Albert Cohen

Matthieu Dolbeault

Optimal pointwise sampling for L 2 approximation

Keywords: 41A65, 41A81, 93E24, 62E17, 94A20

. As a consequence, we obtain for any compact class K ⊂ L 2 that the sampling number ρ rand Cn (K) L 2 in the randomized setting is dominated by the Kolmogorov n-width d n (K) L 2 . While our result shows the existence of a randomized sampling with such near-optimal properties, we discuss remaining issues concerning its generation by a computationally efficient algorithm.

Introduction

We study the approximation of a function u ∈ L 2 (D, µ), where D is a domain in R d and µ a measure on D, by an element ũ of V n , a subspace of L 2 (D, µ) of finite dimension n, based on pointwise data of of u. Therefore, to construct ũ, we are allowed to evaluate u on a sample of m points X = {x 1 , . . . , x m } ∈ D m . In addition, we consider randomized sampling and reconstruction, in the sense that X will be drawn according to a distribution σ over D m , so the error u -ũ should 1 be evaluated in some probabilistic sense. For the sake of notational simplicity, having fixed D and µ, we write thoughout the paper

L 2 := L 2 (D, µ) and v := v L 2 = D |v| 2 dµ 1/2 , (1)
as well as e n (u) := min v∈Vn u -v .

(

) 2
Our main result is the following:

Theorem 1. For some universal constants C, K 1, and for any n-dimensional space V n ⊂ L 2 , there exists a random sampling X = {x 1 , . . . , x m } with m Cn and a reconstruction map R : D m × C m → V n , such that for any u ∈ L 2 ,

E X u -ũ 2 Ke n (u) 2 (3)
where ũ := R(x 1 , . . . , x m , u(x 1 ), . . . , u(x m )).

The reconstruction map R is obtained through a weighted least-squares method introduced in [START_REF] Cohen | Optimal weighted least squares methods[END_REF], which has already been discussed in several papers, see [START_REF] Adcock | Near-optimal sampling strategies for multivariate function approximation on general domains[END_REF][START_REF] Cohen | Optimal sampling and Christoffel functions on general domains[END_REF][START_REF] Haberstich | Boosted optimal weighted leastsquares[END_REF][START_REF] Jakeman | A Christoffel function weighted least squares algorithm for collocation approximations[END_REF][START_REF] Doostan | Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression[END_REF][START_REF] Migliorati | Adaptive approximation by optimal weighted least-squares methods[END_REF][START_REF] Migliorati | Multivariate approximation of functions on irregular domains by weighted least-squares methods[END_REF]. The weights involved are given by the expression

w : x ∈ D → n min v∈Vn v 2 |v(x)| 2 = n n j=1 |L j (x)| 2 , (4)
where the last formula holds for any L 2 -orthonormal basis (L 1 , . . . , L n ) of V n , which, up to the factor n, is the Christoffel function associated to the space V n and the space L 2 (D, µ). The weighted least-squares solution is then simply defined as

ũ := argmin v∈Vn m i=1 w(x i )|u(x i ) -v(x i )| 2 .
(

) 5
Introducing the discrete 2 norm

v 2 X := 1 m m i=1 w(x i )|v(x i )| 2 (6)
and its associated scalar product •, • X , we get a computable formula for ũ:

ũ = argmin v∈Vn u -v 2 X = P X Vn u, (7)
where P X Vn denotes the orthogonal projection on V n with respect to •, • X . Note that strictly speaking • X is not a norm over L 2 , however the existence and uniqueness of P X Vn will be ensured by the second condition in Lemma 1 below, see Remark 2.

Therefore our main achievements lie in the particular choice of the random sample X for ensuring the near-optimal approximation and sampling budget in Theorem 1. Now, the proof of Theorem 1 relies on two conditions: first, the expectation of • 2 X has to be bounded by • 2 up to a constant. Second, an inverse bound should hold almost surely, instead of just in expectation, for functions v in V n . More precisely, one has: Lemma 1. Assume that m and the law σ of X = {x 1 , . . . , x m } are such that

E( v 2 X ) α v 2 , v ∈ V, (8)
and v 2 β v 2 X a.s., v ∈ V n . (9)
Then E( u -ũ 2 ) (1 + αβ)e n (u) 2 . ( 10
)
Proof. Denote u n the orthogonal projection of u on V n with respect to the L 2 (D, µ) norm. Applying Pythagoras theorem both for • and • X , one obtains

E( u -ũ 2 ) = u -u n 2 + E( u n -ũ 2 ) u -u n 2 + βE( u n -ũ 2 X ) = u -u n 2 + βE( u n -u 2 X -u -ũ 2 X ) u -u n 2 + βE( u n -u 2 X ) (1 + αβ) u -u n 2 ,
which proves (10) since u -u n = e n (u).

In section 2, we recall how both conditions ( 8) and ( 9) can be obtained with m quasi-linear in n, that is, of order n log n. We reduce this budget to m of order n in section 3, by randomly subsampling the set of evaluation points, based on results from [START_REF] Marcus | Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem[END_REF][START_REF] Shahaf Nitzan | Exponential frames on unbounded sets[END_REF]. The proof of Theorem 1 follows. We compare it to the recent results [START_REF] Krieg | Function values are enough for L2-approximation: Part II[END_REF][START_REF] Limonova | On sampling discretization in L 2[END_REF][START_REF] Nagel | A new upper bound for sampling numbers[END_REF] in section 4, in particular regarding the domination of sampling numbers by n-widths. We conclude in section 5 by a discussion on the offline computational cost for practically generating the sample X.

Weighted least-squares

A first approach consists in drawing the x i independently according to the same distribution ρ, that is, taking σ = ρ ⊗m . The natural choice for ρ is dρ = 1 w dµ, which is a probability measure since

D 1 w dµ = 1 n n j=1 D |L j (x)| 2 dµ(x) = 1 n n j=1 L j 2 = 1. (11)
We denote by Z = {x 1 , . . . , x m } this first random sample and by • Z the corresponding discrete 2 norm. With this sampling measure,

E( v 2 Z ) = 1 m m i=1 D w(x)|v(x)| 2 dρ = D |v| 2 dµ = v 2 , (12)
so condition ( 8) is ensured for X = Z with α = 1. To study the second condition, we introduce the Hermitian positive semi-definite Gram matrix

G Z := ( L j , L k Z ) j,k=1,...,n (13)
and notice that ( 9) is equivalent to

|ν| 2 = n j=1 ν j L j 2 β n j=1 ν j L j 2 Z = βν * G Z ν, ν ∈ C n , (14)
which in turn rewrites as λ min (G Z ) β -1 . By the central limit theorem, as m tends to infinity, the scalar products L j , L k Z converge almost surely to L j , L k = δ j,k , so G Z converges to the identity matrix, and we expect that λ min (G Z ) β -1 holds for β > 1 with high probability as m gets large. A quantitative formulation can be obtained by studying the concentration of G Z around I in the matrix spectral norm

M 2 := max{|M x| : |x| = 1}.
This is based on the matrix Chernoff bound, see [START_REF] Ahlswede | Strong converse for identification via quantum channels[END_REF][START_REF] Tropp | User-Friendly tail bounds for sums of random matrices[END_REF] for the original inequality and [START_REF] Cohen | Optimal sampling and Christoffel functions on general domains[END_REF], Lemma 2.1, for its application to our problem:

Lemma 2. For m 10n ln( 2n ε ), if X ∼ ρ ⊗m , then P G Z -I 2 1 2 1 -ε. (15)
In particular,

P λ min (G Z ) 1 2 1 -ε.
Thus assumption ( 9) is satisfied with β = 2, but only with probability 1 -ε. As we would like it to hold almost surely, we condition the sampling to the event

E := G Z -I 2 1 2 (16)
which defines a new sample

Y = Z|E. (17)
In practice, Y can be obtained through a rejection method, which consists in drawing successively set of points Z 1 , Z 2 , . . . according to ρ ⊗m , and defining Y = Z k for the first value k such that E holds. We then define ũ as the weighted leastsquare estimator based on this conditioned sample, that is

ũ := P Y Vn u. (18)
This approach was introduced and analyzed in [START_REF] Haberstich | Boosted optimal weighted leastsquares[END_REF], see in particular Theorem 3.6 therein. A simpler version of this result, sufficient for our purposes, is the following:

Lemma 3. For m 10n ln(4n), if Z ∼ ρ ⊗m and Y = Z|E, then G Y -I 2 1 2 , (19)
and

E Y ( u -ũ 2 ) 5e n (u) 2 . ( 20
)
Proof. The first part immediately results from the definition of Y and E, and implies condition (9) with β = 2. Moreover, P(E) 1 2 by Lemma 2 with ε = 1 2 , so for any v ∈ L 2 (D, µ),

E Y ( v 2 Y ) = E Z ( v 2 Z |E) = E Z ( v 2 Z χ E ) P(E) E Z ( v 2 Z ) P(E) 2 v 2 , (21)
so condition (8) holds with α = 2. The conclusion follows from Lemma 1.

Remark 1. The number of redraws k for reaching Y follows a geometric law of expectation

E(k) = P(E) -1 = (1 -ε) -1 , that is E(k)
2 for the particular choice of m in the above lemma. It should be well noted that u is not evaluated at the intermediately generated samples Z 1 , . . . , Z k-1 , which thus enter the offline cost of the samping algorithm.

Remark 2. The fact that the Grammian G Y is non-singular implies that for any u with given values y i at the points x i , we can uniquely define

ũ = P Y Vn u = n j=1 a j L j , (22)
since a = (a 1 , . . . , a n ) * solves the system of normal equations

G Y a = b, (23)
where the right-side vector has coordinates

b j = L j , u Y = 1 m m i=1 w(x i )L j (x i )y i . ( 24
)
If u is in L 2 , the y i are only defined up to a representer, however since two representers u 1 and u 2 coincide µ-almost surely, we find that P Y Vn u is well defined almost surely over the draw of Y .

