Rabah Rabah
email: rabah3.14159@gmail.com

On exact controllability and complete stabilizability for linear systems

Keywords: Exact controllability, complete stabilizability, infinite dimensional systems, neutral type 2010 Mathematics Subject Classification: 93B05, 93D15, 93C23, 93C43

This paper concerns the relation between exact controllability and stabilizability with arbitrary decay rate in infinite dimensional spaces. It appears that in several cases the notions are equivalent, but there are a lot of situations when additional conditions are needed, for example in Banach spaces. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces initiated by V. I. Korobov during the 70th of the past century in Kharkov State University.

Вiсник ХНУ, Сер. Математика, прикладна математика i механiка , том 94 (2021) 5

Introduction

We consider linear systems in the general form

ẋ = Ax + Bu, (1)
where the state x(t) and the control u(t) take values in Banach spaces X and U . The linear operator A is the infinitesimal generator of a C 0 -semigroup S(t) = e At and B is a linear bounded operator. We consider also the cases of Hilbert spaces and some situation when B is an unbounded but admissible operator. The function u(t) is supposed, at least, Bochner integrable. By exact (null) controllability we mean controllability from any state to any state (or zero state). By complete stabilizability we mean exponential stabilizability with arbitrary decay rate by linear state feedback u = Fx. Sometimes the problem is called pole or spectrum assignment.

The well known relation between complete controllability and pole assignment in finite dimensional spaces X and U may be formulated in the following form.

Theorem 1. Let A and B be (real or complex) matrices of dimension n × n and n × m respectively, x and u vector functions of suitable dimension. The system

ẋ = Ax + Bu (2)
is completely controllable, i.e. rank [B AB • • • A n-1 B] = n, if and only if the system is completely stabilizable, i.e. for all λ 1 , . . . , λ n ∈ C, there exists a feedback u = F x such that the spectrum of the closed loop system verifies:

σ(A + BF ) = {λ 1 , . . . , λ n }, i.e. the problem of pole assignment by linear feedback is solvable.

A nice proof in the general case can be found in [START_REF] Wonham | Linear multivariable control: a geometric approach[END_REF]. If the system is assumed to be real (matrices are all real), the spectrum is formed by real or complex conjugate numbers.

Another formulation of the same statement is that all the poles are controllable (Hautus criteria):

∀λ ∈ σ(A), rank[λI -A B] = n.
Our purpose is to discuss some important results on the same problems in infinite dimensional Hilbert or Banach spaces. Since the spectrum is more complicated and the behavior of infinite dimensional systems is not completely determined by the spectrum, solutions of such problems, when they exist, are not so simple.

We recall some classical results concerning the relation between exact controllability and complete stabilizability.

The first important result in this context was given by Slemrod: if e At is a group, then exact controllability implies complete stabilizability. The converse, for a group, was proved by Zabczyk (see for example [START_REF] Zabczyk | Mathematical control theory: an introduction[END_REF]). The result was generalized and precised by several authors for the case of a bounded operator A in Banach spaces, for the case of a semigroup e At and for some classes of systems, governed by partial differential equations or functional-differential equations.

We discuss more precisely the relations between exact (null) controllability and complete stabilizability. In general, for linear systems in Hilbert spaces, exact null controllability implies complete stabilizability, but the converse is not true. We give more recent results on functional-differential systems of neutral type.

Preliminaries

In this section we give some definitions and well known basic results on exact controllability and stabilizability for System [START_REF] Andrew | Range inclusion and factorization of operators on classical Banach spaces[END_REF]. The mild solution (strong for x 0 ∈ D(A) and some restrictions on the class of controls u) is given by

x(t) = x(t, u(•), x 0 ) = e At x 0 + t 0 e A(t-τ ) Bu(τ )dτ. (3)
Definition 1. System (1) is exactly controllable if for all x 0 , x 1 ∈ X, there exists a time T and a control u ∈ L p (0, T ; U ), p ≥ 1 (shortly L p ) such that x(T, u(•), x 0 ) = x 1 . If the same holds for x 1 = 0, we talk about exact null controllability. If e At is a group, or if it is formed by surjective operators, then both notions are equivalent. If necessary, the class of controls L p will be specified.

The notion of stabilizability is based on one concept of asymptotic stability. In infinite dimensional spaces there are several ones. In this paper we consider only exponential stability.

Definition 2.

The semigroup e At , or the uncontrollable system, is said to be exponentially stable if for some real M ≥ 1, ω > 0, and for all t ≥ 0, we have e At ≤ M e -ωt .

In finite dimensional space exponential stability is characterized by the fact that the spectrum of matrix A is in the left half complex plane. The situation in infinite dimensional spaces is much more complicated, see for example [START_REF] Van Neerven | The asymptotic behaviour of semigroups of linear operators. Operator theory advances and application[END_REF] and references therein. Definition 3. System (1) is said exponentially stabilizable if there exists a linear feedback control u(t) = Fx(t) such that the closed loop system becomes exponentially stable, i.e. for some ω > 0 e (A+BF )t ≤ M ω e -ωt , M ω ≥ 1.

The system is said to be completely stabilizable if it is exponentially stabilizable for all ω > 0.

Вiсник ХНУ, Сер. Математика, прикладна математика i механiка , том 94 (2021) 7

In some cases the feedback F is allowed to be not bounded but admissible in some sense.

Complete stabilizability replaces the notion of spectrum assignment which is difficult to be expressed in infinite dimensional spaces. The first reason is that the structure of the spectrum may be complicated and the second one is that, what is noted before, the behavior of the solutions of the systems is not determined by the spectrum only. However, in some cases it is possible to consider this concept also, as we will see later.

Let us now give some general characterizations of the given concepts. We denote by R T the "reachability"operator

R T u(•) = T 0 e A(T -τ ) Bu(τ )dτ.
It is easy to see that this is a linear bounded operator from L p to X. System (1) is exactly controllable if and only if Im R T = X, it is exactly null controllable if and only if Im R T ⊂ Im e AT , for some T > 0.

The surjectivity of the operator R T in Banach spaces can be characterized by the following property of the adjoint operator R * T (see, for instance, [START_REF] Rudin | Functional analysis[END_REF])

∃δ > 0 : ∀x * ∈ X * , R * T x * ≥ δ x * , (4)
where norms are operator or vector norms in the adjoint space. To obtain a more or less explicit condition of exact controllability we need the expression of adjoint operator and the norm R * T x * . We have the following expressions [START_REF] Rabah | Commandabilité des systèmes linéaires à retard constant dans les espaces de Banach (Controllability of linear systems with constant delay in Banach spaces[END_REF]:

         R * T x * = T 0 B * S * (t)x * q dt 1 q , if u ∈ L p , p > 1, R * T x * = ess sup{ B * S * (t)x * , t ∈ [0, T ]}, if u ∈ L 1 , (5)
where ess sup denotes the essential supremum of a set. In fact, it is possible to show that the conditions of exact controllability in the class of controls L p for p > 1 are equivalent and exact controllability in the class L 1 is essentially different. This can be summarized by the following formulation. This gives, in particular, the lack of exact controllability in the class L p for p > 1 for compact or analytic semigroups (see [START_REF] Triggiani | A note on the lack of exact controllability for mild solutions in Banach spaces[END_REF][START_REF] Triggiani | A note on the lack of exact controllability for mild solutions in Banach spaces[END_REF][START_REF] Louis | On exact controllability in Hilbert spaces[END_REF]). The following example shows that there exist systems which are exactly controllable in the class L 1 , but not exactly controllable in classes L p , p > 1.

Example 1 ([19]). Let X = U = 2 , {e i , i = 1, 2, . . . } be the canonical basis in 2 . Let B be defined as the identity operator B = I, and

Ax = ∞ i=1 -i x, e i e i , x = ∞ i=1 x, e i e i , ∞ i=1 i 2 | x, e i | 2 < ∞.
It is easy to see that A is the infinitesimal generator of a C 0 -semigroup

S(t)x = ∞ i=1 e -it
x, e i e i , t ≥ 0 of compact operators for t > 0. The system is exactly controllable in the class L 1 , because R * T x * L∞ = x * , but not in the class L p , p > 1, the operators S(t) being not surjective for t > 0.

2

Let us now consider the problem of null exact controllability. The characterization of this concept is not so clear in general Banach spaces. It is more simple when X and U are Hilbert spaces. This is due to the following classical result.

Theorem 3 [START_REF] Douglas | On majorization, factorization, and range inclusion of operators on Hilbert space[END_REF], [START_REF] Douglas | On majorization, factorization, and range inclusion of operators on Hilbert space[END_REF]). Let C and D be linear bounded operators in Hilbert spaces:

C ∈ L (H 1 , H 2 ), D ∈ L (H 3 , H 2 ).
The following conditions are equivalent:

1. Im C ⊂ Im D.
2. There is a linear bounded operator E such that C = DE.

∃µ ≥

0 : ∀x ∈ H 2 we have C * x 2 ≤ µ 2 D * x 2 .
This means that in Hilbert spaces the condition of exact null controllability Im R T ⊂ e AT may be characterized as follows.

Theorem 4. Let us suppose that X and U are Hilbert spaces. Then the system (1) is exactly null controllable in the class L 2 if and only if

∃δ > 0 : ∀x ∈ X, T 0 B * S * (t)x 2 dt ≥ δ 2 S * (T )x 2 .
Вiсник ХНУ, Сер. Математика, прикладна математика i механiка , том 94 (2021) 9

As it was pointed out by several authors (including Douglas, see for example [START_REF] Embry | Factorization of operators on Banach space[END_REF]), Theorem 3 is not true in this form in general Banach spaces. Some other formulations are possible (see, for instance, [START_REF] Embry | Factorization of operators on Banach space[END_REF][START_REF] Andrew | Range inclusion and factorization of operators on classical Banach spaces[END_REF]). We cannot develop this question here, but we will return to the application of Douglas theorem in Section 4.1.

Exact controllability implies complete stabilizability

In the section we will see how exact controllability can imply complete stabilizability. The main framework here is the Hilbert spaces setting.

