q is locally Lipschitz continuous, h : R q → R m is continuous with h(0) = 0, and F : R n × R m → R n is continuously differentiable with F (0, 0) = ∂F/∂x(0, 0) = 0, A being hyperbolic. Then, there exist neighborhoods V and W of 0 in R n and R q respectively, and a map H :

V × W → R n with H(0, 0) = 0, such that H × Id : V × W → R n × W (x, ζ) → (H(x, ζ), ζ)
is a homeomorphism from V × W onto its image that conjugates [START_REF] Baratchart | A Grobman-Hartman theorem for control systems[END_REF] to

ż = Az + Bh(ζ), ζ = g(ζ). ( 7
)
Note that Theorem 1 tells us about conjugacy of ordinary differential equations rather than control systems. In this connection, the result comes close to the classical Grobman-Hartman theorem asserting that, around a hyperbolic equilibrium point, a differential equation is locally topologically conjugate to its linear approximation [START_REF] Hartman | Ordinary Differential Equations[END_REF]. However, there is one important difference here, namely no hyperbolicity assumption is made on ∂g/∂ζ(0), the derivative of g at zero, and it is not assumed that g(0) = 0, i.e. zero needs not be a fixed point of g. Accordingly, linearization in [START_REF] Hartman | Ordinary Differential Equations[END_REF] is carried out with respect to the variable z only. The fact that ∂g/∂ζ(0) needs not be hyperbolic in Theorem 1 will be crucial to us later on as we shall use periodic ζ(t) for harmonic identification. Let us mention that the linearizing homeomorphism is unique, up to composition with a linear invertible transformation (which induces a change of basis in the state space of the linearized system). We also mention that, even if , g, h are very smooth, the linearizing map H(x, ζ) needs not be differentiable at the origin: only Hölder-continuity will usually hold there.

Autonomous feed-forward of locally stable systems

In this section, we make Theorem 1 global in time for stable A and appropriate g.

Consider system (1) and let us restrict attention to inputs u that are themselves outputs of a (possibly nonlinear) control system:

ζ = g(ζ), u = h(ζ), (8)
where ζ(t) ∈ R q , while g : R q → R q is locally Lipschitz continuous and h : R q → R m is continuous. In particular, u(t) is entirely determined by the finite-dimensional data ζ(0). This operating mode is of course quite special and, from the control viewpoint, it may be described as open loop feed-forward of system (1) by system [START_REF] Jakubczyk | On linearization of control systems[END_REF]. Note that particular use of such a feed-forward occurs in classical harmonic identification of stable linear systems, where the system to be identified is subject to periodic inputs in order to estimate pointwise values of its transfer-function from the steady-state response. In fact, such a description of the steady-state response to a periodic signal at a given frequency as an operator which depends on the frequency is widely used in circuit theory, even in a nonlinear setting [START_REF] Suàrez | Stability Analysis of Nonlinear Microwave Circuits[END_REF]. The discussion in section 4 will dwell on this framework. Now, if we feed forward system (1) with system (8), we obtain system (6). Thus, from Theorem 1, we deduce the result below which gives additional information when system (1) is locally stable. As a piece of terminology, let us agree that the equilibrium (0, 0) of system ( 1) is said to be strictly stable if all eigenvalues of A = ∂f /∂x(0, 0) have strictly negative real parts, that is, if the tangent linear control system at (0, 0) is strictly stable. Moreover, if system (8) happens to have an equilibrium at ζ 0 , that is, if g(ζ 0 ) = 0, then this equilibrium is said to be strongly asymtotically stable if initial conditions sufficiently close to ζ 0 at t = 0 generate trajectories ζ(t) that remain arbitrarily close to ζ 0 for all t ≥ 0.

Theorem 2. Consider system (1) where f is continuously differentiable with respect to both arguments in a neighborhood of (0, 0) and f (0, 0) = 0. Set A = ∂f ∂x (0, 0) and B = ∂f /∂u(0, 0) so that (1) may be rewritten as [START_REF] Baratchart | On the local linearization of control systems[END_REF]. Assume that A is hyperbolic (i.e. has no purely imaginary eigenvalues). Consider also system [START_REF] Jakubczyk | On linearization of control systems[END_REF] where g is Lipschitz continuous and h is continuous, say in the neighborhood of some ζ 0 ∈ R q .