Random subsampling

With Lemma 3, we already have an error bound similar to that of Theorem 1. However, the sampling budget is larger than n by a logarithmic factor, which we seek to remove in this section. To do so, we partition the sample Y into subsets of size comparable to n, and randomly pick one of these subsets to define the new sample. An appropriate choice of the partitioning is needed to circumvent the main obstacle, namely the preservation of condition [START_REF] Jakeman | A Christoffel function weighted least squares algorithm for collocation approximations[END_REF]. It relies on the following lemma, taken from Corollary B of [START_REF] Shahaf Nitzan | Exponential frames on unbounded sets[END_REF], itself a consequence of Corollary 1.5 in [START_REF] Marcus | Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem[END_REF]. The relevance of these two results to sampling problems were exploited in [START_REF] Nagel | A new upper bound for sampling numbers[END_REF] and noticed in [START_REF] Kaemmerer | Worst case recovery guarantees for least squares approximation using random samples[END_REF], respectively. for some constants δ < α β. Then there exists a partition of {1, . . . , m} into two sets S 1 and S 2 such that

1 -5 δ/α 2 αI i∈S j a i a * i 1 + 5 δ/α 2 βI, j = 1, 2. (26)
In Lemma 2 of [START_REF] Shahaf Nitzan | Exponential frames on unbounded sets[END_REF] this result is applied inductively in order to find a smaller set J ⊂ {1, . . . , k} of cardinality |J| cn such that

C -1 I m n i∈J a i a * i CI, (27)
for some universal constants c, C > 1. We adapt this approach in order to obtain a complete partition of {1, . . . , m} by sets having such properties.

Lemma 5. Let a 1 , . . . , a m ∈ C n be vectors of norm |a i | 2 = n m for i = 1, . . . , m, and satisfying

1 2 I m i=1 a i a * i 3 2 I. ( 28
)
Then there exists an integer L and a partition of {1, . . . , m} into 2 L sets J 1 , . . . , J 2 L such that

c 0 I m n i∈J k a i a * i C 0 I, 1 k 2 L , (29)
with universal constants c 0 and C 0 . In addition, each set J k satisfies

|J k | C 0 n. (30)
Proof. The cardinality estimate (30) follows from the upper inequality in (29) by taking the trace

nC 0 = tr(C 0 I) m n i∈J k tr(a i a * i ) = m n |J k | n m = |J k |. ( 31
)
For the proof of (29), if n/m 1/200, then the result holds with L = 0, J 1 = {1, . . . , m} and c 0 = C 0 = 200. Now assuming δ := n/m < 1/200, define by induction α 0 = 1 2 , β 0 = 3 2 , and

α +1 := α 1 -5 δ/α 2 , β +1 := β 1 + 5 δ/α 2 , 0. (32)
As α +1 α 2 , the minimal integer L such that α L 100δ is well defined, and satisfies

α L = α L-1 1 -5 δ/α L-1 2 > 100δ 1 -5 1/100 2 = 25δ. ( 33
)
Moreover α 2 L--1 α L-1 2 L--1 100δ for = 0, . . . , L -1, so

β L = 3α L L-1 =0 1 + 5 δ/α 1 + 5 δ/α Cδ, (34)
with C := 300

2 1+ √ 2 - 1- √ 2
-. Finally, we inductively define partitions {S 1 , . . . , S 2 } for 0 L: start with S 0 1 = {1, . . . , m} and for any , j, noticing that

α I i∈S j a i a * i β I, (35)
apply Lemma 4 to split S j into subsets S +1 2j-1 and S +1 2j satisfying the same property. At the last step, we define

J k = S L k . (36)
The framing (29) thus holds with c 0 = α L /δ 25 and C 0 = β L /δ 11000.

Proof of Theorem 1: Define

a i = w(x i ) m L j (x i ) j=1,...,n (37)
the normalised random vectors corresponding to the sample Y = {x 1 , . . . , x m } introduced in the previous section. As

1 2 I G Y = m i=1 a i a * i 3 2 I (38)
and

|a i | 2 = 1 m w(x i ) n j=1 |L j (x i )| 2 = n m (39)
thanks to the choice of weights (4), the assumptions of Lemma 5 are satisfied. Applying this lemma, we obtain sets J 1 , . . . , J 2 L partitioning {1, . . . , m}. Let κ be a random variable taking value k ∈ {1, . . . , 2 L } with probability p k = |J k |/m, and create a random subsampling X of Y through

X = {x i ∈ Y : i ∈ J κ }. (40)
Then the budget condition |X| = |J κ | C 0 n is satisfied according to (30). Here, we define the discrete norm as

v 2 X := 1 |X| i∈Jκ w(x i )|v(x i )| 2 , (41)
and the associated Gram matrix

G X := ( L j , L k X ) j,k=1,...,n = m |J κ | i∈Jκ a i a * i . (42)
The weighted least-squares estimate is now defined as

ũ := argmin v∈Vn 1 |X| i∈Jκ w(x i )|u(x i ) -v(x i )| 2 , (43)
and it thus depends on the random draws of both Y and κ. Condition (9) follows from the lower inequality in (29

) with β = C 0 c 0 since G X m |J κ | n m c 0 I c 0 C 0 I. (44)
Finally, we have for any v ∈ L 2 (D, µ)

E X ( v 2 X ) = 2 L k=1 p k |J k | i∈J k w(x i )|v(x i )| 2 = E Y ( v 2 Y ) 2e n (u) 2 , ( 45
)
so condition [START_REF] Haberstich | Boosted optimal weighted leastsquares[END_REF] holds with α = 2. Applying Lemma 1, we conclude that [START_REF] Kaemmerer | Worst case recovery guarantees for least squares approximation using random samples[END_REF] holds with C = C 0 and K = 1 + 2 C 0 c 0 .

Comparison with related results

In order to compare Theorem 1 with several recent results [START_REF] Kaemmerer | Worst case recovery guarantees for least squares approximation using random samples[END_REF][START_REF] Krieg | Function values are enough for L2-approximation[END_REF][START_REF] Nagel | A new upper bound for sampling numbers[END_REF][START_REF] Temlyakov | On optimal recovery in L 2[END_REF], we consider its implication when the target function u belongs to a certain class of functions K that describes some prior information on u, such as smoothness.

Recall that if V is a Banach space of functions defined on D and K ⊂ V is a compact set, its Kolmogorov n-width is defined by

d n (K) V := inf dim Vn=n sup u∈K inf v∈Vn u -v V , (46)
where the first infimum is taken over all linear spaces V n ⊂ V of dimension n. This quantity thus describes the best approximation error that can be achieved uniformly over the class K by an n-dimensional linear space.

On the other hand, building a best approximation of u requires in principle full knowledge on u, and we want to consider the situation where we only have access to a limited number of point evaluations. This leads one to consider the sampling numbers, also called optimal recovery numbers, both in the deterministic and randomized settings.

For deterministic samplings, we define the (linear) sampling numbers

ρ det m (K) V := inf X,R X max u∈K u -R X (u(x 1 ), . . . , u(x m )) V , ( 47
)
where the infimum is taken over all samples X = {x 1 , . . . , x m } ∈ D m and linear reconstruction maps R X : C m → V . For random samplings, we may define similar quantities by

ρ rand m (K) 2 V := inf X,R X max u∈K E u -R X (u(x 1 ), . . . , u(x m )) 2 V , (48)
where the infimum is taken over all random variables X = {x 1 , . . . , x m } ∈ D m and linear reconstruction maps R X : C m → V . Note that a deterministic sample can be viewed as a particular choice of random sample following a Dirac distribution in D m , and therefore

ρ rand m (K) V ρ det m (K) V . (49)
Sampling numbers may also be defined without imposing the linearity of R X , leading to smaller quantities. In what follows, we shall establish upper bound on the linear sampling numbers, which in turns are upper bounds for the nonlinear ones. We refer to [START_REF] Novak | Tractability of multivariate problems[END_REF] for an introduction and study of sampling numbers in the context of general linear measurements, and to [START_REF] Novak | Tractability of multivariate problems. Volume III: Standard information for operators[END_REF] that focuses on point evaluation, also termed as standard information.

By optimizing the choice of the space V n used in Theorem 1, we obtain as a consequence that, for V = L 2 = L 2 (D, µ), the sampling numbers in the randomized setting are dominated by the Kolmogorov n-widths.

Corollary 1. For any compact set K ⊂ L 2 , one has

ρ rand Cn (K) L 2 Kd n (K) L 2 , (50)
where C and K are the same constants as in Theorem 1.

Remark 3. The bound (50) cannot be attained with independent and identically distributed sampling points x 1 , . . . , x m . Indeed, consider the simple example, already evoked in [START_REF] Tropp | User-Friendly tail bounds for sums of random matrices[END_REF], where D = [0, 1], µ is the Lebesgue measure,

V n = n i=1 a i χ [ i n , i+1 n [ , (a 1 , . . . , a n ) ∈ C n (51)
is a space of piecewise constant functions, and

K = {u ∈ V n , u L ∞ 1}. Then K ⊂ V n so d n (K) L 2 = 0
, and an exact reconstruction R X u = u is possible if and only if X contains at least one point in each interval i n , i+1 n . Thus ρ det n (K) L 2 = 0, but in the case of i.i.d measurements, m has to grow like n log n to ensure this constraint, due to the coupon collector's problem.

Remark 4. In [START_REF] Krieg | Optimal Monte Carlo methods for L2 -approximation[END_REF], a result similar to Theorem 1 is obtained under the extra assumption of a uniform bound on e n (u)/e 2n (u), yielding the validity of (50) assuming a uniform bound on d n (K) L 2 /d 2n (K) L 2 . The recovery method used in [START_REF] Krieg | Optimal Monte Carlo methods for L2 -approximation[END_REF] is not of least-square type, but rather an elaboration of the pseudo-spectral approach that would simply approximate the inner products u, L j = D uL j dµ by a quadrature, using a hierarchical approach introduced in [START_REF] Wasilkowski | The power of standard information for multivariate approximation in the randomized setting[END_REF].

Ideally, one would like a "worst case" or "uniform" version of Theorem 1, in the form

ρ det Cn (K) L 2 Kd n (K) L 2 , ( 52
)
but it is easily seen that such an estimate cannot be expected for general compacts sets of L 2 , due to the fact that pointwise evaluations are not continuous in L 2 norm.