Consider System (1) in the case of Hilbert spaces X and U . Under the condition that the operator A is the infinitesimal generator of a group e At , B ∈ L (U, X), Slemrod [START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF] obtained the first result in this way:

The condition

∃δ > 0, ∀x ∈ X : T 0 B * e -A * τ x 2 dτ ≥ δ x 2 (6)
implies complete stabilizability

∀ω > 0, ∃F ∈ L (X, U ), e (A+BF )t ≤ M ω e -ωt , M ω ≥ 1,
with a linear bounded feedback F given by F = -B * K -1 ω (T ), where

K ω (T ) = T 0 e -2ωt e -Aτ BB * e -A * τ dτ.
This operator K ω (T ) is, in fact, the Gramian of the system

ẋ = Ax + ωx + Bu,
and the group S ω (t) = e ωt e At has the infinitesimal generator (A + ωI).

The inequality in ( 6) is the condition of exact controllability at time T in the case of a group, and in Hilbert spaces. It corresponds to the Kalman integral criterion:

K 0 (T ) = T 0 e -Aτ BB * e -A * τ dτ
is bounded invertible. The fact that this is a necessary and sufficient condition of exact controllability in Hilbert spaces was published first (to the best of our knowledge) in [START_REF] Megan | On the space of linear controllable systems in Hilbert spaces[END_REF] for the case of a bounded operator A, but the proof, based on Theorem 3 (see [START_REF] Douglas | On majorization, factorization, and range inclusion of operators on Hilbert space[END_REF]), can easily be rewritten in the case of a group e At with unbounded operator A, or directly from [START_REF] Douglas | On majorization, factorization, and range inclusion of operators on Hilbert space[END_REF].

We can formulate this result as follows.

Theorem 5. For System [START_REF] Andrew | Range inclusion and factorization of operators on classical Banach spaces[END_REF], where A is the infinitesimal generator of a group, U and X being Hilbert spaces, exact controllability implies complete stabilizability.

Доведення. This a short proof, see [START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF] for more details. Let ω > 0 be given. Let us take the feedback control

u(t) = Fx(t) = -B * K -1 ω (T )x(t).
We obtain then the closed loop system

ẋ(t) = (A + ωI -BB * K -1 ω (T ))x(t),
where I is the identity operator, and the solution is given by the group S * ωF (t) generated by the operator

A + ωI -BB * K -1 ω (T ).
Let x 0 ∈ D(A * ), then a simple calculation gives (see Appendix in [START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF]):

d dt S * ωF (t)x 0 , K ω (T ) -1 S * ωF (t)x 0 = -e -ωT B * e -A * T S * ωF (t)x 0 -B * S * ωF (t)x 0 .
By density of D(A * ), this implies the uniform boundedness of S ωF (t)

∃M ≥ 1, S ωF (t) ≤ M, t ≥ 0.
This means that for all ω > 0, there exists a feedback law F, such that

S F (t) = e (A+BF )t ≤ M e -ωt , t ≥ 0.
This means that the system is completely stabilizable. 2

Following Slemrod and using some results by Lions, Komornik [START_REF] Komornik | Rapid boundary stabilization of linear distributed systems[END_REF] proved an analogous result for the case when B is unbounded. This was applied to systems governed by PDE with boundary control.

Shortly, with some conditions on the operator B, complete stabilizability is obtained from exact controllability and the feedback is

F = -JB * K -1 ω , K ω = T 0 e -2ωt e -Aτ BJB * e -A * τ dτ,
where J is the canonical Riesz isomorphism.

Вiсник ХНУ, Сер. Математика, прикладна математика i механiка , том 94 (2021) 11

Example 2. This allows to obtain the complete stabilization of systems like

                   ẅ(t, ξ) -∇w(t, ξ) = 0, (t, ξ) ∈ R + × Ω w(0, ξ) = f (ξ), ξ ∈ Ω ẇ(0, ξ) = g(ξ), ξ ∈ Ω w(t, ξ) = u(t, ξ) ξ ∈ ∂Ω.
The boundary control is given by the formula

u(t, ξ) = Fw(t, ξ) = -JB * K -1 ω (w(t, ξ), ẇ(t, ξ)) defined in the space H -1 (Ω) × L 2 (Ω): u(t, ξ) = ∂ ν (P w(t, ξ) + Q ẇ(t, ξ)),
where P and Q are linear and bounded. 2

In the general case of a semigroup, we have the following implication: Theorem 6. For System (1) in Hilbert spaces, with bounded operator B, exact controllability to zero implies complete stabilizability by bounded feedback F.

This is a consequence of the following result implicitly contained in the paper by Datko [START_REF] Datko | Uniform asymptotic stability of evolutionary processes in a Banach space[END_REF] (see also [START_REF] Zabczyk | Mathematical control theory: an introduction[END_REF]Ch. III. 4]).

Theorem 7. Exactly null controllable systems are exponentially stabilizable.

The latter and the following remark give Theorem 6.

Remark 1. Controllability of System (1) is equivalent to controllability of the shifted system:

ẋ = Ax + ωx + Bu.
This result may be extended to the case of an unbounded control operator. We consider the same system (1) in Hilbert spaces, with possible unbounded B and F, they are supposed to be admissible in a certain sense:

B bounded from U to the space X -1 which is the completion of X with the norm (λI -A) -1 x .

Theorem 8 ([24]

). If system (1), in Hilbert spaces, with admissible operator B is exactly null controllable in X -1 , then it is completely stabilizable by an admissible feedback and then, finally, by a bounded feedback F.

The converse is not true, and there are different situations [START_REF] Rabah | Exact null controllability, complete stabilizability and continuous final observability of neutral type systems[END_REF].

Example 3. In the space L 2 (0, +∞), consider the semigroup

S(t)f (ξ) = e -t 2 2 -ξt f (ξ + t), t ≥ 0, ξ ≥ 0.
It is not difficult to see that for this semigroup:

∀ω > 0, ∃M ω ≥ 1, S(t) ≤ M ω e -ωt .
We have σ(S(t)) = {0} and then σ(A) = ∅.

It is easy to see that, for some f = 0 we have S(t)f = 0 for any t ≥ 0. The system is completely stabilizable (B = 0) but not controllable.

2

Example 4. In the space L 2 (0, 1), consider the semigroup

S(t)f (ξ) = f (ξ + t) 0 ≤ t + ξ ≤ 1, 0 t + ξ > 1.
It is not difficult to see that for this semigroup

∀ω > 0, ∃M ω ≥ 1, S(t) ≤ M ω e -ωt .
We have σ(S(t)) = {0} and then σ(A) = ∅.

For any initial function f ∈ L 2 (0, 1), we have S(t)f (x) = 0 for t > 2. This means that S(t) = 0, for all t > 2. Then, for any control operator B, the system is exactly null controllable at time T > 2 with the trivial control u = 0. 2

Complete stabilizability implies exact controllability

In this Section we consider how complete stabilizability may imply exact (null) controllability, and is there an equivalence in some situation as in finite dimensional spaces.

Banach space, bounded case

Consider System (1) where operators A and B are linear, bounded: A ∈ L (X) and B ∈ L (U, X). This means that it is not the case of partial differential or functional-differential equations (with delay). Anyway, it is of historical and theoretical interest.

An important result on complete stabilizability was obtained in [START_REF] Sklyar | The problem of the perturbation of an element of a Banach algebra by a right ideal and its application to the question of the stabilization of linear systems in Banach spaces[END_REF]. First we formulate result about exact controllability.

Theorem 9 (Korobov-Rabah (1979), [START_REF] Korobov | Exact controllability in Banach space[END_REF]). System (1) is exactly controllable if and only if there exists k ∈ N such that

Im [B AB • • • A k B] = X, (7)
where

[B AB • • • A k B] is acting from U × • • • × U to X
and the condition is independent on the class of controls.

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In fact the exact controllability holds in the class of piece-wise controls, when the condition of the theorem is verified.

Theorem 10 (Sklyar (1982), [START_REF] Sklyar | The problem of the perturbation of an element of a Banach algebra by a right ideal and its application to the question of the stabilization of linear systems in Banach spaces[END_REF]). System (1) with bounded operators is completely stabilizable if and only if the block operator [B AB • • • A k B] is right invertible for some k:

∃P i ∈ L (X, U ), i = 0, 1, . . . , k : BP 0 + ABP 1 + • • • + A k BP k = I. (8)
There is a spectral formulation, which is a consequence of the following result.

Theorem 11 [START_REF] Kaashoek | Analytic operator functions with compact spectrum. III: Hilbert space case: Inverse problem and applications[END_REF], [START_REF] Kaashoek | Analytic operator functions with compact spectrum. III: Hilbert space case: Inverse problem and applications[END_REF]). The operator

[B AB • • • A k B] is right invertible for some k if and only if operators [λI -A B], from X × U to X are right invertible for all λ ∈ C, i.e. ∀λ ∈ C, ∃Q λ ∈ L (X), ∃P λ ∈ L (X, U ) : (λI -A)Q λ + BP λ = I. ( 9
)
This gives a spectral formulation of the characterization of complete stabilization.

Theorem 12. System (1) is completely stabilizable if and only if operators [λI -A B] are right invertible for all λ ∈ C.

If U, X are Banach spaces, not isomorphic to Hilbert spaces, the condition (8) (or the spectral formulation ( 9)) is more than exact controllability, it is equivalent to the following two conditions:

1. Im [B AB • • • A k B] = X, i.

Ker [B AB • • • A

k B] is complemented.
This means that in Banach spaces (not isomorphic to Hilbert spaces) there are exactly controllable systems, generated by a group, which are not completely stabilizable.

Example 5 [START_REF] Sklyar | The problem of the perturbation of an element of a Banach algebra by a right ideal and its application to the question of the stabilization of linear systems in Banach spaces[END_REF]). Consider the Banach space U which has at least a subspace L which cannot be complemented in U . Let X be the factor space U/L and B be the canonical projection of U onto X = U/L. The operator B is then onto (surjective). This means that the system ẋ = x + Bu, is exactly controllable because Im B = X (here A = I). If we suppose that it is completely stabilizable, then, according to Theorem 8, there exists P ∈ L (X, U ) such that FP = I, and Ker P = L can be complemented. This is in contradiction with the construction of L.

2

The result of Theorem 10 was proved in [START_REF] Markus | Complete controllability of spectrum assignment in infinite dimensional spaces[END_REF] without citing the original result [START_REF] Sklyar | The problem of the perturbation of an element of a Banach algebra by a right ideal and its application to the question of the stabilization of linear systems in Banach spaces[END_REF].

The situation is different in Hilbert spaces: every closed subspace can be complemented and the equivalence between range condition [START_REF] Kaashoek | Analytic operator functions with compact spectrum. III: Hilbert space case: Inverse problem and applications[END_REF] and right invertibility (8) can be given by Douglas theorem.