(i) If we feed system (1) with initial state x(0) by the output of system [START_REF] Jakubczyk | On linearization of control systems[END_REF] with initial state ζ(0), and if we denote by x(t, x(0), ζ(0)) the solution at time t of the first equation in [START_REF] Baratchart | A Grobman-Hartman theorem for control systems[END_REF], then there exist two neighborhoods V and W of 0 and ζ 0 in R n and R q respectively, and a map H :

V × W → R n with H(0, 0) = 0, of the form H × Id : V × W → R n × W (x, ζ) → (H(x, ζ), ζ) ,
which is a homeomorphism from V × W onto its image such that

H(x(t, x(0), ζ(0)), ζ(t)) = e At H(x(0), ζ(0)) + t 0 e A(t-τ ) Bh(ζ(τ )) dτ (9) for all x(0) ∈ V , ζ(0) ∈ W , and all t such that x(t, x(0), ζ(0)) ∈ V and ζ(t) ∈ W .
(ii) When the equilibrium of ( 1) is strictly stable, and if system (8) has an equilibrium at ζ 0 which is strongly asymtotically stable with h(ζ 0 ) = 0, then (i) holds for all t ≥ 0, possibly at the cost of shrinking V and W .

Proof: to deduce (i) from Theorem 1, first extend f , g,and h to the whole of R n ×R m and R q respectively, keeping the same regularity, using a partition of unity. This step is purely formal, since Theorem 2 is local and therefore the result will not depend on the precise extension which is made. It is necessary, though, because Theorem 1 was stated for globally defined maps. We may also assume without loss of generality that ζ 0 = 0. Then, we observe that the right hand side of ( 9) is a solution to

ż = Az + Bu, u(t) = h(ζ(t)), ( 10
)
with initial condition z(0) = H(x(0), ζ(0)), and we appeal to the local conjugacy of ( 6) to [START_REF] Hartman | Ordinary Differential Equations[END_REF]. As to (ii), note that if the equilibrium of ( 1) is strictly stable and the equilibrium of ( 8) is strongly asymtotically stable, and if moreover h(ζ(0)) = 0, then (x(t, x(0), ζ(0)), ζ(t)) will stay in V × W for all t ≥ 0 provided that x(0) and ζ(0) are small enough. Indeed, a strictly stable linear system subject to an input which is sufficiently small for all t ≥ 0 (namely (10)) has a state-trajectory that remains arbitrarily close to the origin if the initial state is small enough. Thus, on a possibly smaller neighborhood V × W of the origin, the linearization in equation ( 9) will be valid for all t ≥ 0. Assumptions and notation being the same as in Theorem 2, let Ω ⊂ R n × W be the image of the homeorphism H × Id. Its inverse is clearly of the form Ψ × Id : Ω → V × W . Now, if we let ẑ(t, z(0), u) indicate the state at time t of the linear control system ż = Az + Bu with initial state z(0) and control u(t), then (9) expresses that In the special case where system (1) is of Hammerstein type, so that we can write

x(t, x(0), ζ(0)) = Ψ ẑ t, H(x(0), ζ(0)), h(ζ) , ζ(t) . ( 11
ẋ = A 1 x + B 1 ϕ(u) (12)
where (A 1 , B 1 ) is a controllable pair in R n×(n+m1) with A strictly stable and ϕ a map from R m into R m1 assumed to be continuous, we see from (12), upon setting in (11 In particular, a Hammerstein system fed with the state of an arbitrary nonlinear system is a cascade of the form (11) where Ψ (thus also H) is independent of its second argument. This indicates that such systems are extremely special and highly non-generic among all nonlinear systems, already locally around a strictly stable equilibrium point.