It is however possible to recover such uniform estimates by mitigating the non-achievable estimate (52) in various ways. One first approach, developed in [START_REF] Limonova | On sampling discretization in L 2[END_REF][START_REF] Temlyakov | On optimal recovery in L 2[END_REF], gives an inequality similar to (52), with d n (K) L 2 replaced by d n (K) L ∞ . It is based on the following lemma, see Theorem 2.1 in [START_REF] Temlyakov | On optimal recovery in L 2[END_REF], which we recall for comparison with our Lemma 1: Lemma 6. Assume that µ is a finite measure of mass µ(D) = M < ∞, that the constant functions belong to V n , and that there exists a sample X = {x 1 , . . . , x m } and weights w i such that the discrete norm

v 2 X = 1 |X| m i=1 w i |v(x i )| 2 (53)
n for certain choices of spaces V n . Results similar to Lemma 3 have been obtained in the above references, with (L j ) j=1,...,n replaced by ( L j ) j=1,...,n .

One could stop at this point and compute the approximation ũ = P Y Vn u, which satisfies error bounds both in expectation when comparing to e n (u), see Lemma 3, or uniformly when comparing to e n (u) L ∞ , see Theorem 1 (iii) in [START_REF] Cohen | Optimal weighted least squares methods[END_REF]. Once the measurements of u are performed, the computation of ũ requires to solve a n × n linear system as in Remark 2, so the online stage takes a time O(τ n log n + n 3 ), where τ is the cost of each measurement of u.

However, in applications where the evaluation cost τ becomes very high (for example when each evaluation x → u(x) requires solving a PDE by some numerical code, or running a physical experiment), further reduction of the size of the sample may prove interesting, and justifies the interest for sparsification methods. The greedy selection method from [START_REF] Batson | Twice-Ramanujan Sparsifiers[END_REF], which is used in [START_REF] Temlyakov | On optimal recovery in L 2[END_REF] and leads to (58), has a complexity in O(mn 3 ) = O(n 4 log n), but it can only be applied to the worst-case setting, with the uniform error bound e n (u) L ∞ .

On the other hand, the iterative splitting method that we have used in this paper following the ideas from [START_REF] Marcus | Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem[END_REF][START_REF] Nagel | A new upper bound for sampling numbers[END_REF] is not easily implemented, and one obvious method consists in testing all partitions of {1, . . . , m} into sets S 1 and S 2 when applying Lemma 4. Note that this lemma is in practice used L times, with

L = O(log log n) since 2 L = O( m n ) = O(log n).
The algorithm consisting in subdividing the sample L times, each time checking that the Gram matrices corresponding to S 1 and S 2 are well conditioned, and keeping one such subset at random, thus has an exponential complexity O(2 m n 3 ) = O(n cn ). Having a different strategy that would produce the random sample in polynomial time is currently an open problem to us.

We summarise these computational observations in the following table, which illustrates the conflicts between reducing the sampling budget, ensuring optimal approximation results, and maintaining a reasonable cost for sample generation. + random sparsification [START_REF] Marcus | Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem[END_REF] Cn O(n cn ) → O(n r ) ?

As a final remark, let us to emphasize that although the results presented in our paper are mainly theorical and not practically satisfactory, due both to the computational complexity of the sparsification, and to the high values of the numerical constants C and K in Theorem 1, they provide some intuitive justification to the boosted least-squares methods presented in [START_REF] Haberstich | Boosted optimal weighted leastsquares[END_REF], which consist in removing points from the initial sample as long as the corresponding Gram matrix G X remains well conditioned. For instance, Lemma 4 allows to keep splitting the sample even after L steps, if one still has a framing 1 2 I G X 3 2 I and a sufficiently large ration |X| n . Nevertheless, it would be of much interest to find a randomized version of [START_REF] Batson | Twice-Ramanujan Sparsifiers[END_REF] giving a bound of the form (50), since this would give algorithmic tractability, smaller values for C and K, and the possibility to balance these constants in Theorem 1.

Lemma 4 .

 4 Let a 1 , . . . , a m ∈ C n be vectors of norm |a i | 2 δ for i = 1, . . . , m,

-ũ 2 ) 2 u -ũ 2

 222 Ce n (u) Ce n (u)

satisfies a framing

Then u -P X Vn u

where e n (u)

Proof. For any v ∈ L 2 , we have v 2 M v 2 L ∞ , and as

Hence

and we conclude by optimizing over v ∈ V n .

Here, in contrast to the derivation of [START_REF] Kaemmerer | Worst case recovery guarantees for least squares approximation using random samples[END_REF] in Lemma 1, one only uses the framing property (54), and does not need the condition E( v 2 X ) α v 2 . For this reason, one may achieve the above objective with a simpler sparsification approach proposed in [START_REF] Batson | Twice-Ramanujan Sparsifiers[END_REF] and adapted in [START_REF] Limonova | On sampling discretization in L 2[END_REF], which performs a greedy selection of the points x i within the sample Y , together with the definition of weights w i associated with these points. If the initial sample Y satisfies

then, for any c > 1 the selection algorithm produces a sample X of at most cn points such that (54) holds with α

Optimizing the choice of V n (but imposing that constant functions are contained in this space), this leads to the following comparison result between deterministic optimal recovery numbers in L 2 and n-widths in L ∞ : for any compact set K ∈ C(D), one has

where C depends on c > 1. For c = 2, one can take C = 11. We refer to [START_REF] Limonova | On sampling discretization in L 2[END_REF][START_REF] Temlyakov | On optimal recovery in L 2[END_REF] where this type of result is established. Another approach consists in making pointwise evaluations continuous by restriction to the case where K = B H is the unit ball of a reproducing kernel Hilbert space H ⊂ L 2 , and assuming that the sequence

The following result from [START_REF] Nagel | A new upper bound for sampling numbers[END_REF], also based on the sparsification techniques from [START_REF] Marcus | Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem[END_REF], improves on a bound found in [START_REF] Krieg | Function values are enough for L2-approximation[END_REF]

In [START_REF] Krieg | Function values are enough for L2-approximation: Part II[END_REF], a similar inequality is established for more general compact classes K of L 2 such that point evaluation is well defined on functions of K, however now with p sums with p < 2 replacing 2 sums.

In the above inequality, the logarithmic factor appears as a residual of the result obtained before sparsification, contrarily to the bounds (50) and (58), which do not explicitely depend on the size of the initial sample Y . This results in a gap of a factor log n between (59) and known lower bounds for ρ det Cn (B H ) 2 , see [START_REF] Nagel | A new upper bound for sampling numbers[END_REF].

Computational aspects

The various results (50), (58), (59) ensure the existence of good sampling and reconstructions algorithms in various settings. We end by a discussion on the computational cost of these strategies.

For the weighted least-squares methods corresponding to samples Z and Y , the most expensive step consists in assembling the matrix G Z as a sum of m matrices of size n, so the algorithmic complexity is of order O(mn 2 ) = O(n 3 log n). Besides, to obtain G Y , this step may need to be repeated a few times, as explained in Remark 1, but this only affects the offline complexity by a small random factor.

Note that we assumed that an orthogonal basis (L 1 , . . . , L n ) of V n is explicitly known, which might not be the case for irregular domains D. However, under reasonable assumptions on D or V n , one can compute an approximately orthogonal basis ( L 1 , . . . , L n ), either by performing a first discretization of D with a large number of points, or by using a hierarchical method on a sequence of nested spaces [START_REF] Adcock | Near-optimal sampling strategies for multivariate function approximation on general domains[END_REF][START_REF] Arras | Sequential sampling for optimal weighted least squares approximations in hierarchical spaces[END_REF][START_REF] Cohen | Optimal sampling and Christoffel functions on general domains[END_REF][START_REF] Haberstich | Boosted optimal weighted leastsquares[END_REF][START_REF] Migliorati | Adaptive approximation by optimal weighted least-squares methods[END_REF][START_REF] Migliorati | Multivariate approximation of functions on irregular domains by weighted least-squares methods[END_REF]. These additional steps have complexities O(K n n 2 ) and O(n 4 ) respectively, where K n is the maximal value of the inverse Christoffel function n j=1 |L j | 2 which might grow more than linearly with


Amaury Hayat
email: amaury.hayat@enpc.fr.

Peipei Shang
email: shang@tongji.edu.cn.

Exponential stability of density-velocity systems with boundary conditions and source term for the H 2 norm

In this paper, we address the problem of the exponential stability of density-velocity systems with boundary conditions. Density-velocity systems are typical hyperbolic systems that are omnipresent in physics as they encompass all systems that consist in a flux conservation and a momentum equation. In this paper we show that any such system can be stabilized exponentially quickly in the H 2 norm using simple local boundary feedbacks, provided a condition on the source term is valid. This condition holds for most physical systems, even when the source term is not dissipative. Besides, the feedback laws obtained only depend on the target values at the boundaries, which implies that they do not depend on the expression of the source term or the force applied on the system. This makes them both very easy to implement in practice and robust to model errors. For instance, for a river modeled by Saint-Venant equations this means that the feedback law does not require any information on the friction model, the slope or the shape of the channel considered. This feat is obtained by showing the existence of a basic H 2 Lyapunov function. We apply it to several systems: the general Saint-Venant equations, the isentropic Euler equations, the motion of water in rigid-pipe, the osmosis phenomenon, the traffic flow, etc.