Remark 2. When this paper was under review, appeared a conference paper [START_REF] Makarov | On Assignment of the Upper Bohl Exponent for Linear Discrete-Time Systems in Infinite-Dimensional Spaces[END_REF] on exact controllability and complete stabilizability (in fact arbitrary assignability of the Bohl exponent) of a class of linear non-stationary discrete systems in Banach spaces. The main result concerns a case when the state operator is constant, the control operator is periodic, the space X is a reflexive Banach space and U is a separable Hilbert space: it is claimed that exact controllability implies arbitrary assignability of Bohl exponents. No proofs are given in this paper.

Hilbert spaces, bounded case, spectrum assignment

In Hilbert spaces, exact controllability is equivalent to complete stabilizability, but more precisely we have a pole assignment result which may be interesting, at least in operator theory.

Theorem 13 [START_REF] Eckstein | Exact controllability and spectrum assignment[END_REF] [START_REF] Eckstein | Exact controllability and spectrum assignment[END_REF]). The system (1) is exactly controllable if and only if for every nonempty compact set Λ ⊂ C, there is a bounded linear feedback F, such that σ(A + BF) = Λ, i.e. the problem of spectrum assignment is solvable.

The proof uses explicitly the Hilbert space structure, in particular the fact that the subspace Ker[B AB • • • A k B] can always be complemented. There is also a dual formulation of this result in [START_REF] Kaashoek | Analytic operator functions with compact spectrum. III: Hilbert space case: Inverse problem and applications[END_REF] and a constructive approach of the dual operator corresponding to F: let C be an operator from X to the Hilbert space Y , find K such that σ(A + KC) = Λ. This appears in the problem of detectability and construction of asymptotic observers.

To summarize the characterization in Hilbert spaces, when the operators are bounded, we can formulate as follows.

Theorem 14. Let U and X be Hilbert spaces, A and B be linear bounded operators. The following statements are equivalent 1. The system (1) is exactly controllable in the class L 2 .

2. For every nonempty compact set Λ ⊂ C, there is a bounded linear feedback F, such that σ(A + BF) = Λ, i.e. the problem of spectrum assignment is solvable.

Im [B AB • • • A

k B] = X for some k ∈ N,
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4. Im [λI -A B] = X for all λ ∈ C.
This is a complete analog of Theorem 1.

Unbounded case in Hilbert spaces

We return to the system (1) with unbounded A and bounded B.

Theorem 15 [START_REF] Megan | On the space of linear controllable systems in Hilbert spaces[END_REF], Zabczyk (1976), see [START_REF] Zabczyk | Mathematical control theory: an introduction[END_REF]). If e At is a group and the system (1) in Hilbert spaces is completely stabilizable, then it is exactly controllable.

This result was extended to the case of surjective operators e At , t ≥ 0 by Rabah & Karrakchou (1997) and Zeng & Xie & Guo (2013), see [START_REF] Rabah | On Exact Controllability and Complete Stabilizability for Linear Systems in Hilbert Spaces[END_REF]35].

We can summarize as: Theorem 16. The system (1) in Hilbert spaces, with a bounded operator B, is exactly controllable in the class L 2 if and only if 1. Operators e At are surjective for t ≥ 0.

The system is completely stabilizable.

Exact controllability is equivalent to exact null controllability when e At are surjective for t ≥ 0.

But what about exact controllability in the class L 1 in this context? Surjectivity of operators e At is not needed. Let us return to Example 1.

Example 6 (Example 1 continued). As we saw, the system is exactly controllable in the class L 1 but not in the class L p for p > 1. Let us show that it is completely stabilizable by a bounded feedback. Let us take arbitrary ω > 0 and n ∈ N such that n > ω. Define operator F from X = 2 to U = 2 by

Fx = n-1 i=1 (i -n) x, e i e i .
Then

(A + BF)x = n-1 i=1 -n x, e i e i + ∞ i=n -i x, e i e i , which gives for the semigroup S F (t) = e (A+BF )t e (A+BF ) x 2 = n-1 i=1 e -2nt | x, e i | 2 + ∞ i=n e -2it | x, e i | 2 ≤ e -2nt x 2 .
And this means that e (A+BF ) ≤ e -nt ≤ e -ωt , i.e. the system is completely stabilizable.

The case of neutral type systems

We consider some large class of neutral type systems

ż(t) = A -1 ż(t -1) + Lz t (•) + Bu(t), (10)
where

Lz t (•) = 0 -1 [A 2 (θ) ż(t + θ) + A 3 (θ)z(t + θ)] dθ,
with z(t) taking values in R n and u(t) in R p . The system can be written as

ẋ = Ax + Bu, (11)
where A is the infinitesimal generator of a C 0 -semigroup e At given in the product space

M 2 (-1, 0; R n ) def = R n ×L 2 (-1, 0; R n )
, noted shortly M 2 , and A, B are defined by

Ax(t) =   Lz t (•) dz t (θ) dθ   , x(t) = z(t) z t (•) , Bu(t) = u(t) 0 ,
with the domain D(A) given by

D(A) = (v, ϕ) ∈ M 2 : ϕ(•) ∈ H 1 , v = ϕ(0) -A -1 ϕ(-1) .
The characteristic matrix is noted by

∆ A (λ) = λI -A -1 e -λ -λ 0 -1 A 2 (θ)e sθ dθ - 0 -1 A 3 (θ)e sθ dθ.
The behavior of the dynamical system is described by the spectrum:

σ(A) = {λ : det ∆ A (λ) = 0}.
The set σ(A) is formed by eigenvalues of finite multiplicity, see for example [START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF] and references therein.

Complete stabilizability

It is characterized by two conditions: one on the spectrum σ(A) , the second on the neutral term (here A -1 ).

Theorem 17 ([20]). The system [START_REF] Korobov | Exact controllability in Banach space[END_REF], in abstract form [START_REF] Louis | On exact controllability in Hilbert spaces[END_REF], is completely stabilizable if and only if

1. For all λ ∈ C, rank ∆ A (λ) B = n. 2. For all µ ∈ C, µ = 0, rank µI -A -1 B = n.
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The condition 2 may be formulated as

Im A -1 ⊂ Im [B A -1 B • • • A n-1 -1 B].
Proof. It is important to check the meaning of the second condition. A large part of the spectrum of A may be obtained from the non zero spectrum of the matrix A -1 :

λ = log |µ| + i(arg µ + 2kπ) + o(1/|k|), k ∈ Z.
The eigenvalues are in some circles centered at log µ with decreasing radii, when |k| is growing [START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF]. The pole assignment is possible in the interior of the circles under Condition 1, which means the controllability of poles.

The spectrum of the operator A is close to some vertical axes defined by the spectrum of the neutral term, the matrix A -1 . We need to move these axes by the feedback in order to obtain complete stabilizability, and then the corresponding values µ must be controllable. The necessity of both conditions is then clear enough.

Let us shortly consider the sufficient part. The condition 2 allows to move arbitrary all the non zeros eigenvalues of A -1 , and to make it simple. The condition 1 allows to control an eigenvalue in the interior of circles (for k large). It remains then to move a finite number of eigenvalues, it is possible as shown in [START_REF] Rabah | On strong regular stabilizability for linear neutral type systems[END_REF][START_REF] Rabah | Stability and stabilizability of mixed retarded-neutral type systems[END_REF].

It is of interest to show the form of the feedback needed for exponential and then for complete stabilizability:

u(t) = Fx(t) = F -1 ż(t -1) + T 0 [F 2 (θ) ż(t + θ) + F 3 (θ)z(t + θ)] dθ.
This means that F is not bounded and domains of A + BF and A are not the same. However this form is necessary in order to move the spectrum [START_REF] O'connor | On stabilization by state feedback for neutral differential equations[END_REF][START_REF] Pandolfi | Stabilization of neutral functional differential equations[END_REF].

Complete stabilizability and exact controllability

The notion of exact controllability for System (10) must be modified. As here the control u is finite dimensional, exact controllability in the sense of Definition 1 is not possible. In fact it may be shown that, in this case, the reachability set R T is in D(A) [START_REF] Rabah | The analysis of exact controllability of neutral-type systems by the moment problem approach[END_REF][START_REF] Rabah | Stability and stabilizability of mixed retarded-neutral type systems[END_REF]. Then by exact controllability at time T , we mean R T = D(A), and exact null controllability means to reach zero from all initial condition in D(A). Results given in [START_REF] Rabah | The analysis of exact controllability of neutral-type systems by the moment problem approach[END_REF][START_REF] Rabah | Stability and stabilizability of mixed retarded-neutral type systems[END_REF] may be summarized in the following theorem. Theorem 18. If A is the infinitesimal generator of a group or, equivalently, if the matrix A -1 is not singular, 0 / ∈ σ(A -1 ), then exact controllability is equivalent to complete stabilizability.

Proof. If µ = 0 is not in σ(A -1 ), conditions 1 and 2 of Theorem 17 give a criteria of exact controllability. We refer to [START_REF] Rabah | The analysis of exact controllability of neutral-type systems by the moment problem approach[END_REF][START_REF] Rabah | Stability and stabilizability of mixed retarded-neutral type systems[END_REF] for detailed proofs. In our case of System [START_REF] Korobov | Exact controllability in Banach space[END_REF], for exact null controllability we can obtain only a partial result.

Theorem 19. Exact controllability to zero of System (10) implies complete stabilizability.

Proof. If Condition 1 in Theorem 17 is not verified, it means that some eigenvalues are not controllable. Taking a corresponding eigenvectors as an initial condition, we can see that exact controllability does not hold. To prove Condition 2 in Theorem 17, we can proceed by derivation of the solution z(t) = 0 for t ≥ T ≥ n. The calculation gives [START_REF] Rabah | Exact null controllability, complete stabilizability and continuous final observability of neutral type systems[END_REF]), which means that Condition 2 in Theorem 17 is verified.