) m = m 1 , q = m, A = A 1 , B = B 1 and h = ϕ, that H = ψ =

Harmonic-like identification

In this section, we assume throughout that f is continuously differentiable and that (1) has a strictly stable equilibrium at (0, 0). For simplicity, we also assume that (1) has a single input. Let us pick system [START_REF] Jakubczyk | On linearization of control systems[END_REF] to be a simple harmonic oscillator whose frequency is parametrized by the initial condition of the system's equations:

ζ = (ζ 1 , ζ 2 , ω) h(ζ) = ζ 1 ζ1 = -λζ 2 , ζ2 = λ ζ 1 , λ = 0. ( 13
)
Subject to initial conditions ζ 1 (0) = a and ζ 2 (0) = b, and λ(0) = ω, the solution to (13) is given by λ

(t) = ω and ζ 1 (t) = a cos(ωt) -b sin(ωt), ζ 2 (t) = a sin(ωt) + b cos(ωt), ( 14
) so that ζ 1 (t) + iζ 2 (t) = (a + ib)e iωt . Since ζ 2 = ζ 2 1 + ζ 2
2 is independent of t, we certainly have that system (13) is strongly asymptotically stable in the neighborhood of an equilibrium (0, 0, ω 0 ). Therefore, by Theorem 2 point (ii), ( 9) is valid for all t ≥ 0 if a, b are small enough and ω varies in the neighborhood of some reference frequency ω 0 .

By (13), ( 14) and the commutativity of convolution, we may rewrite the right-hand side of (9) in the case at hand as

e At H(x(0), ζ(0)) + t 0 e Aτ B a cos(ω(t -τ )) -b sin(ω(t -τ ) dτ.
Now, granted that A is strictly stable so that e At goes to zero exponentially fast as t → +∞, the above quantity is equivalent for large positive t to where F (iω) = (iωId -A) -1 B is the Fourier transform of e Aτ B evaluated at ω, that is to say F is the transfer function of the linear control system ż = Az + Bu.

From ( 9) and the previous calculation, we now see that for large positive t

H(x(t, x(0), ζ(0)), ζ(t)) ∼ Re (a + ib)e iωt F (iω) , ( 15
)
where the symbol "∼" means that the difference of the two quantities involved goes to zero uniformly (in fact exponentially fast) as t → +∞. Recalling from (11) the definition of Ψ and using that it is Hölder continuous, we deduce from (15) that

x(t, x(0), ζ(0)) ∼ Ψ Re (a + ib)e iωt F (iω) , ζ(t) , (16)
as t → +∞. Because ζ(t) is 2π/ω-periodic by ( 14), this entails that x(t, x(0), ζ(0)) tends uniformly to a limit cycle given by the right-hand side of (16). In particular, this limit cycle is independent of x(0). We will denote it by Γ(t, ζ(0)). Note that, from the identification point of view, Γ(t, ζ(0)) can be estimated from the steady-state output of (1) when the state of the latter is observed, at least for large enough t. Hereafter, we assume that this is indeed the case and that Γ(t, ζ(0)) is known for all t sufficiently large. The relation to be identified, which generalizes standard harmonic identification for linear systems, may then be written as:

Ψ Re (a + ib)e iωt F (iω) , ζ(t) = Γ(t, ζ(0)), ( 17
)
where ζ(0) = (a, b, ω) is a vector of control variables which can be chosen arbitrarily in a neighborhhod of (0, 0, ω 0 ), while (A, B, Ψ) is the set of unknown quantities to be determined. Observe, due to the presence of Ψ, that this set of unknowns lies in no finite dimensional space.

To better understand the situation, let us consider first the simplest but already interesting case where n = 1 so that F is complex-valued. Let V 0 be a ball around (0, 0) in R 2 and W 0 be a real interval centered at some ω 0 = 0, such that (16), thus also (17) are valid for all (a, b) ∈ V 0 and ω ∈ W 0 . Pick such a triple (a, b, ω) = ζ(0) with (a, b) = (0, 0), thereby giving rise to a limit cycle Γ(t, ζ(0)) through (17). Choose ω ∈ W 0 , with ω = ω, and define for each τ ∈ R a pair (a τ , b τ ) ∈ V 0 by the formula

a τ + ib τ = (a + ib)e i(ω-ω )τ . ( 18
)
Let ζ τ (t) be the solution to (13) with initial conditions a τ , b τ , ω . We claim that there are arbitrary large τ 0 for which