Introduction

Density-velocity systems are important 2 × 2 hyperbolic systems as they represent the physical phenomena where the flux is conserved, while the energy can be either increased or decreased. In physics they are found in fluid mechanics, electromagnetism, etc. The problem of stabilization of 2 × 2 hyperbolic systems was studied in the literature for a long time. To our knowledge, the first result was obtained by Greenberg and Li in [START_REF] Greenberg | The effect of boundary damping for the quasilinear wave equation[END_REF], where the stabilization of quasilinear wave equations was discussed in the framework of C 1 solutions using the characteristic method. This result is then generalized by Qin [START_REF] Hu | Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems[END_REF] to n × n homogeneous systems. Later on, Coron et al. [START_REF] Coron | A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations[END_REF] introduced Lyapunov approach to analyze the asymptotic behavior of linear 1 hyperbolic systems in the L 2 norm, and then generalized for nonlinear hyperbolic systems in the framework of C 1 and H 2 solutions [START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 -norm[END_REF][START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF][START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]. Both of these two methods guarantee the exponential stability of the nonlinear homogeneous hyperbolic systems when the boundary conditions satisfy an appropriate sufficient dissipativity property. Such boundary conditions are the so-called static boundary feedback control and lead to feedbacks that only depend on the measures at the boundaries. Different from the stabilization of the hyperbolic systems without source terms [START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 -norm[END_REF][START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF][START_REF] Coron | A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations[END_REF][START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF][START_REF] Greenberg | The effect of boundary damping for the quasilinear wave equation[END_REF], the increase or decrease of the energy leads to nonuniform steady-states with sometimes large variations in space. In this paper, we address the exponential stability of such nonlinear systems for the H 2 norm, although the result is also true for the H p norm for any p ≥ 2. Mentioning the norm is not superfluous as, for nonlinear systems, the stability for different norms are not equivalent [START_REF] Coron | Dissipative boundary conditions for nonlinear 1-d hyperbolic systems: sharp conditions through an approach via time-delay systems[END_REF]. In particular it has been shown in [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2× 2 hyperbolic systems over a bounded interval[END_REF] that the basic quadratic Lyapunov functions fail to ensure the stabilization in the L 2 norm for nonlinear hyperbolic systems systems and that one has to study the H 2 norm instead. Other attempt of basic Lyapunov functions have been constructed to ensure the stability of hyperbolic systems in the C 1 norm, for instance [START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 -norm[END_REF][START_REF] Hayat | Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the C 1 norm[END_REF][START_REF] Hayat | On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C 1 norm[END_REF].

Physical density-velocity systems often have well-known conservative or dissipative energy or entropy functions when no source term occurs [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]. These dissipative energy or entropy functions are quite useful for the analysis of such system and enable to obtain stability results (see for instance [START_REF] Bastin | A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states[END_REF][START_REF] Coron | A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations[END_REF][START_REF] Coron | On boundary control design for quasilinear hyperbolic systems with entropies as lyapunov functions[END_REF] for the use of entropy as control Lyapunov function for Saint-Venant equations and [START_REF] Bastin | A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states[END_REF] for the Euler equations). When source terms appear, however, no such function is usually known, especially when the source term is not dissipative. In the previous contribution [START_REF] Bastin | A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states[END_REF], the authors also studied the stabilization of hyperbolic densityvelocity equations, but with dissipative source terms only depending on the unknown functions. This is the case for Saint-Venant equations with no slope and with a constant friction, or for the isentropic Euler equations when the gas pressure is simply assumed to be a power function of the gas density and the friction proportional to the square of the gas velocity. However, the source terms may also depend on the space variable in practice and may not be dissipative. This is the case for example for the Saint-Venant equations with both slope and arbitrary friction, or Euler equations with arbitrary friction and slope, and general gas pressure, which are more realistic.

For general density-velocity systems, we find that for any H 2 steady-state, there always exists a basic quadratic Lyapunov function for the H 2 norm that guarantees the exponential stability of the steady-states for the H 2 norm provided suitable boundary conditions and a reasonable physical condition on the source term. Our result in this paper is quite generic and can be widely used in applications, we illustrate it by applying it to several physical systems: the general nonlinear Saint-Venant equations, the general isentropic Euler equations, the motion of water in a rigid pipe, a flow model under osmosis phenomenon and a traffic model.

Moreover, our method has many advantages when applying it in the real world. For example, to stabilize the Saint-Venant equations, we require only some information on the section and the velocities at the boundaries. No information is required on the internal section profile, on the slope, or even on the friction. This is very convenient in practice, as this feedback law can be applied without a clear information of the inner state of the channel (bathymetry, material, profile, etc.) since there may be no way to know properly the precise shape or material of the channel. Besides, while many friction models exist (see e.g. [START_REF] Chanson | Hydraulics of open channel flow[END_REF]Section 4.5]), it also ends the debate about which friction model to use in this context as this feedback law works for any of them, without having to know it.

Another contribution of this paper is that we also study the stabilization of general density-velocity systems with a single boundary control. This corresponds for instance to the regulation of navigable rivers, where one usually applies only one boundary control at the downstream of the channel. We remark here that, in general, it is not obvious at all that a system can be stabilized using less boundary controls. Indeed, when one boundary condition is fixed, there exist examples where one cannot stabilize the system using static feedback controls at the other boundaries. See [START_REF] Bastin | Stability and Boundary Stabilisation of 1-D Hyperbolic Systems[END_REF]Proposition 5.12], or [START_REF] Gugat | On the limits of stabilizability for networks of strings[END_REF] for a concrete example. When studying an inhomogeneous system with less controls (so-called underactuated system), the backstepping method introduced by Krstic et al. in [START_REF] Krstic | Boundary Control of PDEs: A Course on Backstepping Designs[END_REF] is a powerful tool. Initially developed for parabolic equations [START_REF] Smyshlyaev | Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations[END_REF], this method has been applied to first-order linear hyperbolic equations in [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], then to general bidirectional linear systems in [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF]. For the nonlinear case, one can refer to [START_REF] Coron | Local exponential H 2 stabilization of a 2×2 quasilinear hyperbolic system using backstepping[END_REF], where the authors designed a full-state feedback control actuated on only one boundary and achieved exponential stability for the closed-loop 2 × 2 quasilinear hyperbolic systems in H 2 norm. However, when using this method, one requires a full-state feedback control rather than a local static boundary feedback control. In some cases, it is possible to design an observer to tackle this issue [START_REF] Di Meglio | A backstepping boundary observer for a class of linear first-order hyperbolic systems[END_REF][START_REF] Diagne | Backstepping stabilization of the linearized saint-venant-exner model[END_REF][START_REF] Vazquez | Backstepping boundary stabilization and state estimation of a 2 × 2 linear hyperbolic system[END_REF] and to avoid measuring the whole state. Another issue that sometimes occurs with the backstepping method is the delay-robustness of the control law, see for instance [START_REF] Auriol | Delay-robust control design for two heterodirectional linear coupled hyperbolic PDEs[END_REF] where this problem is studied. The advantage in our work is that using a Lyapunov approach enables us to derive very simple static and even local control laws and we only need to measure the value of the state at one boundary. This means that most of the state can be unknown. Compared to the backstepping method, we also only need to know the value of the steady-state targeted at one boundary, and the rest of its profile can remain unknown.

The optimality issue is also addressed in this paper: while we do not look at the optimality of the boundary control with respect to the decay rate (see Remark 3.1 for more details), we aim at giving the largest possible range of feedback laws such that there exists a basic quadratic Lyapunov function to stabilize the system. For single boundary control case, i.e. one boundary condition is fixed, we show in Theorem 2.6 that the dissipative condition on the boundary we imposed is optimal in the sense that one can not give strictly less restrictive boundary condition to guarantee the existence of basic quadratic Lyapunov function. For two boundary controls case, the situation is more complicated, as it has more freedom to control the system. However, we can still present some optimality results on several aspects (see Theorems 2.2-2.4). The organization of the paper is as follows. In Section 2, we present the main results: the exponential stabilization of general density-velocity systems with two boundary controls (Theorem 2.1), and with a single boundary control (Theorem 2.5). Then, Theorems 2.2-2.4 and Theorem 2.6 show the optimality of these two theorems. At the end of this section, we apply these results to several physical models. In Section 3, we give the proof of Theorem 2.1 by showing the existence of a basic quadratic Lyapunov function for the H 2 norm. Theorem 2.5 is deduced as a special case of Theorem 2.1. The proof of Theorems 2.2-2.4 and Theorem 2.6 are presented in Section 4. Finally, some detailed computations are provided in the appendices.

Model considered and main results

A nonlinear hyperbolic density-velocity system is composed of a mass conservation law and a balance of momentum [START_REF] Bastin | A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states[END_REF] and is thus given by

∂ t H + ∂ x (HV ) = 0, (2.1)
∂ t V + V ∂ x V + ∂ x (P (H, x)) + S(H, V, x) = 0, (2.2)
where t ∈ [0, +∞), x ∈ [0, L] with L > 0 any arbitrary constant. In many applications, H denotes the density and V is the velocity. HV is the flow density and S(H, V, x) is a source term resulting of non-conservative forces acting on the system, such as slope or friction. The first equation expresses the flux conservation and is often known as continuity equation, while the second equation is usually referred as dynamical or momentum equation. In this second equation, V ∂ x V represents the variation of the kinetic energy, while ∂ x (P (H, x)) represents the variation of the potential energy and corresponds to a conservative force (e.g. pressure, gravitation, etc.). As we are interested in physical systems, we assume in the whole paper that

S ∈ C 2 ((0, +∞) 2 × [0, L]; R), P ∈ C 2 ((0, +∞) × [0, L]; R
) and here and hereafter, we also assume that

H > 0, V ≥ 0, ∂ H P (H, x) > 0. (2.3)
However in this paper we also extend the results to the situations where the flow is not uniquely determined (this could happen for instance in horizontal pipes or channels). The steady-states (H * , V * ) of (2.1)-(2.2) are the solutions of

(H * V * ) x = 0, V * V * x = -S(H * , V * , x) -∂ x (P (H * , x))
and thus solutions of

H * V * = Q * , (2.4) V * x = V * S(H * , V * , •) + ∂ x P (H * , •) ∂ H P (H * , •)H * -V * 2 , (2.5)
where Q * ≥ 0 is any given constant set point. For each initial condition (H * (0), V * (0)) ∈ (0, +∞) × [0, +∞) satisfying ∂ H P (H * (0), 0)H * (0) -V * (0) 2 > 0, there exists a unique maximal solution to (2.4)-(2.5), and this maximal solution exists as soon as the condition

∂ H P (H * , •)H * > V * 2 is satisfied.
Of course this would also be true with ∂ H P (H * (0), 0)H * (0) -V * (0) 2 < 0, note in this case that the propagation velocities of the system (2.1)-(2.2) would have the same sign. We assume in the following that the propagation velocities of this system have opposite signs, which, from (2.1)-(2.2), means that

∂ H P (H * , •)H * > V * 2 . (2.6)
This assumption is usually known as subcritical regime or subsonic flows (see [START_REF] Gugat | Existence of classical solutions and feedback stabilization for the flow in gas networks[END_REF][START_REF] Gugat | Global boundary controllability of the Saint-Venant system for sloped canals with friction[END_REF] for instance). This holds for example in the case of the Saint-Venant equations when the Froude number is strictly less than 1, or in a congested regime in traffic flows. In this case the system needs to have a prescribed boundary condition at x = 0 and a boundary condition at x = L to be well-posed. When, on the contrary, the regime is supercritical and the propagation velocities have the same sign, the situation is different: two boundary conditions have to be prescribed at x = 0 and none at x = L to guarantee the well-posedness (see [START_REF] Gugat | Global controllability between steady supercritical flows in channel networks[END_REF]). As hyperbolic systems with propagation velocities of the same sign can always be stabilized by the means of proportional boundary feedback (see e.g. [START_REF] Hayat | On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C 1 norm[END_REF]), we do not consider this situation here and focus on the more challenging case of subcritical flows.