Im A -1 ⊂ Im [B A -1 B • • • A n-1 -1 B] (see
The converse (in fact the criteria of exact null controllability without relation with stabilizability) has been proved in some particular cases of neutral type systems with discrete delays [START_REF] Metel Skiȋ | Criteria for the constructive identifiability and complete controllability of linear time-independent systems of neutral type[END_REF][START_REF] Khartovskiȋ | Complete controllability and controllability for linear autonomous systems of neutral type[END_REF] and in the case of retarded systems [START_REF] Olbrot | Null controllability of a class of functional differential systems[END_REF]. In the case of our general system [START_REF] Korobov | Exact controllability in Banach space[END_REF], the question is still open, see Conjecture in [START_REF] Rabah | Exact null controllability, complete stabilizability and continuous final observability of neutral type systems[END_REF] and [START_REF] Barkhayev | Conditions of exact null controllability and the problem of complete stabilizability for time-delay systems[END_REF].

Conclusion

We gave a short review of the relations between exact controllability and pole assignment (complete stabilizability). There are a lot of papers concerning different classes of systems, in particular, governed by partial differential equations. This paper gives our point of view to some questions concerning abstract infinite dimensional systems, based on results close to ours, in particular for functional differential equations of neutral type. Several problems are still open concerning the general abstract case or some particular classes. Добре вiдома еквiвалентнiсть мiж повною керованiстю та розташуванням полюсiв у ситуацiї скiнченновимiрних просторiв в загальному випадку не має мiсця в нескiнченномiрних просторах. Точної керованостi недостатньо для повної стабiлiзацiї, якщо U i X є банаховими просторами. У постановцi Гiльбертового простору цей результат є справедливим. Зворотньє твердження також не є простим: в деяких ситуацiях повна стабiлiзацiя передбачає точну керованiсть (у просторi Банаха з обмеженими операторами), в iнших ситуацiях це не вiдповiдає дiйсностi. Вiдповiднi результати наведенi з деякими iдеями доведення. Технiчнi деталi можна знайти в цитованiй лiтературi. Наведено кiлька прикладiв. Особлива увага придiляється випадку нескiнченновимiрних систем, збудованих за системами iз загаюваннями нейтрального типу iз розподiленими загаюваннями. Бiльш детально дослiджується питання про зв'язок мiж точною нуль-керованiстю та повною стабiлiзацiєю. Загалом мiж цими поняттями немає еквiвалентностi. Однак для деяких класiв рiвнянь нейтрального типу iснує еквiвалентнiсть. Питання про те, чи є еквiвалентнiсть для бiльш загальних систем, залишається вiдкритим. Це короткий i невичерпний огляд деяких дослiджень теорiї керування у нескiнченновимiрних просторах. Нашi роботи в цiй сферi були iнiцiйованi В. I. Коробовим протягом 70-х рокiв минулого сторiччя у Харкiвському державному унiверситетi.

Ключовi слова: Точна керованiсть; повна стабiлiзацiя; нескiнченновимiрнi системи; нейтральний тип.

On exact controllability and complete stabilizability for linear systems Rabah Rabah IRCCYN, Ecole des Mines de Nantes 1 Rue de la Noë, 44300 Nantes, France In this paper we consider linear systems with control described by the equation ẋ = Ax + Bu where functions u and x take values in U and X respectively. For such object, a short review of results concerning relations between exact controllability and complete stabilizability (stabilizability with arbitrary decay rate) is given. The analysis is done in various situations: bounded or unbounded state and control operators A and B, Banach or Hilbert spaces U and X.

Вiсник ХНУ, Сер. Математика, прикладна математика i механiка , том 94 (2021) 23

The well known equivalence between complete controllability and pole assignment in the situation of finite dimensional spaces is no longer true in general in infinite dimensional spaces. Exact controllability is not sufficient for complete stabilizability if U and X are Banach spaces. In Hilbert space setting this implication holds true. The converse also is not so simple: in some situations, complete stabilizability implies exact controllability (Banach space setting with bounded operators), in other situation, it is not true. The corresponding results are given with some ideas for the proofs. Complete technical development are indicated in the cited literature. Several examples are given. Special attention is paid to the case of infinite dimensional systems generated by delay systems of neutral type in some general form (distributed delays). The question of the relation between exact null controllability and complete stabilizability is more precisely investigated. In general there is no equivalence between the two notions. However for some classes of neutral type equations there is an equivalence. The question how the equivalence occurs for more general systems is still open. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces. Our works in this area were initiated by V. I. Korobov

  e. exact controllability by Theorem 9.

Fig. 1 .

 1 Fig. 1. Spectrum of A, for each µ ∈ σ(A -1 ), µ = 0.

  This implies a necessary condition of exact controllability in the class L p for p > 1: operators S(t) = e At are surjective for all t ≥ 0.

	Theorem 2 ([19]). System (1) is exactly controllable in the class L p if and only
	if
	1. For p = 1, if lim n→∞	B * S * (t)x * n = 0 uniformly almost everywhere on [0, T ]
	implies lim n→∞	x * n = 0.
	2. For p > 1, if lim n→∞	B * S * (t)x * n = 0 almost everywhere on [0, T ] implies
	lim n→∞	x * n = 0.

  35. Yi. Zeng, Z. Xie, F. Guo. On exact controllability and complete stabilizability for linear systems. Appl. Math. Lett., -2013. -7. 26. -P. 766-768, DOI: 10.1016/j.aml.2013.02.008. Ax + Bu, де функцiї u та x приймають значення U та X вiдповiдно. Для такого об'єкта подано короткий огляд результатiв, що стосуються зв'язкiв мiж точною керованiстю та повною стабiлiзацiєю (стабiлiзацiя з довiльною швидкiстю спаду). Аналiз проводиться в рiзних ситуацiях: обмежений чи необмежений стан та оператори керування A та B, простори Банаха або Гiльберта U та X.

	Про точну керованiсть та повну стабiлiзацiю
	для лiнiйних систем
	Рабах Рабах
	НДIЗКН, Вища гiрнича школа м. Нант
	вул. Ное 1, Нант, Францiя, 44300
	У цiй роботi ми розглядаємо лiнiйнi системи з керуванням, описанi рiвнянням
	ẋ =

  during the 70th of the past century in Kharkov State University. Key words: Exact controllability; complete stabilizability; infinite dimensional systems; neutral type. Article history: Received: 31 August 2021; Final form: 19 October 2021; Accepted: 20 October 2021.


Keywords: promises, futures, regimes of promising, emerging science and technologies, innovation policy, mission-oriented policy

This editorial introduces the basic idea and content of the special section "Enemies of the future? Questioning the regimes of promising in emerging science and technology". It sheds light on how visions and stories of technological futures are being produced and, in combination, have stabilized into 'regimes of promising' that shape emerging technoscientific domains and guide research, innovation and governance within these domains. The special section zooms in to particular illustrative instances that reveal the dynamics of the dominant regime of promising and also reveal how they are breaking down, and in some cases being replaced by alternatives. The purpose of this special section is to show the importance of regimes of promising for research and innovation policy and to stimulate further discussions on alternative regimes of promising and their ramifications.

Introduction

Promises, future visions, expectations and imaginaries are not merely accompanying, they are also shaping modern science and technology. They support scientific enthusiasm and stimulate competition, raise financial resources, and orient research funds (Van Lente 2012). It has been argued that emerging science and technological fields rely heavily on the production and circulation of promises (Borup et al. 2006, Pollock and[START_REF] Pollock | The business of expectations: How promissory organizations shape technology and innovation[END_REF]. As a result, future visions and promises about emerging science and technologies such as nano-, bio-, info-and cognitive (NBIC) technologies, personalised medicine and synthetic biology, have been increasingly forged and brought to public attention [START_REF] Robinson | Distinguishing the umbrella promise of Converging Technology from the dynamics of Technology Convergence[END_REF], OECD 2014). A large flux of visions and stories of technological futures are being produced, are circulating and, together, these efforts and concomitant interdependencies have stabilized into 'regimes of promising' that provide rules and mechanisms in current technoscientific domainshow futures are being produced and how they are mobilised to shape the present.

More than visions or imaginaries, which highlight the content of potential futures, techno-scientific promises establish a relationship between the future and the present [START_REF] Joly | On the Economics of Techno-scientific Promises. Débordements : Mélanges offerts à Michel Callon[END_REF][START_REF] Audétat | Pourquoi tant de promesses? Sciences et Technologies émergentes[END_REF]. Promises can be mobilised to construct "horizons of hope" that fuel and forge the expectations of diverse technoscientific actors. In turn, such expectations shape agenda setting and action; as a result, promises can create requirements to be fulfilled by technoscientific actors and other corners of society [START_REF] Parandian | Dual dynamics of promises, and waiting games around emerging nanotechnologies[END_REF]Rip 2012, Robinson, Le Masson and[START_REF] Robinson | Waiting games: innovation impasses in situations of high uncertainty[END_REF]. Importantly, technoscientific promises are relational. As a rule, technoscientists promise better futures to diverse audiences (venture capital, stakeholders, policy makers, the wider publics) who are supposed to act upon them. In some cases, they genuinely act as "promise entrepreneurs". Elon Musk is a prominent example of a promise entrepreneur [START_REF] Tutton | Sociotechnical Imaginaries and Techno-Optimism: Examining Outer Space Utopias of Silicon Valley[END_REF].

Over the past five decades, the phenomenon of technoscientific promising has become systematised and embedded in the governance of new and emerging science and technologies. In other words, a regime of technoscientific promising has emerged. Understanding the regime(s) of technoscientific promising1 is important for public policy. Speculative bubbles associated to hype/hope/disappointment may generate instabilities, for instance, and one may fear that current challenge-driven research and innovation policy will generate distrust and a crisis of legitimacy of researchparticularly when research and innovation are presented as the only way to achieve "grand societal challenges". Moreover, regimes of technoscientific promising are no longer just a feature of "applied" or "strategic" technoscience, they have come to dominate the whole system of research since the start of big national programs on biotechnology, nanotechnology and brain research and the growing importance of project funding for the support of science. Moreover, the trend in new public management of research places as a requirement that researchers and research organisations must demonstrate how their research will impact society in a beneficial wayhype and hope are being institutionalised, requiring a counterbalance of controlled speculation through a variety of anticipation and reflexivity support systems [START_REF] Joly | Research Impact Assessment: From ex-post to real-time assessment[END_REF].

Likewise, regimes of promising are challenged by other ways to articulate research with societal expectations, like the slow science movement (Alleva 2006), participatory research and community-based research, frugal innovation [START_REF] Pisoni | Frugal approach to innovation: State of the art and future perspectives[END_REF]) and citizen-driven co-creation [START_REF] Robinson | RRI legacies: co-creation for responsible, equitable and fair innovation in Horizon Europe[END_REF]) and calls for "Responsible Stagnation" (de Saille & Medvecky 2016).