Γ(τ 0 , ζ(0)) = Γ(τ 0 , ζ τ0 (0)). ( 19
(iω + d)(iω + d) = 0, (24)
which is equivalent to d being the positive root of the equation

z 2 -z ω + ω tan(ωτ 0 + θ 0 ) -ωω = 0. ( 25
)
in the unknown z. Note that tan(ωτ 0 + θ 0 ) = 0, since d > 0 hence e i(ωτ0+θ0) cannot be real by (24). Solving (25), the coefficients of which are known, gives us d as announced. Altogether, we were able to recover F up to a multiplicative constant which is the best we can hope for since pre-multiplication by a constant can be incorporated to the definition of Ψ. Section 5 will illustrate this procedure numerically. We now discuss the case where the dimension n of system (1) is arbitrary. Then F (iω) is of the form

F (iω) = 1 q(iω)     p 1 (iω) p 2 (iω) • • • p n (iω)     (26)
where q is a monic polynomial of degree n with roots in the open left half-plane, since F is assumed to be stable, and p 1 , • • • p n are polynomials of degree at most n -1 with real coefficients which are jointly coprime with q.

We pick the dynamics of system (8) to be a concatenation of n harmonic oscillators of the form (13) with respective states (ζ 1,j , ζ 2,j , ω j ) for 1 ≤ j ≤ n. This results in a global state

ζ = (ζ 1,1 , ζ 2,1 , ω 1 , ζ 1,2 , ζ 2,2 , ω 2 , • • • , ζ 1,n , ζ 2,n , ω n ), (27)
with initial condition

ζ(0) = (a 1 , b 1 , ω 1 , a 2 , b 2 , ω 2 , • • • , a n , b n , ω n ), (28)
and we set h(ζ) = Σ n j=1 ζ 1,j . This time, computations similar to those which led us to (16) yield

x(t, x(0), ζ(0)) ∼ Ψ Re Σ n j=1 (a j + ib j )e iωj t F (iω j ) , ζ(t) , (29)
as t → +∞ and, instead of (17), we get

Ψ Re Σ n j=1 (a j + ib j )e iωj t F (iω j ) , ζ(t) = Γ(t, ζ(0)), (30)
where Γ(t, ζ(0)) indicates a limit cycle in R n which we suppose again to be known, since it can in principle be estimated by observing the steady state of system (1) fed by system [START_REF] Jakubczyk | On linearization of control systems[END_REF], provided that the ζ k,j (0) are sufficiently small and that the ω j are sufficiently close to reference frequencies ω 0,j . Extending a previous notation, we let V 0 be a ball around (0, 0) in R 2 and W 0,j be a real interval centered at ω 0,j such that (29) and (30) are valid as soon as all (a j , b j ) ∈ V 0 and ω 0,j ∈ W 0,j for all j ∈ {1, • • • , n}.

Pick (ω 1 , • • • , ω n ) ∈ W 0,1 × • • • × W 0,
n so that they are all distinct, nonzero, and the diference between any two of them is a rational multiple of 2π: ω i -ω j = π2n i,j /d i,j for some relative integers n i,j , d i,j . Choose also (ω

1 , • • • , ω n ) ∈ W 0,1 × • • • × W 0,n with ω j = ω j . The n + 1 vectors Re(F (iω j ) -F (ω j )), Im(F (iω n ) -F (ω n )) 1≤j≤n
are linearly dependent over R, hence there are real numbers ρ j , 1 ≤ j ≤ n, not all zero, and a real number

θ 0 such that Re Σ n-1 j=1 ρ j F (iω j ) -F (iω j ) +ρ n e iθ0 F (iω n ) -F (iω n ) = 0.
Set τ 0 = θ 0 /ω n , a n = ρ n , b n = 0, and (a j + ib j ) = ρ j e -iωj τ0 , 1 ≤ j ≤ n -1. Then, we have that

Re Σ n j=1 (a j + ib j )e iωj τ0 F (iω j ) -F (iω j ) = 0, (31)
and in (31) we may replace τ 0 by any number of the form τ 0 + kd where k is an interger and d a common multiple of the d i,j . In addition, we may assume that ρ 2 1 +• • •+ρ 2 n is fixed and small enough that (a j , b j ) ∈ V 0 for j ∈ {1, • • • , n}. Next, paralleling (18), we define for τ ∈ R: a j,τ + ib j,τ = (a j + ib j )e i(ωj -ω j )τ , 1 ≤ j ≤ n.