In the following, we give two strategies of boundary controls. As a first strategy, Theorem 2.1 relies on two boundary controls, i.e. the number of controls are equal to the number of the unknown functions. While in practice, one may control only one boundary. In the regulation of navigable rivers, for instance, one usually applies only one control at the downstream of the channel. Theorem 2.5 is thus concerned with the stabilization of general density-velocity systems with a single boundary control. Finally, Theorems 2.2-2.4 and Theorem 2.6 illustrate the sharpness of these two theorems.

Two boundary controls

We aim at stabilizing the steady-states of (2.1)-(2.2) with boundary feedback controls. We suppose that in general, the boundary conditions have the following form

V (t, 0) = B 1 (H(t, 0)), V (t, L) = B 2 (H(t, L)), (2.7)
where the control function

B = (B 1 , B 2 ) : R 2 → R 2 is of class C 2 .
As mentioned above, this kind of feedback control is one of the most simple potential feedback law as one does not need to know the full-state. Moreover, this control is local in the sense that one only needs to measure the value at the same end where the control acts.

Taking Saint-Venant equations as an example, these kind of boundary conditions are usually imposed by physical devices in engineering system (e.g. sluice gates, feeding valves, pumps, etc.). If the control actions are provided by two underflow gates located at the left end x = 0 and the right end x = L of the reach. A standard discharge relationship for underflow gates is as follows

H 2 (t, 0)V 2 (t, 0) = u 0 (t)(H up -H(t, 0)), H 2 (t, L)V 2 (t, L) = u L (t)(H(t, L) -H do ),
where H up and H do are the water levels outside the reach that are supposed to be constant and satisfy the inequality H up > H do , u 0 (t) and u L (t) represent the height of the aperture for underflow gates which correspond to our control actions. For more details of the applications, as for instance mobile spillways or tunable hydraulic gates as in irrigation canals and navigable rivers, one can refer to [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF].

For any given initial condition

H(0, x) = H 0 (x), V (0, x) = V 0 (x), x ∈ [0, L], (2.8)
with (H 0 , V 0 ) ∈ H 2 ((0, L); R 2 ), we define the following first-order compatibility conditions (see [START_REF] Bastin | Stability and Boundary Stabilisation of 1-D Hyperbolic Systems[END_REF])

V 0 (0) =B 1 (H 0 (0)), V 0 (L) =B 2 (H 0 (L)), (V 0 ∂ x V 0 + ∂ H P (H 0 , •)∂ x H 0 + ∂ x P (H 0 , •) + S(H 0 , V 0 , •)) (0) =B 1 (H 0 (0))∂ x (H 0 V 0 )(0), (V 0 ∂ x V 0 + ∂ H P (H 0 , •)∂ x H 0 + ∂ x P (H 0 , •) + S(H 0 , V 0 , •)) (L) =B 2 (H 0 (L))∂ x (H 0 V 0 )(L).
(2.9)

We recall now the definition of the exponential stability for the H 2 norm.

, V 0 ) ∈ H 2 ((0, L); R 2 ) satisfying H 0 -H * H 2 ((0,L);R) + V 0 -V * H 2 ((0,L);R) < δ (2.10)
and the compatibility conditions (2.9), and for any T > 0, the Cauchy problem (2.1)-(2.2), (2.7) and (2.8) has a unique solution (H(t, •),

V (t, •)) ∈ H 2 ((0, L); R 2 ) satisfying H(t, •) -H * H 2 ((0,L);R) + V (t, •) -V * H 2 ((0,L);R) ≤Ce -γt H 0 -H * H 2 ((0,L);R) + V 0 -V * H 2 ((0,L);R) , ∀ t ∈ [0, T ). (2.11)
The special case of a steady-state with V * = 0 is physically important as it covers the case where the flow is bidirectional and the direction of the flow is not uniquely determined. Thus, we also include this special case in what follows.

We have the following main result for two boundary controls.

Theorem 2.1. Let (H * , V * ) be a steady-state of the nonlinear hyperbolic density-velocity system (2.1)-(2.2), (2.7) satisfying

∂ V S(H * , V * , •) -V * ∂ H S(H * , V * , •) ∂ H P (H * , •) ≥ 0, ∀x ∈ [0, L].
(2.12)

If the boundary conditions satisfy:

B 1 (H * (0)) ∈ - ∂ H P (H * (0), 0) V * (0) , - V * (0) H * (0) , (2.13)
B 2 (H * (L)) ∈ R \ - ∂ H P (H * (L), L) V * (L) , - V * (L) H * (L) (2.14) if V * ≡ 0 and B 1 (H * (0)) ≤ 0, (2.15) B 2 (H * (L)) > 0 (2.16) if V * ≡ 0. Then the steady-state (H * , V * ) is exponentially stable for the H 2 norm.
Theorem 2.1 is proved in Section 3.

Remark 2.1. The remarkable feature of this result is that it holds for any length L > 0, while this is impossible in general for 2 × 2 systems [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2× 2 hyperbolic systems over a bounded interval[END_REF][START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF][START_REF] Gugat | On the limits of stabilizability for networks of strings[END_REF].

Remark 2.2. Conditions (2.13)-(2.14) can also be alternatively given as

B 1 (H * (0)) ∈ - ∂ H P (H * (0), 0) V * (0) , - V * (0) H * (0) , B 2 (H * (L)) ∈ R \ - ∂ H P (H * (L), L) V * (L) , - V * (L) H * (L) if V * ≡ 0, (2.17)
while (2.15)-(2.16) can be alternatively given as

B 1 (H * (0)) < 0, B 2 (H * (L)) ≥ 0 if V * ≡ 0. (2.18)
The proof of this remark is given in Appendix C.

Remark 2.3. As announced in the introduction, Theorem 2.1 is proved by showing the existence of a basic quadratic Lyapunov function for the H 2 norm (or basic H 2 Lyapunov function). The definition of the basic H 2 Lyapunov function can be found in [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2× 2 hyperbolic systems over a bounded interval[END_REF] for linear systems and in [START_REF] Hayat | On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C 1 norm[END_REF] for quasilinear systems. In particular, for solutions of the system (2.1)-(2.2), the basic H 2 Lyapunov functions are the Lyapunov functions that can be written as

V (H, V ) = L 0 q 1 (x)(H -H * ) 2 + q 2 (x)(H -H * )(V -V * ) + q 3 (x)(V -V * ) 2 dx + L 0 q 1 (x)(∂ t H) 2 + q 2 (x)∂ t H∂ t V + q 3 (x)(∂ t V ) 2 dx + L 0 q 1 (x)(∂ 2 t H) 2 + q 2 (x)∂ 2 t H∂ 2 t V + q 3 (x)(∂ 2 t V ) 2 dx. (2.19)
Remark 2.4. One could wonder how optimal Conditions (2.12)-(2.16) are. We start by showing in Theorem 2.2 that (2.13)-(2.16) are in fact optimal in several aspects, regardless of Condition (2.12). Concerning Condition (2.12), the question of its physical signification is still open, although one could note that Condition (2.12) holds naturally for most physical systems studied in the literature (e.g. Saint-Venant equations, isentropic Euler equations, traffic flows, etc.), as illustrated in the physical examples at the end of this section (see Subsection 2.3). In fact, it holds for any source that consists of friction, slope, electric field or external force. Indeed, slope, electric fields and external forces do not depend on H or V and therefore (2.12) holds directly. Besides, whatever the model chosen, friction forces in S are non-decreasing with V and are non-increasing with H (this last feature comes from the fact that the friction increases at most linearly with the volume and S is a volumic force), which implies also directly (2.12). In particular, it is important to note that Condition (2.12) also holds even when the source term is not dissipative, as S could be negative.

Concerning the sharpness of the conditions in two boundary controls case, we can show the following theorems. On one hand, one has Theorem 2.2 (Optimality of Conditions (2.13)-(2.16)). If there exists a basic quadratic Lyapunov function for the H 2 norm, then either

B 1 (H * (0)) ∈ - ∂ H P (H * (0), 0) V * (0) , - V * (0) H * (0) , or B 2 (H * (L)) ∈ R \ - ∂ H P (H * (L), L) V * (L) , - V * (L) H * (L) if V * ≡ 0, (2.20) and B 1 (H * (0)) ≤ 0 or B 2 (H * (L)) > 0 if V * ≡ 0. ( 2
B 2 (H * (L)) ∈ R \ ε - ∂ H P (H * (L), L) V * (L) , -ε - V * (L) H * (L) if V * ≡ 0, (2.22) B 2 (H * (L)) > -ε if V * ≡ 0. (2.23)
This shows that in this case, the condition on x = L cannot be significantly improved if one wants a basic quadratic Lyapunov function that exists for any length.

Remark 2.5. We also point out that in some applications, the steady-state may not exist on [0, +∞) (see e.g. [START_REF] Gugat | H 2 -stabilization of the isothermal Euler equations: a Lyapunov function approach[END_REF] for the Euler equations when the source term does not depend on the slope and [START_REF] Gugat | Stationary states in gas networks[END_REF] when constant slope is considered). See also [START_REF] Gugat | Global boundary controllability of the Saint-Venant system for sloped canals with friction[END_REF]

B 1 (H * (0)) ∈ - ∂ H P (H * (0), 0) V * (0) , - V * (0) H * (0) (resp. B 1 (H * (0)) ≤ 0) . (2.24)
The proof of Theorems 2.2-2.4 are given in Section 4.