In this special section, four articles that critically examine the dominant regime of technoscientific promising are brought together to highlight the limits of the dominant regime of technoscientific promising and to shed light on what could be called alternative regimes of technoscientific promising.

Four contributions

Gaia Barazzetti and colleagues observe that at the end of the race to map the human genome, a new cycle of technoscientific promising is being geared towards the prospect of personalized medicine. Drawing on the increasing speed of DNA sequencing at decreasing costs, personal medicine has gained attention in many countries keen to support biomedical research like Switzerland. The article shows that, while competing to obtain public support for large research infrastructures with ambitious discourse of preventing and curing diseases, the overall promise of personalized medicine is actually being reformulated by stakeholders. Using the concept of "ethical boundary work", Barazzetti et al. focus on the negotiation between promoters, researchers, clinicians, the authorities and public health actors, concerning the uncertainties emerging around the promise of personalized medicine. Challenged to be explicit about whom personalized medicine is going to benefit, and also challenged to define and clearly articulate the responsiblities and warrants which need to be established for biobanking, the promise is renamed "personalized health" in order to satisfy ethical and governance principles, and to gain legitimation by patients and citizens. The prospect of personalised health sustains a stable network of stakeholders and clears the way to larger projects of funding at the national level. As a consequence, a new cycle of promise and requirements is in play, bound to (and conditioned by) the Swiss context, displaying a distinct regime of technoscientific promising -shaped by the local and national circumstances.

Matthias Wienroth explores the regime of technoscientific promising concerning genome editing for human health, in the United Kingdom context. Focusing on the emergence and governance of genomeediting and how the regime drives and shapes the articulation of public goods and their beneficiaries, Wienroth offers a critique on how specific publics are created, leading to a specific lock in, and lock out, of a variety of publics and the associated norms and values. This, Wienroth argues, has led to a restricted articulation of the "moral futures", associated with genome-editing and amplified through the regime of promising, leading to potential challenges for the anticipatory governance of genome editing in the United Kingdom.

In general, regimes of promising structure the ways societies connect to the future. The paper by Pierre-Benoit Joly and Claire Le Renard investigates this phenomenon in-depth by historicizing the societal relationship to the future. They draw on the concept of "regime of historicity", defined as an organizational structure that is imposed on the societal experience of time and which is mobilized to articulate the present, the past and the future. Following this line, Joly and Le Renard suggest that, so to say, the nature of "future" has changed. In what they call a presentist regime, the future is much more fragile and problematic than in the modernist regime which prevailed until the 1990s.

Drawing from four case studies, Joly and Le Renard argue that the way techno-scientific promises align with regimes of historicity is crucial, showing that promising in the presentist regime using the modernist frame of innovation (creative destruction) raises many problems. When the future is fragile, while confidence in technological progress has faded away on the grounds of many controversies about their social and environmental consequences, the foundations of promises can no longer be the same. Instead, they argue, that the building of "horizons of hope" may rest on processes of collective experimentation, and thus, an alternative regime of technoscientific promising.

The contribution of Ferretti and Van Lente delves deep into the promises of science and technology to include citizens in its developments, to achieve a better alignment between science and society. In the last decade, the dominant regime of techoscientific promising has increasingly included values and norms linked to democracy and representation. In policy circles, for instance, Ferretti and Van Lente argue that the hope is that 'citizen science' and its various guises will help to connect science and society better and that this will sustain the legitimacy and efficacy of public investments in science and technology. In other settings, such as in hacker communities, alternative notions of citizen engagement with science and technology prevail, creating a strong contrast to the dominant regime of technoscientific promising.

Focusing on the Maker Movement, Ferretti and Van Lente map the rise across Europe of citizen engagement, studying the practices, aspirations, values and motivations of the participants. A key theme of the paper is the apparent contrast between policy expectations of the Maker Movement and the expectations of the Maker Movement itself. While EU Policy documents frame the rise of the Maker Movement as supporting the promise that including citizens will reinforce the development and uptake of public science and technology, the Maker Movement, on the other hand, stresses that it is much more than just including citizens in science and technology. Other motivations and drivers, such as fun through technology development and curiosity, for instance, are highly valued and seen as missing in established science and technology. The promise of the Maker Movement, thus, can be read as a criticism of mainstream science and innovation systems, instead of a support of policies to further include citizens. In this way, the contribution of Ferretti and Van Lente provides insights into what might be seen as an alternative regime of promising regarding the development and deployment of new and emerging technologiesa regime not operating as a monolithic force but being shaped by ongoing contestations and reinforcements from a variety of innovating stakeholders.

Discussion and outlook

The thesis of the existence of a dominant regime of technoscientific promising refers to the intensification and systematisation of the use of expectations, visions and hype to govern technical research and development over the past five decades. The regime of technoscientific promising as a key driver for mobilising resources and driving technology development has come about through a tri-partite move of (a) the future being more of a threat that a hope for humanity [START_REF] Innerarity | The future and its enemies: in defense of political hope[END_REF], (b) research and technology development appearing as the solution to making the future a safer place and (c) the increasing pressure to show how public investments into new science and technologies will lead to desirable societal impacts [START_REF] Joly | Research Impact Assessment: From ex-post to real-time assessment[END_REF]. All four contributions in this special section reveal that this dominant regime of technoscientific promising still remains very visible today, but that it is evolving and, in some cases, breaking down.

Presently we are witnessing a shift towards challenge-oriented policies for research and innovation. Sustainable Development Goals and mission-oriented policies [START_REF] Mazzucato | Mission-oriented innovation policies: challenges and opportunities[END_REF]) are shifting the perspective from what technologies can promise towards what society actually needs. Eventually, this "directionality turn" may require new regimes of promising, such as the "regime of collective experimentation" as argued by Joly and Le Renard, or the promises of the hacker culture, as studied by Ferretti and Van Lente. Yet, there is a danger also that the directionality turn may lead to repackaging the business-as-usual activities of new and emerging science and technologies, thus reinforcing the current regime of technoscientific promisingbigger promises for bigger goals. In this way, the dominant regime would block novel engagements and creative new pathsand thus appear as an 'enemy of the future'. This unfortunate possibility is already visible in the promotion of research and innovation to tackle challenges such as climate change. Here, an emphasis is placed on the role of new and emerging technologies in the transitions from present situations, such as pesticide-based agriculture or fossil fuel heavy energy systems, towards greener situations, such as agroecology approaches to agriculture and clean energy. What is receiving less attention in this discourse, and also in green transition policies, are the painful effects of destabilising the incumbent situation, for example in agriculture and energy supply and use. The need to discontinue existing systems is not included in the narratives bandied about through the regime of technoscientific promising.

The persistence of the current regime of promising and the emergence of alternatives creates both a challenge and an opportunity for policy: the large promising surrounding new and emerging technologies should be balanced with the support for alternatives regimes of promising, such as collective experimentation and distributed science and innovation. Also, the current moves by public agencies such as the European Commission to focus on missions might be a step in this direction.

With [START_REF] Stengers | Le champignon de la fin du monde: sur la possibilité de vivre dans les ruines du capitalisme, Les Empêcheurs de penser en rond[END_REF], Joly and Le Renard argue for building "a horizon of hope", that recognises that the future is fragile but emphasises that it can be both contested and co-constructed. In this special section we show that in a post-progress society, where societal grand challenges are recognised as important, there are indeed attempts to reimagine the relationship of new and emerging technologies with the future. We argue that questioning the regimes of technoscientific promising is needed, to avoid that they turn into enemies of the future.

Whilst we argue that a dominant regime of technoscientific promising has emerged over the past five decades, as you will read in this special section, alternative regimes of technoscientific promising are emerging, therefore we use regimes in the plural.


D Sauzin
email: david.sauzin@obspm.fr.

VARIATIONS ON THE RESURGENCE OF THE GAMMA FUNCTION

Keywords: Resurgence, Stirling series, Lambert

whether they are published or not. The documents may come

Introduction

Let C denote the Riemann surface of the logarithm. The functions

λ(z) := Γ(z) √ 2π z z-1 2 e -z , λ c (z) := z -c λ(z) (c ∈ C) (1.1)
are meromorphic in C with their principal branches holomorphic in C -R ≤0 . We have λ(z) ---→ We are interested in the asymptotic expansion of λ(z) itself, as well as that of λ c (z) or 1/λ c (z), in the framework of Écalle's Resurgence Theory. Our study was prompted by M. Kontsevich, who communicated us an explicit formula for the Laplace transform of a function related to λ, which entailed resurgence [8]. Our purpose here is to give various formulas for "minors", "natural-majors" and "real-majors" of λ c , explaining along the way the meaning and interest of this Resurgence Theory jargon, how real-majors relate to Kontsevich's formula and why these explicit formulas prove the resurgent character of λ or λ c independently of that of the Stirling series. We will also discuss the Stokes phenomenon for Γ in the language of Écalle's alien calculus.

Resurgence Theory originated with dynamical systems but has recently pervaded many areas of mathematics and matematical physics.

Beginning of a synopsis of Resurgence Theory

2.1 A resurgent function is a function ϕ(z) that is holomorphic in an unbounded domain, in which it can be obtained by Borel-Laplace summation from a resurgent series φ(z).

2.2 A resurgent series is a power series φ(z) in z -1 whose Borel transform ∧ ϕ(ξ) (the formal series obtained by termwise application of B : z -c → ξ c-1 /Γ(c)) is convergent near ξ = 0 and is an endlessly continuable germ, which we call "the minor".

2.3

Being endlessly continuable, roughly speaking, means that ∧ ϕ(ξ) analytically continues to a (possibly multivalued) holomorphic function without any natural boundary: the analytic continuation is possible along any path except for a discrete set of singularities along the path, that can be circumvented, at will, to the left or to the right (with some stipulations-see [START_REF] Écalle | Les fonctions résurgentes[END_REF] for the most general definition, or [START_REF] Candelpergher | Approche de la résurgence[END_REF][START_REF] Kamimoto | Iterated convolutions and endless Riemann surfaces[END_REF]).