(32)

Let ζ(t), ζ τ (t) be defined by ( 27) with initial condition

ζ(0) = (a 1 , b 1 , ω 1 , a 2 , b 2 , ω 2 , • • • , a n , b n , ω n ) and ζ τ (0) = (a 1,τ , b 1,τ , ω 1 , a 2,τ , b 2,τ , ω 2 , • • • , a nτ , b n,τ , ω n )
respectively. From (32), ( 31) and (30), we deduce as before (cf. ( 19)) that there are (a j , b j ) ∈ V 0 , 1 ≤ j ≤ n, and arbitrary large τ 0 for which

Γ(τ 0 , ζ(0)) = Γ(τ 0 , ζ τ0 (0)). ( 33
)
Since both sides of (33) can be observed for τ 0 large enough by our very assumptions, it is in principle possible to find adequate values for a j , b j , and τ 0 . This time of course, determining the (a j , b j ) and τ 0 no longer reduces to a one-dimensional search that can be solved by dichotomy: it amounts to find zeros of a function of n + 1 variables on a rectangular domain (the numbers ρ j and τ 0 as introduced before (31)). Granted the normalization ρ 2 1 +• • •+ρ 2 n = ε small enough, this is indeed the correct number of unknowns to fulfill the n equations in (31)). The problem is difficult, of the same type as black-box optimization, and we shall not discuss here which algorithm or heuristics is adapted to solve it. We merely assume hereafter that the (a j , b j ) and τ 0 can be found.

Subsequently, we obtain a family of equations of the form (31), parametrized by the ω j (we fix hereafter the ω j ). We explain below how these equations allow us to determine the dynamics of the linearized system, that is, the polynomial q in (26). For the sake of simplicity, we only discuss the generic case where the p j in 29) and (30) by composing it to the right with a linear invertible transformation, we may assume that p j (z) = z j-1 so that (26) can be rewritten as

F (iω) = 1 q(iω)     1 iω • • • (iω) n-1     . (34) Then, F (iω j ) -F (iω j ) = q(-iω j )q(-iω j ) |q(iω j )| 2 |q(iω j )| 2 (35) ×     q(iω j ) -q(iω j ) iωq(iω j ) -iω j q(iω j ) • • • (iω) n-1 q(iω j ) -(iω j ) n-1 q(iω j )     ,
and substituting in (31) gives us, after chasing denominators, a system of n polynomial equations of degree 4n -1 in the n unknows coefficients of q (the fact that when n = 1 we found in (25) an equation for d of degree 2 rather than 3 is because the constant coefficient factors out in (35) when n = 1). This system can be solved by elimination, using for instance a Gröbner basis algorithm. The fact that, for generic choices of ω 1 , • • • , ω j ) at least, the set of solution indeed consists of isolated n-tuples (i.e. has dimension 0) and that exactly one of them yields a stable polynomial requires an argument that is beyond the scope of the present paper. The reconstruction of the map Ψ in (17), which could in principle be carried out pointwise once q is known, is also left for future study.

In the previous analysis, we did not take into account uncertainty arising from measurement errors nor numerical sensitivity. In this connection, let us mention that the ω j are essentially (locally) free parameters of the procedure described above, that can be used to cross-correlate the results and compute averages. We shall not dwell on this issue.

Numerical experiments

In this section, we illustrate the procedure described in the previous section in the case of a first order system, i.e. when n = 1. Although this is the simplest instance of our method, it is already interesting and it displays essential features of the algorithm without having to grapple with a heavy numerical search. The system under consideration is the simple circuit shown in figure 1, with input u(t) and output y(t) which is equal to the state x(t). The nonlinear map f is given by f (u) = u -10u 2 , and the values of the resistor and capacitor are R = 100Ω and C = 15.9µF respectively. The system has a fixed point at (0, 0) with strictly stable linear approximation, and in fact A ∼ -628.9. Note that the system is of Hammerstein type although this played no role in our analysis. With the notation of (20), we pick ω = 100π and ω = 150π, corresponding to frequencies 50Hz and 75Hz respectively. The vector (a, b) is chosen to be (0.001, 0) which corresponds to a small amplitude signal. Figure 2 shows the difference between both sides of (20) (i.e. the left hand side of (20)) when τ 0 gets replaced by a generic time instant τ , as a function of τ over a period. The seeked values for τ 0 , that is, time instants at which the right hand side of (23) vanishes, are those τ for which this difference is zero. It must be said that the use of harmonic balance techniques to simulate the output of the system gets us rid from the need to wait for the transient to vanish and the output to become in steady state. If we select for instance the value τ 0 = 0.5351 and solve (25) (with θ 0 = 0 and ρ = 0.001), we find that the positive root is 627.4 which is a rather good approximation of the theoretical d, namely 628.9.