Single boundary control

Suppose now that we have only a single feedback control, the other boundary condition being imposed, for instance by a constant but unknown upstream flow rate on which we cannot act. One notices that now the boundary conditions are special cases of (2.7), which read as follows

H(t, 0)V (t, 0) = Q 0 , V (t, L) = B 2 (H(t, L)), (2.25)
where Q 0 is the unknown constant inflow upstream. Using the same basic quadratic Lyapunov function for the H 2 norm we can still achieve the exponential stability which is a direct application of Theorem 2.1 by noticing now that B 1 (H(t, 0)) = Q 0 /H(t, 0) and that the steady-state satisfies H * (x)V * (x) = Q 0 . In the special case of V * = 0, the fixed boundary condition becomes H(t, 0)V (t, 0) = 0 which implies (when looking locally around the steady-state) that V (t, 0) = V * (0) ≡ 0. Physically, it represents a close-end or a wall with no possible penetration at one side. We have the following main result for single boundary control.

Theorem 2.5. Let (H * , V * ) be a steady-state of the nonlinear hyperbolic density-velocity system (2.1)-(2.2), (2.25) satisfying (2.12). If the boundary control satisfies:

B 2 (H * (L)) ∈ R \ - ∂ H P (H * (L), L) V * (L) , - V * (L) H * (L) if V * ≡ 0 and B 2 (H * (L)) > 0 if V * ≡ 0, (2.26)
then the steady-state (H * , V * ) is exponentially stable for the H 2 norm.

Remark 2.6. Once again, the result holds for any length L > 0, while this is impossible in general for 2 × 2 systems.

Remark 2.7. Note that Q 0 is assumed to be constant otherwise no steady-state (H * , V * ) exists. However, the stabilization of slowly-varying target-state when Q 0 can vary, possibly a lot, but slowly would hold under the same condition by adapting the control as in [START_REF] Hayat | PI controller for the general Saint-Venant equations[END_REF].

Remark 2.8. From Remark 2.2, one can deduce easily that Theorem 2.5 still holds if the control is located on x = 0 instead, while the imposed flow is located on x = L, i.e.

V (t, 0) = B 1 (H(t, 0)), H(t, L)V (t, L) = Q L , (2.27)
where

Q L is the unknown constant outflow downstream, thus, B 2 (H(t, L)) = Q L /H(t, L), while B 1 : R → R of class C 2
is the control function. In this case, the condition on the control becomes

B 1 (H * (0)) ∈ - ∂ H P (H * (0), 0) V * (0) , - V * (0) H * (0) if V * ≡ 0 and B 1 (H * (0)) < 0 if V * ≡ 0. ( 2
B 2 (H * (L)) ∈ R \ - ∂ H P (H * (L), L) V * (L) , - V * (L) H * (L) if V * ≡ 0 and B 2 (H * (L)) > 0 if V * ≡ 0.
(2.29)

The proof of Theorem 2.6 is shown in Section 4.

The abstract density-velocity system (2.1)-(2.2) covers many well-known systems in the literature where Condition (2.12) holds automatically and we give now a few examples.

Examples

General Saint-Venant equations The Saint-Venant equations are the basis model for the regulation of navigable rivers and irrigation networks in agriculture. The stabilization of the Saint-Venant equations by means of local boundary feedbacks has been widely studied [START_REF] Bastin | A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states[END_REF][START_REF] Bastin | On Lyapunov stability of linearised saint-venant equations for a sloping channel[END_REF][START_REF] Coron | A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations[END_REF][START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF][START_REF] Valérie | Boundary control of open channels with numerical and experimental validations[END_REF][START_REF] Prieur | Robust boundary control of systems of conservation laws[END_REF]. Recently in [START_REF] Hayat | A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope[END_REF], the authors obtained the stabilization of the Saint-Venant equations with non-negligible friction and arbitrary slope. However, this result is obtained under the assumption of a rectangular cross section with a constant width and a known friction model. In the following, we show that our result applies to the most general 1D Saint-Venant equations with arbitrary varying slope, section profile and friction model [START_REF] De Saint-Venant | Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit[END_REF]:

∂ t A + ∂ x (AV ) = 0, (2.30)
∂ t (AV ) + ∂ x (AV 2 ) + gA(∂ x H -S b (x) + S f (A, V, x)) = 0, (2.31)
where A is the cross-sectional area of the section, V is the velocity, AV is consequently the flux, H is the height of the water, S b is the slope, S f is the friction and g is the gravity acceleration. Note that the friction logically depends on H and V but can also depend on x for external reasons, for instance if the material of the channel changes. We assume naturally that H is monotonic increasing with A, thus there exists a function G strictly increasing with A such that H = G(A, x) and consequently (2.31) can be written as

∂ t V + V ∂ x V + g∂ A G(A, x)∂ x A + g∂ x G(A, x) + g(S f (A, V, x) -S b (x)) = 0.
(2.32)

Thus, system (2.30)-(2.31) has the form (2.1)-(2.2) with P = gG(A, x) and S = g(S f -S b ). Besides, to be physically acceptable, the friction term has to be increasing with V and decreasing with A. Hence

∂ V S = g∂ V S f > 0 and ∂ A S = g∂ A S f < 0, noticing that ∂ A P = g∂ A G(A, x) > 0,
B 1 (A * (0)) ∈ - g∂ A G(A * (0), 0) V * (0) , - V * (0) A * (0) , B 2 (A * (L)) ∈ R \ - g∂ A G(A * (L), L) V * (L) , - V * (L) A * (L) .
(2.33)

Remark 2.9. We recall here that we only consider subcritical steady-states, i.e. steady-states satisfying (2.6) (which means that the system has two propagation velocities of opposite sign). The case of supercritical steady-states where the system has two propagation velocities of the same sign is easier and can be found for instance in [START_REF] Hayat | On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C 1 norm[END_REF].

Remark 2.10. One can note that if the influence of the slope is larger than that of the friction for the steadystate considered, i.e. S f (A * , V * , x) < S b (x), then this steady-state exists for any length L > 0, and Theorem 2.3 applies. This happens for instance when the channel is prismatic with a constant rectangular section, the slope is constant and S f (A, V, x) = kV 2 /A with kV * 2 (0)/A * (0) < S b (see [START_REF] Hayat | On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C 1 norm[END_REF]Section 5]) or the example of the Saint-Venant and is referred by hydraulic engineers as S1 curves. Finally, the above condition that guarantees a steady-state existing for any length is not necessary in general. Note for instance that in some situations subcritical constant stationary states where the influence of the slope is exactly equal to that of the friction could exist [START_REF] Bastin | On Lyapunov stability of linearised saint-venant equations for a sloping channel[END_REF][START_REF] Gugat | Global boundary controllability of the Saint-Venant system for sloped canals with friction[END_REF].

Remark 2.11. This example also covers the so-called Savage-Hutter equations for granular flows (see [START_REF] Stuart | The motion of a finite mass of granular material down a rough incline[END_REF] or [START_REF] Gugat | Boundary stabilization of quasilinear hyperbolic systems of balance laws: Exponential decay for small source terms[END_REF] for instance).

Water motion in a rigid pipe The water motion in a rigid pipe is a common example for engineering system, whose equations are given in [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF] as follows

∂ t exp gH c 2 + ∂ x V exp gH c 2 = 0, ∂ t V + V ∂ x V + ∂ x (gH) + S f (V, x) = 0, (2.34)
where H is the piezometric head, V > 0 is the water velocity, c is the sound velocity in water, g is the gravity acceleration, and S f is the friction term. Denoting H = exp gH/c 2 , this system has the form of (2.1)-(2.2) with P = c 2 ln H. As previously, to be physically acceptable, the friction term S f has to be non-decreasing with V , thus (2.12) holds and Theorem 2.1 applies again.

The isentropic Euler equations The isentropic Euler equations are used to model the gas transportation in pipelines. There are many literatures on the stabilization of the isentropic Euler equations [START_REF] Bastin | A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states[END_REF][START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF][START_REF] Dick | A strict H 1 -Lyapunov function and feedback stabilization for the isothermal Euler equations with friction[END_REF][START_REF] Gugat | Gas flow in fan-shaped networks: classical solutions and feedback stabilization[END_REF][START_REF] Gugat | Existence of classical solutions and feedback stabilization for the flow in gas networks[END_REF][START_REF] Gugat | H 2 -stabilization of the isothermal Euler equations: a Lyapunov function approach[END_REF][START_REF] Gugat | Boundary feedback stabilization of the isothermal Euler equations with uncertain boundary data[END_REF]. But all those results are obtained without considering the pipeline slope and using the polytropic gas assumption or the isothermal assumption. The isentropic Euler equations with slope and friction have exactly the form (2.1)-(2.2) as (see e.g. [4, Section 1.8.1] or [START_REF] Gugat | The isothermal Euler equations for ideal gas with source term: product solutions, flow reversal and no blow up[END_REF])

∂ t +∂ x ( V ) = 0, ∂ t V +V ∂ x V + ∂ x (P( )) + 1 2 θV |V | + g sin α(x) = 0, (2.35)
where is the gas density, V is the velocity, α ∈ C 2 (R) is the slope of the pipe, g is the acceleration gravity, P( ) is the pressure (increasing with ) with P ( ) > 0 being the sound speed in the gas, θ = λ/D with λ > 0 is the friction coefficient and D > 0 the diameter of the pipe. In this case, P := 0 P (s)/s ds and S = θV |V |/2 + g sin α. Thus, ∂ P ( ) > 0, ∂ S = 0, ∂ V S > 0 as long as V > 0, which implies that (2.12) holds and that Theorem 2.1 applies. Note that this holds in particular in the case where the gas is polytropic, i.e. P = a 2 γ with γ > 1 (as in [START_REF] Bastin | A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states[END_REF]), and in the case of the isothermal Euler equation, i.e. P = a 2 (as in [START_REF] Gugat | The isothermal Euler equations for ideal gas with source term: product solutions, flow reversal and no blow up[END_REF]).

Flow under osmosis

Osmosis is a spontaneous movement of solvant or solute through a semipermeable membrane in a solute/solvant mix. This phenomenon is extremely important in chemistry and biology as it is the main way by which water is transported out of cells in living organisms. Besides, biological membranes allow much faster filtration than any artificial mechanical membrane, thus attempts have been recently made to design active membranes that would mimic this behavior and a mechanical model for this phenomenon can be found in [START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF].