Borel-Laplace summation i.e. Laplace transform (or some variant) composed with Borel transform, thus gives the function ϕ(z); the formal series φ(z) (which usually is divergent everywhere) then appears as the asymptotic expansion at infinity of ϕ(z):

Resurgent series φ(z) ∧ ϕ(ξ) endlessly continuable
Resurgent function ϕ(z) Here, we assume ∧ ϕ(ξ) to have analytic continuation with at most exponential growth along the ray e iθ R >0 for some θ ∈ R and

ϕ(z) = L θ ∧ ϕ(z) = e iθ ∞ 0 e -zξ ∧ ϕ(ξ) dξ in Π θ τ := { z ∈ C | -θ -π 2 < arg z < -θ + π
2 , e(ze iθ ) > τ } (2.1) with τ > 0 large enough so that the exponential decay of the Laplace kernel ensures the integrability of e -zξ ∧ ϕ(ξ) at infinity. 1

2.4

The most elementary examples of endlessly continuable functions are the meromorphic functions-the analytic continuation has only one branch in this case-and algebraic functions. Another example is

∧ U (a, b, ξ) = ξ a-1 Γ(a) (1 + ξ) b-a-1
, whose Laplace transform is the confluent hypergeometric function U (a, b, z) ([10, Ex. 6.105]). The Stirling series μ(z) is a resurgent series, since its Borel transform

∧ µ(ξ) = n≥0 B 2n+2 (2n+2)! ξ 2n = ξ -2 ξ 2 coth ξ 2 -1 ∈ C{ξ} (2.2)
analytically continues as a meromorphic function on C (with simple poles on 2πiZ * ). The Laplace transform

L (-π 2 , π 2 ) ∧
µ is none other than log λ(z) , which is thus a resurgent function (see [4, pp. 244-246] or [START_REF] Mitschi | Divergent series, summability and resurgence[END_REF]Th. 5.41]).

2.5

We said power series in § 2.2, but we may accept non-integer powers (or even more general monomials, e.g. involving powers of log z). However, we need all monomials z -c of φ(z) to have e c > 0 for the minor B φ(ξ) to be integrable at 0 and thus have a meaningful Laplace transform. The integrability constraint is bypassed in Écalle's "singularity theory" ( [START_REF] Écalle | Les fonctions résurgentes[END_REF], [START_REF] Écalle | Six lectures on transseries, analysable functions and the constructive proof of Dulac's conjecture[END_REF], [12, §3.1-3.2] or [10, Sec. 6.8]), which not only widens the scope of Borel-Laplace summation but also provides the appropriate framework for alien calculus-see Section 4.

A natural-major is any endlessly continuable holomorphic function ∨ ϕ(ξ) initially defined in a finite sector Σ R,J := { ξ = r e iθ ∈ C | 0 < r < R, θ ∈ J }, with some R > 0 and some interval J of length > 2π. Singularities are defined by quotienting the space of naturalmajors by the subspace C{ξ} of regular germs. The equivalence class of (

The image ∧ ϕ is the minor of the singularity ϕ; it is initially defined in a sector Σ R,I with I := J ∩ (2π + J) but is endlessly continuable too.

A singularity ϕ is called an integrable singularity if its minor ∧ ϕ is uniformly integrable at the origin in every sector Σ R,I and if it has a representative

ϕ(ξ) ∈ C{ξ} resp. ξ 1/2 C{ξ} ⇒ a representative of ∧ ϕ is ∨ ϕ(ξ) := ∧ ϕ(ξ) log ξ 2πi resp. 1 2 ∧ ϕ(ξ).
L θ ϕ(z) := C δ,θ e -zξ ∨ ϕ(ξ) dξ + e iθ ∞ δ e iθ e -zξ ∧ ϕ(ξ) dξ holomorphic for z ∈ Π θ τ , (2.4)
which does not depend on δ nor on the chosen natural-major ∨ ϕ(ξ) and can be rewritten in terms of the sole natural-major as an integral L θ ∨ ϕ(z) := H θ e -zξ ∨ ϕ(ξ) dξ over a θ-rotated Hankel contour H θ running from e i(θ-2π) ∞ to e iθ ∞ and circling anticlockwise around 0 at distance δ (at least if ∨ ϕ(ξ) itself has at most exponential growth at ∞ in directions θ and θ -2π). Moreover, as in Footnote 1, the various L θ ϕ, θ ∈ I, can be glued together, giving rise to L I ϕ holomorphic in D I τ . This is an extension of the usual Laplace transform in the sense that L I ∧ ϕ = L I ∧ ϕ for any integrable minor ∧ ϕ. For example, the Laplace transform of the singularity defined by

∨ I c (ξ) := ξ c-1 (1-e -2πic )Γ(c) for c ∈ C -Z >0 , ∨ I c (ξ) := ξ c-1 (c-1)! log ξ 2πi for c ∈ Z >0 (2.5)
is L I I c = z -c for any I. In general, the asymptotic behaviour at ∞ of L I ϕ(z) only depends on the asymptotic behaviour at 0 of a natural-major: if ∨ ϕ(ξ) can be represented as a convergent series of monomials proportional to ∨ I cn (ξ), then L I ϕ(z) has an asymptotic expansion at ∞ given by the corresponding series of monomials proportional to z -cn .

2.7

Resurgence and Borel summation are compatible with multiplication, the corresponding operation on singularities being the "convolution of singularities" * .

Main results

We now state our main results as two theorems, which will be proved (in Sections 4-5) independently one of the other and independently of the Stirling series.

3.1

The Lambert function x → W (x) is implicitly defined by W (x) e W (x) = x; the canonical reference for the description of its analytic continuation is [START_REF] Corless | On the Lambert W function[END_REF], the only singularities are located at x = -e -1 and 0. One denotes by W 0 and W -1 its real branches so that -e -1 < x < 0 ⇒ W 0 (x) > W -1 (x), both have square root branch points at -e -1 .

Theorem 1. (i) λ 3/2 (z) is a resurgent function obtained by L (-π 2 , π
2 ) from the integrable singularity λ 3/2 that has natural-major and minor explicitly given, for arg ξ = 0, by

∨ λ 3/2 (ξ) = 1 √ 2π W 0 (-e -1-ξ ), ∧ λ 3/2 (ξ) = 1 √ 2π W 0 (-e -1-ξ ) -W -1 (-e -1-ξ ) . (3.1) (ii) χ(z) := 1 λ -3/2 (z) is a resurgent function obtained by L (-π 2 , π
2 ) from the integrable singularity χ that has natural-major and minor explicitly given, for arg ξ = -π, by

∨ χ(ξ) = i √ 2π W -1 (-e -1+ξ ), ∧ χ(ξ) = i √ 2π W -1 (-e -1+ξ ) -W 0 (-e -1+ξ ) . (3.2) (iii) Define inductively a 1 = 1 and a k = 1 k+1 a k-1 -k-1 =2 a a k+1-for k ≥ 2, so that a 2 = 1 3 , a 3 = 1 36 , a 4 = -1 270 , a 5 = 1 4320 , a 6 = 1 17010 , a 7 = -139 5443200 , . . . (3.3) Then ∨ λ 3/2 (ξ) = -1 √ 2π 1 + k≥1 (-1) k a k (2ξ) k/2 , ∧ λ 3/2 (ξ) = 1 √ π n≥0 a 2n+1 2 n+1 ξ n+ 1 2 (3.4) and ∨ χ(ξ) = -i √ 2π 1 + k≥1 i k a k (2ξ) k/2 , ∧ χ(ξ) = 1 √ π n≥0 (-1) n a 2n+1 2 n+1 ξ n+ 1 2 . (3.5)
Notice that, as a consequence of the above statements,

∨ χ(ξ) = i ∨ λ 3/2 (e -iπ ξ), ∧ χ(ξ) = i ∧ λ 3/2 (e -iπ ξ) = -i ∧ λ 3/2 (e iπ ξ). (3.6)
Since the analytic continuation of the Lambert W function has no other singularities than those located at -e -1 and 0, it follows that the minors

∧ λ 3/2 (ξ), ∧ χ(ξ) ∈ ξ 1 2
C{ξ} as well as the natural-majors

∨ λ 3/2 (ξ), ∨ χ(ξ) ∈ C{ξ 1 2
} are endlessly continuable; all their singularities are located above 2πiZ and are square root branch points. On the Riemann surface of ξ 12 , we thus have four singularity-free sectors, corresponding to arguments in

J 1 := (-π 2 , π 2 ), J 2 := ( π 2 , 3π 2 ), J 3 := ( 3π 2 , 5π 2 ), J 4 := (-3π 2 , -π 2 ).
The functions

λ 3/2 = z -3 2 λ and χ = z -3 2
λ are the Laplace transforms of their minors as in Formula (2.1) with any θ ∈ J 1 and τ > 0, hence, in the notation of Footnote 1,

z -3 2 λ(z) = L J 1 ∧ λ 3/2 (z), z -3 2 λ(z) = L J 1 ∧ χ(z) in θ∈J 1 , τ >0 Π θ τ = {-π < arg z < π} (3.7)
(using J 3 instead of J 1 amounts to a simple change of branch of the square roots). The functions L J 2 ∧ λ 3/2 and L J 2 ∧ χ are defined in {-2π < arg z < 0} and differ from the previous ones by the Stokes phenomenon through θ = π 2 , which in this case is equivalent to Euler's reflection formula-see Section 4.

It follows from § 2.7 that, for every c ∈ C,

λ c = z -c+3/2 λ 3/2 is resurgent with λ c = I c-3/2 * λ 3/2 .
Note that λ c is an integrable singularity if and only if e c > 0.

The sequence (a n ) was defined in [START_REF] Marsaglia | A new derivation of Stirling's approximation to n![END_REF] in relation with the asymptotics of Γ. Since Γ(n +

λ(z) ∼ λ(z) := n≥0 (2n + 1)!! a 2n+1 z -n , 1 λ(z) ∼ λ(-z) in C -R ≤0 . (3.8)
3.2 Real-majors 2 constitute a variant of the notion of natural-majors (see [6, pp. 81-82]): a real-major of a singularity ϕ is a function

ν ϕ(ξ) = -2πi ∨ ϕ(e -πi ξ)
where ∨ ϕ(ξ) is any naturalmajor. The minor and the Laplace transform of ϕ can be retrieved as

∧ ϕ(ξ) = -1 2πi ν ϕ(e πi ξ) - ν ϕ(e -πi ξ) , L θ ϕ(z) = ν L θ ν ϕ(z) := 1 2πi H θ+π e -zξ ν ϕ(ξ) dξ. (3.9)
By Fourier-Laplace inversion, there are also integral formulas for a natural-major ∨ ϕ(ξ) or a real-major ν ϕ(ξ) in terms of ϕ = L θ ϕ, as well as for the minor ∧ ϕ(ξ) when ϕ is an integrable singularity. For instance with θ = 0, in which case ϕ is defined at least in the half-plane Π 0 τ , a real-major is given by for any u ∈ Π 0 τ . If ϕ(z) happens to be analytic along R >0 and integrable at z = 0, then one can take u = 0, i.e.