We should mention that if we perform the same experiment except that (a, b) = (0.1, 0), that is, if we use a much larger amplitude signal, the error on d becomes as bad as 1000 (to be compared with 628.9), which seems to indicate that we leave the domain where the exact linearization asserted in Theorem 2 is valid.

The system in figure 1 has infinitely many other equilibrium points than (0, 0), associated with constant inputs. We performed experiments similar to the above for the value u ≡ 2. In figure 3, we show the diagram corresponding to figure 2 for this case. If we select for τ 0 from this diagram, whichever zero we choose, we find that the distance to the theoretical d when solving (25) is consistently of the order of 10 -5 (to be compared with 628.9), which is excellent.

  ) Thus, for fixed input u(t) = h(ζ(t)) resulting from a given ζ(0), the state trajectory x(t, x(0), ζ(0)) is obtained by applying the nonlinear map Ψ(., ζ(t)) to the state of the linear control system ż = Az + Bu initialized at z(0) = H(x(0), ζ(0)).

  Id will satisfy the latter equation when u = ζ, no matter what the function ζ is.

∞ 0 e 0 e 0 e

 000 Aτ B(a cos(ω(t -τ )) -b sin(ω(t -τ ))) dτ = ∞ Aτ B Re (a + ib)e iω(t-τ ) dτ = Re (a + ib)e iωt ∞ Aτ B e -iτ dτ = Re (a + ib)e iωt F (iω) ,

Figure 1 :

 1 Figure 1: System

Figure 2 :

 2 Figure 2: Γ(τ, ζ(0)) -Γ(τ, ζ τ (0)) as a function of τ .

Figure 3 :

 3 Figure 3: Γ(τ, ζ(0)) -Γ(τ, ζ τ (0)) as a function of τ .

  ib)e iωτ0 = (a τ0 + ib τ0 )e iωτ0 . Clearly, relation (21) holds if and only if interval of length at least π/ω whose center is a sufficiently large real number. As both Γ(τ, ζ(0)) and Γ(τ, ζ τ (0)) can be evaluated for each τ large enough, τ 0 can in principle be found by dichtomy because of the continuity of the map in (23). Note in passing that the two estimates involved have different degree of complexity: Γ(τ, ζ(0)) can be obtained for all τ in one stroke by recording the output of system (1) with input h(ζ) where ζ is the solution to (13) initialized at (a, b, ω), whereas each value Γ(τ, ζ τ (0)) requires to feed system (1) with input h(ζ τ ) where ζ τ is the solution to (13) initialized at (a τ , b τ , ω ), and then to evaluate the corresponding output at time τ .Once τ 0 is found, the pole of F is easily computed. For if we write F (z) = c/(z +d) where c, d are nonzero real numbers and d > 0 by the assumed stability of F , and if we set (a + ib) = ρe iθ0 where ρ > 0, then we can rewrite (21) as

	which amounts to
	Re (a + ib)e iωτ0 F (iω) -F (iω ) = 0	(21)
	since (a + τ 0 = -	arg{(a + ib)(F (iω) -F (iω ))} + kπ ω	, k ∈ N,	(22)
	which provides us with infinitely many equally spaced positive solutions to (19), thereby proving the claim.
	By (22), finding τ 0 meeting (19) reduces to a 1-dimensional search for the zeros of
		τ → Γ(τ, ζ(0)) -Γ(τ, ζ τ (0))		(23)
	over any real Im	e i(ωτ0+θ0)
					(20)

) Indeed, since Ψ is injective and ζ(τ ) = ζ τ (τ ) for each τ by (18), relation (19) is equivalent to the equality Re (a + ib)e iωτ0 F (iω) = Re (a τ0 + ib τ0 )e iωτ0 F (iω )