Osmosis phenomenon through a membrane permeable to the solute but not to the solvant can be modeled by a potential barrier which acts on the solute. This creates, from Newton's law, a volume force on the fluid -c(x)∂ x U, where U is the profile of the potential barrier, compactly supported, c is the concentration and x is the space variable [START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF]. In an inviscid fluid modeled by the isentropic Euler equations (2.35), this reduces to adding an external compactly supported pressure term. Therefore, we still have ∂ P ( ) > 0, ∂ S = 0, ∂ V S > 0 as long as V > 0, and Theorem 2.1 applies. Note that any external potential acting on a fluid modeled by the isentropic Euler equations would fit in our framework, osmosis is only an example.

A second order traffic flow model Traffic flows can be modelled at a macro-scale by hyperbolic balance laws. In the last 20 years many so called second-order model were developed. An example can be found in [START_REF] Li | Well-posedness theory of an inhomogeneous traffic flow model[END_REF] where the model is derived with respect to a given speed v( ) and writes this way

∂ t + ∂ x ( v) = 0, ∂ t v + v∂ x v + (∂ v( )) 2 ∂ x = v( ) -v τ , (2.36)
where τ > 0 is the relaxation time, v is strictly decreasing with and f : → v( ) is a concave function defined on [0, max ] with c > 0 such that f ( c ) = max (f ( )) > 0 and f ( max ) = 0. When v ∈ C 1 (R×[0, L]) and for any 0 < C < f ( c ) there exists * such that * v( * ) = C and > c on [0, L]. Then one can check that ( * , v * ) with v * := v( * ) is a steady-state of (2.36) in subcritical regime (called congested regime in the literature of traffic models). Besides this system has exactly the form (2.1)-(2.2), with

P = 0 u(∂ u v(u)) 2 du, and
S( , v, x) = v -v( ) τ .
Therefore

∂ v S( * , v * , x) = 1 τ , ∂ S( * , v * , x) = - ∂ v( * ) τ (2.37)
and

∂ v S( * , v * , x) -v * ∂ S( * , v * , x) ∂ (P ( * )) = * ∂ v( * ) + v(ρ * ) τ * ∂ v( * ) . ( 2

.38)

As v is decreasing with , we have ∂ v( * ) < 0. And as f is concave and > c , then ∂ f ( * ) < 0 which implies that v( * ) + * ∂ v( * ) ≤ 0 and therefore Condition (2.12) holds and Theorem 2.1 can be applied. What is striking is that, in this case, Condition (2.12) corresponds exactly to the physical condition of congested regime on the system [START_REF] Treiber | Traffic flow dynamics. Traffic Flow Dynamics: Data, Models and Simulation[END_REF].

3 Exponential stability of density-velocity hyperbolic systems

In this section we prove Theorem 2.1. Let (H * , V * ) be a steady-state of (2.1)-(2.2), (2.7). We start by proving the exponential stability of the linearized system around this steady-state for the L 2 norm to give an idea of how the proof works and then, we show that the same type of Lyapunov function can be applied to ensure the exponential stability of the nonlinear system for the H 2 norm.

Exponential stability of the linearized system

Around the steady-state (H * , V * ), the linearized system of (2.1)-(2.2), (2.7) is given by:

∂ t h + V * ∂ x h + H * ∂ x v + V * x h + H * x v = 0, ∂ t v + V * ∂ x v + V * x v + ∂ H P (H * , x)∂ x h + ∂ 2 HH P (H * , x)H * x h + ∂ 2 Hx P (H * , x)h + ∂ H S(H * , V * , x)h + ∂ V S(H * , V * , x)v = 0. (3.1) and v(t, 0) = c 1 h(t, 0), v(t, L) = c 2 h(t, L), (3.2)
where h = H -H * and v = V -V * are the perturbations and

c 1 = B 1 (H * (0)), c 2 = B 2 (H * (L)). (3.3)
To simplify the notations, we denote from now on

S H * := ∂ H S(H * , V * , x), S V * := ∂ V S(H * , V * , x), S H * := ∂ 2 Hx P (H * , x) + S H * . (3.4)
Thus, the linearized system of (2.1)-(2.2), (2.7) around the steady-state (H * , V * ) given by (3.1) becomes

h v t + V * H * ∂ H P (H * , x) V * h v x + V * x H * x S H * + ∂ 2 HH P (H * , x)H * x S V * + V * x h v = 0 0 . (3.5)
We recall that the steady-state 0 ∈ L 2 ((0, L); R 2 ) is said exponentially stable (for the L 2 norm) if there exist ν > 0 and C > 0 such that, for every (h 0 (x), v 0 (x)) ∈ L 2 ((0, L); R 2 ), the Cauchy problem (3.5) with initial condition

h(0, x) = h 0 (x), v(0, x) = v 0 (x) (3.6)
is well-posed and its solution satisfies

(h(t, •), v(t, •)) L 2 ((0,L);R 2 ) ≤ Ce -νt (h 0 , v 0 ) L 2 ((0,L);R 2 ) , ∀t ∈ [0, +∞). (3.7)
We prove the following proposition Proposition 3.1. Let (H * , V * ) be any given steady-state of (2.1)-(2.2), (2.7) such that (2.12) holds, if the boundary conditions satisfy:

c 1 ∈ - ∂ H P (H * (0), 0) V * (0) , - V * (0) H * (0) , c 2 ∈ R \ - ∂ H P (H * (L), L) V * (L) , - V * (L) H * (L) if V * ≡ 0 (3.8)
and

c 1 ≤ 0, c 2 > 0 if V * ≡ 0, (3.9)
then the null steady-state h = 0, v = 0 of the system (3.5) and (3.2) is exponentially stable for the L 2 norm.

Proof. Observe that the matrix V * H * ∂ H P (H * , •) V * can be diagonalized, therefore the system can be put under the Riemann invariant form by the following change of variables

z 1 z 2 =   ∂ H P (H * ,x) H * 1 -∂ H P (H * ,x) H * 1   h v . ( 3
∂ t z 1 + λ 1 ∂ x z 1 + γ 1 z 1 + δ 1 z 2 = 0, ∂ t z 2 -λ 2 ∂ x z 2 + γ 2 z 1 + δ 2 z 2 = 0, (3.11)
where

λ 1 = V * + ∂ H P (H * , x)H * > 0, λ 2 = ∂ H P (H * , x)H * -V * > 0 (3.12)
and

γ 1 = 1 4 2S V * + 2S H * H * ∂ H P (H * , x) + 3λ 2 H * x H * -λ 1 ∂ 2 Hx P (H * , x) ∂ H P (H * , x) + λ 2 ∂ 2 HH P (H * x)H * x ∂ H P (H * , x) , γ 2 = 1 4 2S V * + 2S H * H * ∂ H P (H * , x) -λ 1 H * x H * -λ 2 ∂ 2 Hx P (H * , x) ∂ H P (H * , x) + λ 1 ∂ 2 HH P (H * , x)H * x ∂ H P (H * , x) , δ 1 = 1 4 2S V * -2S H * H * ∂ H P (H * , x) + λ 2 H * x H * + λ 1 ∂ 2 Hx P (H * , x) ∂ H P (H * , x) -λ 2 ∂ 2 HH P (H * , x)H * x ∂ H P (H * , x) , δ 2 = 1 4 2S V * -2S H * H * ∂ H P (H * , x) -3λ 1 H * x H * + λ 2 ∂ 2 Hx P (H * , x) ∂ H P (H * , x) -λ 1 ∂ 2 HH P (H * , x)H * x ∂ H P (H * , x) .
(3.13)

The boundary conditions (3.2) become

z 1 (t, 0) = k 1 z 2 (t, 0), z 2 (t, L) = k 2 z 1 (t, L) (3.14)
with

k 1 = c 1 + ∂ H P (H * (0),0) H * (0) c 1 -∂ H P (H * (0),0) H * (0) , k 2 = c 2 -∂ H P (H * (L),L) H * (L) c 2 + ∂ H P (H * (L),L) H * (L) . (3.15)
With these conditions, the Cauchy problem (3.11), (3.14) with any given initial condition z(0,

x) = (z 10 , z 20 ) ∈ L 2 ((0, L); R 2 ) is well-posed (see [4, Appendix A]
), which implies that the original system in physical coordinates is also well-posed. We define the function φ by

φ(x) = exp x 0 γ 1 (s) λ 1 (s) + δ 2 (s) λ 2 (s) ds (3.16)
and we introduce the following lemma, that can also be found in [START_REF] Hayat | A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope[END_REF] in the particular case of the Saint-Venant equations with constant rectangular section and friction given by

S f = kV 2 A -1
where k > 0 is a constant friction coefficient.

Lemma 3.1. Denote by

η := λ 2 λ 1 φ, (3.17) then for any x ∈ [0, L], η (x) = φδ 1 λ 1 + γ 2 λ 2 φ η 2 . (3.18)
The proof of this lemma is given in Appendix B.