ν ϕ [0] = L 0 ϕ. Theorem 2. If e c < 1 2 , then λ c = ν L (-π 2 , π 2 ) ν λ c in C -R ≤0 , with a real-major ν λ c := L 0 λ c explicitly given by ν λ c (ξ) = Γ( 3 2 -c) √ 2π R (ξ + e Q -Q -1) c-3 2 dQ for ξ ∈ C -R ≤0 . (3.11)
The function ν λ c (ξ) extends analytically to the Riemann surface of the logarithm except at the points that project onto 2πiZ. Is principal branch extends without any singularity to the sector

{-3π 2 < arg ξ < 3π 2 }. Note that for ξ ∈ C -R ≤0 the integrand in (3.11) is well-defined, because e Q -Q -1 ≥ 0 for all Q ∈ R, (3.12)
and has asymptotic equivalent e (c-3 2 )Q as Q → +∞ and |Q| c-3 2 as Q → -∞, whence integrability follows under our condition e c < 1 2 . The analytic continuation of ν λ c (ξ) along any path of C which avoids 2πiZ can be explicitly obtained by continuously deforming the integration path in the right-hand side of (3.11). Theorem 2 thus entails a constructive proof of the resurgent character of λ c (z), independently of Theorem 1.

Remark 3.1. ξ / ∈ 2πiZ means that Σ ξ := (Q, P ) ∈ C 2 | P 2 = ξ + e Q -Q -1 is a
regular complex curve (of infinite genus). When c ∈ Z ≤0 , the consideration of ν λ c (ξ) or of its monodromy around a point of 2πiZ leads to the periods γ P -3+2c dQ, γ ⊂ Σ ξ . Remark 3.2. One can do everything with ν(z) := Γ(z+ 1 2 ) √ 2π z z e -z and ν c (z) := z -c ν(z) instead of λ and λ c . One obtains slightly different formulas, e.g.

e c < 1 ⇒ L 0 ν c (ξ) = 2 -3 2 R (ξ + e Q -Q -1) c-3 2 e Q/2 dQ for ξ ∈ C -R ≤0 . (3.13)
This formula with c = 0 is the one obtained by M. Kontsevich [8].

Remark 3.3. One can compute the minor ∧ λ c (ξ) from (3.9a) and (3.11) using analytic continuation in c. One gets nice integral formulas for c integer, for instance

∧ λ 1 (ξ) = 1 2πi √ 2 γ (-ξ + e Q -Q -1) -1/2 e Q dQ, γ encircling 0. (3.14)
When c is half-integer, the link with Theorem 1 and the Lambert W function is that, for 0 < |ξ| < < 1, the two local solutions to -ξ + e Q -Q -1 = 0 are given by

Q ± (ξ) = -1 -ξ -W 0,-1 (-e -1-ξ ) = ±(2ξ) 1/2 + O(ξ).
(3.15)

3.3

The space of resurgent series is stable under multiplication ( § 2.7) and nonlinear operations like composition or substitution into a convergent series [START_REF] Écalle | Les fonctions résurgentes[END_REF], [START_REF] Sauzin | Nonlinear analysis with resurgent functions[END_REF], [START_REF] Kamimoto | Iterated convolutions and endless Riemann surfaces[END_REF]. On the other hand, nonlinear analysis is also compatible with Borel-Laplace summation, but that is much more elementary [11, § 3.5]. For instance, since log λ = L J 1 (B μ) as mentioned earlier, it follows that λ = 1 + L J 1 B(-1 + exp μ) ∼ exp μ. Comparing with (3.8), by uniqueness of the asymptotic expansion, we get λ = exp μ. The resurgent character of λ could thus be deduced from that of the Stirling series μ: it is always the case that the exponential of a series whose Borel transform is meromorphic, thus resurgent, has an endlessly continuable Borel transform. However, the resulting germ usually admits no closed form formula, even if one started with an explicit meromorphic function: one obtains a function that is guaranteed to be endlessly continuable, the singularities of which can be located and analyzed by means of Écalle's alien calculus, but there is no reason why this function should be susceptible of an expression in terms of elementary ones. As testified by Theorems 1 and 2, the Borel transform of the exponential of the Stirling series is a noticeable exception.

Proof of Theorem 1. Alien Calculus for λ

Lemma 4.1. For any z ∈ C with e z > 0,

√ 2π z -3 2 λ(z) = +∞ 0 e -zξ q + (ξ) -q -(ξ) dξ, (4.1)
where 0 < q -(ξ) < q + (ξ) are the roots of the equation q -ln q -1 = ξ for ξ > 0.

Remark 4.1. We noticed later that this result and its proof are essentially in [START_REF] Marsaglia | A new derivation of Stirling's approximation to n![END_REF]. That paper goes on making use of the expansion of q ± at ξ = 0 but does not refer to the Lambert W function which, however, allows one to write

q + (ξ) = -W -1 (-e -1-ξ ), q -(ξ) = -W 0 (-e -1-ξ ). (4.2)
The connection between the expansion of q ± at 0 and the Lambert function was noticed in [START_REF] Borwein | Emerging tools for experimental mathematics[END_REF] and used as an illustration of "experimental mathematics", as we later discovered.

Proof. We start with the formula

z -c-1 = +∞ 0 e -zζ ζ c Γ(c + 1)
dζ for e c > -1 and e z > 0 (4.3)

(the very basis of Borel-Laplace summation method!). Choosing c = z, we get

z -z Γ(z) = z -z-1 Γ(z + 1) = +∞ 0
e -zq q z dq = +∞ 0 e -z(q-ln q) dq.

The map q → ζ = q -ln q induces a decreasing diffeomorphism (0, 1) → (1, +∞) and an increasing diffeomorphism (1, +∞) → (1, +∞). The inverse of the decreasing diffeomorphism is ζ → q -(ζ -1) and the inverse of the increasing one is ζ → q + (ζ -1), both have a square root singularity at ζ = 1. Thus, splitting the integral in two parts and changing variable, we get

z -z Γ(z) = - +∞ 1 e -zζ q -(ζ -1) dζ + +∞ 1 e -zζ q + (ζ -1) dζ = e -z
+∞ 0 e -zξ q + (ξ) -q -(ξ) dξ.

We obtain (4.1) by multiplying by z e z and integrating by parts. One gets (4.2) from q-ln q = ζ ⇔ q e -q = e -ζ ⇔ -q = W -1 (-e -ζ ) or W 0 (-e -ζ ).

We thus have obtained λ 3/2 = L 0 ∧ λ 3/2 with ∧ λ 3/2 as in (3.1). As already mentioned, the endless continuability and the location of the singularities of ∧ λ 3/2 are simple consequences of the complex analytic structure of W . The existence of constants A, B > 0 such that

| ∧ λ 3/2 (ξ)| ≤ A|ξ| + B for arg ξ ∈ J 1 ∪ J 2 ∪ J 3 ∪ J 4
allows us to vary the integration direction in the Laplace representation of λ 3/2 (z) and get

λ 3/2 = L (-π 2 , π 2 ) ∧ λ 3/2 . Since q = 1
is a critical point of multiplicity 1 for the holomorphic function q → qlog q -1 = ξ, the critical value ξ = 0 is a square root branch point for the two local inverse branches, q -(ξ) and q + (ξ) = q -(e -2πi ξ); these are two holomorphic functions of ξ 1/2 , one for each choice of the square root, which admit convergent Puiseux expansions in ξ 1/2 C{ξ 1/2 } (equivalently: W 0 and W -1 have a square root branch point at -1/e). Following [START_REF] Marsaglia | A new derivation of Stirling's approximation to n![END_REF] and [START_REF] Borwein | Emerging tools for experimental mathematics[END_REF], we write the Puiseux expansion in terms of 2ξ so as to get rational coefficients:

q -(ξ) = -W 0 (-exp(-1 -ξ)) = 1 + k≥1 (-1) k a k (2ξ) k/2 , (4.4)
q + (ξ) = -W -1 (-exp(-1 -ξ)) = -W 0 (-exp(-1 -e -2πi ξ)) = 1 + k≥1 a k (2ξ) k/2 (4.5)
with the sequence (a k ) k≥1 of Part (iii) of Theorem 1 (the induction formula is obtained in [START_REF] Marsaglia | A new derivation of Stirling's approximation to n![END_REF] from the differential equation satisfied by q ± -1 with respect to (2ξ) 1/2 ). The Puiseux expansion (3.4 b) for

∧ λ 3/2 (ξ) follows. The relation q + (ξ) = q -(e -2πi ξ) shows that ∧ λ 3/2 is the monodromy variation of ∨ λ 3/2 (ξ) := - 1 √ 2π q -(ξ) = 1 √ 2π W 0 (-exp(-1 -ξ)) (4.6)
which certainly is o(1/|ξ|) in view of (4.4) and is thus a natural-major for ∧ λ 3/2 . This concludes the proof of Part (i) of Theorem 1 as well as (3.4).

We will find a natural-major for e qζ q -c-1 dq for any u > 0 (

(which can be viewed as a particular case of integral Borel transform, obtained by Fourier-Laplace inversion from (4.3)). Fix z > 1. Choosing ζ = z and c = z -1:

z z-1 Γ(z) = 1 2πi u+i∞ u-i∞ e qz q -z dq = 1 2πi u+i∞ u-i∞
e -z(-q+log q) dq for any u > 0. (

Let us fix u > 1 and perform the change of variable ζ = -q + log q along the integration contour {q = u + it | t ∈ R}. The new contour is

γ u := {ζ(t) = ζ 1 (t) + iζ 2 (t) | t ∈ R} with ζ 1 (t) := -u + 1 2 ln(u 2 + t 2 ), ζ 2 (t) := -t + arctan t u . (4.9)
We have -q e -q = -e ζ , hence -q = W (-e ζ ) is a branch of the Lambert W function.

The even function t → ζ 1 (t) is decreasing on (-∞, 0], while the odd function t → ζ 2 (t) is decreasing; the contour γ u comes from infinity in the first quadrant, reaches the real point ζ(0) = -u + ln u < -1, and goes to infinity in the fourth quadrant. Since -e ζ(0) > -e -1 and -u < -1, we see that -u = W -1 (-e ζ(0) ), hence the inverse change of variable is given by -q = -(u + it) = W -1 (-e ζ(t) ), at least for t close to 0.

One sees that, by taking u close enough to 1, we can ensure that the contour γ u crosses the line { m ζ = -1} the first time between -1 and -1 + 2πi and the second time between -1-2πi and -1, hence the inverse change of variable q(ζ) := -W -1 (-e ζ ) extends analytically to a neighbourhood of γ u , and (4.8) yields

z z-1 Γ(z) = 1 2πi γu e -zζ q (ζ) dζ = z 2πi γu e -zζ q(ζ) dζ = z e z 2πi H 0 e -zξ q(-1 + ξ) dξ. (4.10)
In the last step, we have used the change of variable ξ = 1 + ζ with the convention arg(1 + ζ(0)) = -π, as well as the fact that the contour 1+γ u is homotopic to the Hankel contour H 0 in C -2πiZ relatively to e ξ → +∞.

Viewed as a natural-major,

1 i √ 2π q(-1 + ξ), which is none other that ∨ χ, is thus mapped by L 0 to √ 2πz z-2 e -z Γ(z)
, which is

1 λ -3/2 . Finally, since arg ξ = -π ⇒ ∨ χ(ξ) = i √ 2π W -1 (-e -1+ξ ) = -i √ 2π q + (
e iπ ξ), we can use the Puiseux expansion (4.5), q + (e iπ ξ) = 1 + k≥1 i k a k (2ξ) k/2 ∈ C{ξ 1/2 }, which shows that χ := sing 0 ∨ χ(ξ) is an integrable singularity with a minor

∧ χ(ξ) = - i √ 2π q + (e iπ ξ) -q + (e -iπ ξ) = i √ 2π W -1 (-e -1+ξ ) -W 0 (-e -1+ξ )
whose Puiseux expansion is (3.5 b). The proof of Theorem 1 is complete.

Alien calculus and Stokes phenomenon

A point on the line-segment (0, ω) close to ω ∈ C can be written ω + ξ with the convention arg ξ = arg ω -π. Given ϕ, there are finitely many points ω 1 < ω 2 < • • • < ω r-1 on (0, ω) such that the analytic continuation of the minor ∧ ϕ exists along every path γ ε , ε ∈ {+, -} r-1 , obtained by following (0, ω) and circumventing each ω j to the right (resp. left) if ε j = + (resp. -) without going backwards. Then Écalle defines two operators:

∆ + ω ϕ := sing 0 cont γ +•••+ ∧ ϕ(ω + ξ) , ∆ ω ϕ := ε p(ε)!q(ε)! r! sing 0 cont γε ∧ ϕ(ω + ξ) (4.11)
with p(ε) := number of '+' in ε and q(ε) := r -1 -p(ε). The operators ∆ ω are derivations for * and generate an ∞-dimensional free Lie algebra of derivations acting on the algebra of singularities, giving rise to interesting algebraic combinatorics questions [START_REF] Écalle | Les fonctions résurgentes[END_REF]. The operators ∆ + ω (which can be expressed in terms of ∆ ω , ω ∈ (0, ω)) satisfy a modified Leibniz rule and have a natural interpretation in terms of Stokes phenomenon [START_REF] Écalle | Les fonctions résurgentes. Tomes I[END_REF][START_REF] Écalle | Les fonctions résurgentes[END_REF][START_REF] Écalle | Six lectures on transseries, analysable functions and the constructive proof of Dulac's conjecture[END_REF][START_REF] Mitschi | Divergent series, summability and resurgence[END_REF], which we now illustrate on λ c . The analytic continuation of W 0 and W -1 is known well enough so that (3.1) allows one to compute directly, with ω ± m := 2πm e ± iπ 2 for m ≥ 1,

∆ ω ± m λ c = ± 1 m λ c , ∆ + ω + m λ c = λ c , ∆ + ω - m λ c = -λ c for m = 1 0 for m ≥ 2 (4.12)
with c = 3/2 and thus also 3 with arbitrary c. It follows that

-π < arg z < 0 ⇒ L J 1 λ c (z) = L J 2 λ c + m≥1 e -ω + m z ∆ + ω + m λ c = 1 1-e -2πiz L J 2 λ c (z) 0 < arg z < π ⇒ L J 4 λ c (z) = L J 1 λ c + m≥1 e -ω - m z ∆ + ω - m λ c = (1 -e 2πiz )L J 1 λ c (z)
(which can be viewed as the solution to a Riemann-Hilbert problem-the one induced by the wall-crossing formula in Donaldson-Thomas "Doubled A 1 " theory if c = 0). Recall that

L J 1 λ 3/2 = λ 3/2 . Since J 1 = -π + J 2 = π + J 4 , (3.6) implies L J 2 λ 3/2 (z) = -iL J 1 ∧ χ(e iπ z) = -i/λ -3/2 (e iπ z) and L J 4 λ 3/2 (z) = iL J 1 ∧
χ(e -iπ z) = i/λ -3/2 (e -iπ z), hence both Stokes phenomena amount to the reflection formula Γ(z)Γ(1 -z) = π sin(πz) .

Proof of Theorem 2

Consider λ(z) for z ∈ R >0 . We know that λ(z) is bounded on [u, +∞) for any u > 0 hence, for any c ∈ C, the real-major for λ c (z) = z -c λ(z) that is defined by (3.10) is holomorphic for e ξ > 0 and we get its analytic continuation to a sector of opening 3π in the Riemann surface of the logarithm C by varying the integration half-line.

Let us do it under the assumption e c < 1 2 . Since λ c (z) ∼ 3 A proof can also be derived from the relation λ = exp μ as in [START_REF] Mitschi | Divergent series, summability and resurgence[END_REF]-beware of the typo in [10, (6.99)].

for any 0 < ε < π, hence we can vary the integration ray in the above Laplace transform and obtain the analytic continuation of To conclude the proof of Theorem 2, we observe that, to follow the analytic continuation of ν λ c (ξ) along a path Γ of C that starts on {arg ξ = 0}, it is sufficient to deform continuously the integration path in (3.11) so that the integration variable Q avoids the zeroes of the expression ξ + e Q -Q -1. Such a continuous deformation exists as long as 0 is not a critical value of Q → ξ + e Q -Q -1, i.e. as long as the path Γ avoids any critical value of Q → -e Q + Q + 1. These critical values are precisely the points of 2πiZ.

z→∞ 1 along

 1 R >0 , with log λ(z) asymptotic to the Stirling series μ

  has analytic continuation with uniform exponential bound in a sector arg ξ ∈ I, then the Laplace transforms L θ ∧ ϕ(z), θ ∈ I, can be glued together (the Cauchy theorem entail that they match) and give rise to a function L I ∧ ϕ(z) analytic in D I τ := θ∈I Π θ τ (sectorial neighbourhood of ∞ of opening |I| + π).

∨

  ϕ(ξ) is denoted by ϕ = sing 0 ∨ ϕ(ξ) . The monodromy variation induces an operator var : ϕ → var ϕ = ∧

∨

  ϕ(ξ) that is o(1/|ξ|) uniformly in every sector Σ R,I . The latter condition implies that ϕ is uniquely determined by ∧ ϕ; we use the notation ϕ = ∧ ϕ. Conversely, any integrable minor ∧ ϕ(ξ) is the minor of an integrable singularity; e.g.

∧

2. 6

 6 Suppose ϕ is an arbitrary singularity whose minor extends analytically to a sector Σ ∞,I , with | ∧ ϕ(ξ)| ≤ C e τ |ξ| on Σ ∞,I ∩ {|ξ| ≥ 1}. Then, given θ ∈ I, for any natural-major ∨ ϕ(ξ) and any δ > 0 small enough, we can consider the contour C δ,θ := {θ ∈ [θ -2π, θ] → δ e iθ } and the function

3 2 )

 2 = 2 -n-1 (2n+1)!! √ π, the expansions (3.4)-(3.5) imply asymptotic expansions in the same domain as in (3.7):

νϕ

  [u] (ξ) = +∞ u e -zξ ϕ(z) dz for arg ξ = 0 (3.10)

1 λ - 3 / 2

 132 (z) starting with the formulae c > 0 and ζ > 0 ⇒ ζ c Γ(c + 1)

z→0 + 1 √ 1 √ 1 2

 111 2π z -c-1 2 is then integrable at z = 0, u = 0 yields a real-majorν λ c (ξ) = L 0 λ c (ξ). Moreover, λ c (z) ∼ |z|→∞ z -c and λ c (z) ∼ |z|→0 2π z -c-uniformly for |arg z| < π -ε (5.1)

ν 1 √ 2π z -c+ 1 2 ReRz -c+ 1 2 e 2 by ( 4 . 3 )

 122243 λ c (ξ) to a sector of opening 3π in C in the formν λ c (ξ) = L J 1 λ c (ξ) for -3π 2 < arg ξ < 3π 2 , with J 1 := (-π 2 , π 2 ). (5.2)Moreover, in view of (5.1), this representation of ν λ c (ξ) shows that it is bounded in the domain {|ξ| > τ and |arg ξ| < 3π 2 -ε} for every τ, ε > 0. Therefore Fourier-Laplace inversion yields λ c (z) = ν L θ ν λ c (z) for all z ∈ C -R ≤0 and θ ∈ J 1 such that θ + arg z ∈ J 1 . We now prove Formula (3.11). For each z > 0,Γ(z) = +∞ 0 e -x x z dx x = z z R e -ze Q e zQ dQ (change of variable x = z e Q ), hence λ c (z) = -z(e Q -Q-1) dQ and ν λ c (ξ) = 1 dQ z -c+ 1 2 e -z(ξ+e Q -Q-1).Now e(ξ + e Q -Q -1) > 0 by (3.12) and e( 3 2 -c) > 0, thus +∞ 0 -z(ξ+e Q -Q-1) dz = Γ( 3 2 -c)(ξ + e Q -Q -1) c-3, and the conclusion follows.

They owe their name to their usefulness in the summation of divergent series with real coefficients.

Acknowledgements. The author thanks Capital Normal University for their hospitality during the period September 2019-February 2020, where this research was started. This paper is partly a result of the ERC-SyG project, Recursive and Exact New Quantum Theory (Re-NewQuantum) which received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 810573.