We introduce now the following Lyapunov function candidate for the L 2 norm

V = L 0 f 1 (x)e 2 x 0 γ 1 (s) λ 1 (s) ds z 2 1 (t, x) + f 2 (x)e -2 x 0 δ 2 (s) λ 2 (s) ds z 2 2 (t, x) dx, (3.19)
where f 1 and f 2 are positive C 1 functions to be chosen later on. From the positivity of f 1 and f 2 , there exist a 1 and a 2 positive constants such that

a 2 (z 1 , z 2 ) L 2 ((0,L);R 2 ) ≤ V ≤ a 1 (z 1 , z 2 ) L 2 ((0,L);R 2 ) (3.20)
which means that V is equivalent to the L 2 norm of (z 1 , z 2 ), thus is equivalent to the L 2 norm of (h, v) from the linear change of variables (3.10). Therefore, it suffices to show the exponential decay of V to obtain the exponential stability of (3.5) and (3.2) for the L 2 norm. Differentiating (3.19) with time along the C 1 -solutions of (3.11) and (3.14), one has

dV dt = -λ 1 f 1 e 2 x 0 γ 1 (s) λ 1 (s) ds z 2 1 -λ 2 f 2 e -2 x 0 δ 2 (s) λ 2 (s) ds z 2 2 L 0 - L 0 -(λ 1 f 1 ) e 2 x 0 γ 1 (s) λ 1 (s) ds z 2 1 + (λ 2 f 2 ) e -2 x 0 δ 2 (s) λ 2 (s) ds z 2 2 + 2(f 1 δ 1 e 2 x 0 γ 1 (s) λ 1 (s) ds + f 2 γ 2 e -2 x 0 δ 2 (s) λ 2 (s) ds )z 1 z 2 dx. (3.21)
Using the boundary conditions (3.14), we get

dV dt = -λ 1 (L)f 1 (L)e 2 L 0 γ 1 (s) λ 1 (s) ds -k 2 2 λ 2 (L)f 2 (L)e -2 L 0 δ 2 (s) λ 2 (s) ds z 2 1 (t, L) -λ 2 (0)f 2 (0) -k 2 1 λ 1 (0)f 1 (0) z 2 2 (t, 0) - L 0 e x 0 γ 1 (s) λ 1 (s) ds z 1 e -x 0 δ 2 (s) λ 2 (s) ds z 2 T Ĩ e x 0 γ 1 (s) λ 1 (s) ds z 1 e -x 0 δ 2 (s) λ 2 (s) ds z 2 dx, (3.22)
where

Ĩ =    -(λ 1 f 1 ) f 1 δ 1 φ + f 2 γ 2 φ f 1 δ 1 φ + f 2 γ 2 φ (λ 2 f 2 )    . (3.23)
Therefore, it suffices using the definition of φ given in (3.16) to show that there exist f 1 and f 2 , such that the matrix Ĩ is positive definite and that

λ 1 (L)f 1 (L)φ 2 (L) -k 2 2 λ 2 (L)f 2 (L) > 0, λ 2 (0)f 2 (0) -k 2 1 λ 1 (0)f 1 (0) > 0 (3.24)
to prove the exponential decay of V. Indeed, if Ĩ is positive definite, there exist a constant θ > 0 and a small µ > 0 such that for any (z 1 , z 2 ) ∈ L 2 ((0, L); R 2 ), one has

L 0 e x 0 γ 1 (s) λ 1 (s) ds z 1 e -x 0 δ 2 (s) λ 2 (s) ds z 2 T Ĩ e x 0 γ 1 (s) λ 1 (s) ds z 1 e -x 0 δ 2 (s) λ 2 (s) ds z 2 dx ≥ θ L 0 e 2 x 0 γ 1 (s) λ 1 (s) ds z 2 1 + e -2 x 0 δ 2 (s) λ 2 (s) ds z 2 2 dx ≥ µV. (3.25)
Before going any further, observe that under the assumption (2.12), from (3.13), (3.12) and noticing the notations (3.4), one has

φδ 1 λ 1 + γ 2 λ 2 φ λ 2 λ 1 φ 2 = φ λ 2 1 (λ 1 δ 1 + λ 2 γ 2 ) = φ λ 2 1 (λ 1 + λ 2 ) 2 S V * + (λ 2 -λ 1 ) 2 S H * H * ∂ H P (H * , x) + (λ 2 1 -λ 2 2 ) 4 ∂ 2 Hx P (H * , x) ∂ H P (H * , x) = φ λ 2 1 ∂ H P (H * , x)H * S V * -V * H * ∂ H P (H * , x) S H * ≥ 0, (3.26)
which together with Lemma 3.1 implies that (3.17) is a solution to the differential equation

η = δ 1 λ 1 φ + γ 2 λ 2 φ η 2 , η(0) = λ 2 (0) λ 1 (0) (3.27)
on [0, L]. Thus, there exists ε 1 > 0 such that for any ε ∈ [0, ε 1 ), there exists a solution η ε on [0, L] to

η ε = δ 1 λ 1 φ + γ 2 λ 2 φ η 2 ε + ε, η ε (0) = λ 2 (0) λ 1 (0) + ε (3.28)
and such that we can define a map ε → η ε which is C 0 on [0, ε 1 ) (see [START_REF] Hartman | Ordinary differential equations[END_REF]). Let us define

f 1 = 1 λ 1 η ε and f 2 = η ε λ 2 , (3.29)
where ε ∈ (0, ε 1 ) can be chosen later on. One has from (3.8) (resp. (3.9)) and (3.15) that

k 2 1 ≤ λ 2 (0) λ 1 (0) 2 , k 2 2 < λ 1 (L) λ 2 (L) 2 . (3.30)
From the definition of f 1 and f 2 in (3.29) and noticing (3.28), one has

k 2 1 < λ 2 (0)f 2 (0) λ 1 (0)f 1 (0) . (3.31)
Moreover, noticing the definition of η in (3.17) and that η ε (x) > η(x), ∀x ∈ [0, L]. From the continuity of ε → η ε , there exists 0 < ε 2 < ε 1 such that for all ε ∈ (0, ε 2 )

k 2 2 < λ 1 (L)f 1 (L) λ 2 (L)f 2 (L) φ 2 (L), (3.32)
which, together with (3.31) is exactly the same as condition (3.24). We choose such ε ∈ (0, ε 2 ), and we are left to prove that Ĩ defined by (3.23) is positive definite. We have from (3.23), (3.29) and (3.28) that

det( Ĩ) = -(λ 1 f 1 ) (λ 2 f 2 ) -f 1 δ 1 φ + f 2 γ 2 φ 2 = 1 η 2 ε (η ε ) 2 - δ 1 λ 1 φ + γ 2 λ 2 φ η 2 ε 2 > 0. (3.33)
Besides, from (3.28) and (3.29), one has -(λ 1 f 1 ) > 0 and (λ 2 f 2 ) > 0, hence Ĩ is positive definite. Thus, there exists µ 1 > 0 sufficiently small such that

dV dt ≤ -µV (3.34)
along the C 1 -solutions of the system (3.11) and (3.14) for any µ ∈ (0, µ 1 ). Using Gronwall's inequality, one obtains V(t) ≤ e -µt V(0).

Since the C 1 -solutions are dense in the set of L 2 -solutions and the decay rate µ does not depend on the derivatives, inequality (3.34) also holds in the sense of distributions for the L 2 -solutions (see [4, Section 2.1]) for the details). Thus, the exponential stability of (3.5) and (3.2) in the L 2 norm is also guaranteed thanks to the linear change of variables (3.10). This ends the proof of Proposition 3.1.

Remark 3.1. Note that here µ exists but can be arbitrarily small, and therefore does not guarantee a minimal exponential decay rate. This, however, is unavoidable as long as we try to find the largest possible range of stability on the boundary conditions using boundary feedback controls, since at the border of this stability domain, the decay rate is expected to tend to 0.

Exponential stability of the nonlinear system

For the exponential stability of nonlinear system, the proof will be similar to the linearized case. For a given steady-state (H * , V * ) defined on [0, L], we can still define h = H -H * and v = V -V * as previously and (z 1 , z 2 ) using the same change of variables (3.10). Then, for (z 1 , z 2 ) small enough, the system (2.1)-(2.2), (2.7) is equivalent to

z t + A(z, x)z x + M (z, x)z = 0, (3.35)
where

A(0, x) = λ 1 (x) 0 0 -λ 2 (x) , M (0, x) = γ 1 (x) δ 1 (x) γ 2 (x) δ 2 (x) , (3.36)
and

z 1 (t, 0) =m 1 (z 2 (t, 0)), z 2 (t, L) =m 2 (z 1 (t, L)), (3.37)
with

m 1 (0) = m 2 (0) = 0, m 1 (0) = k 1 , m 2 (0) = k 2 , (3.38)
here, k 1 and k 2 are defined as (3.15). In (3.37), m 1 and m 2 are found by the implicit function theorem around 0, for z 1 and z 2 small enough (see [37, A.2] for more details in a similar case). Noticing that the exponential stability of the steady-state (H * , V * ) of system (2.1)-(2.2), (2.7) is therefore equivalent to the exponential stability of the null steady-state (z 1 = 0, z 2 = 0) of system (3.35)-(3.38), we use the following theorem, which is a direct application of [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]Theorem 6.10].

Theorem 3.2. If there exist C 1 functions g 1 (x) > 0 and g 2 (x) > 0 such that, with Q = diag(g 1 (x), g 2 (x)), one has

-(QA(0, •)) + QM (0, x) + M T (0, x)Q (3.39)
is positive definite on [0, L] and the following inequalities hold

k 2 1 < λ 2 (0)g 2 (0) λ 1 (0)g 1 (0) , k 2 2 < λ 1 (L)g 1 (L) λ 2 (L)g 2 (L) , ( 3
V = L 0 (f 1 (E(z, x)z) 2 1 + f 2 (E(z, x)z) 2 2 )dx + L 0 (f 1 (E(z, x)z x ) 2 1 + f 2 (E(z, x)z x ) 2 2 )dx + L 0 (f 1 (E(z, x)z xx ) 2 1 + f 2 (E(z, x)z xx ) 2 2 )dx, (3.41)
where E(0, •) = Id (see [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]Chapter 6] for more details). This is the reason why we claim that this proof is actually the same as the proof of the exponential stability in the linearized case, and we will now see that we can use a similar Lyapunov function but for the H 2 norm.

Proof of Theorem 2.1. Let

g 1 := e 2 x 0 γ 1 (s) λ 1 (s) ds f 1 , g 2 := e -2 x 0 δ 2 (s) λ 2 (s) ds f 2 ,
where f 1 and f 2 are defined in (3.29). One can directly check that -(QA(0, •)) + QM (0, x) + M T (0, x)Q = e

B 1 (H * (0)) ∈ R \ - ∂ H P (H * (0), 0) V * (0) , - V * (0) H * (0) and B 2 (H * (L)) ∈ - ∂ H P (H * (L), L) V * (L) , - V * (L) H * (L) . ( 4
k 2 1 > η 2 (0) and k 2 2 ≥ φ 2 (L) η 2 (L) , (4.2)
where φ is defined by (3.16) and η by (3.17). We define now

a = δ 1 φ, b = γ 2 φ -1 . (4.3)
On the other hand, by assumption we know that there exists a basic quadratic Lyapunov function for the H 2 norm, i.e. there exist C 1 functions f 1 > 0 and f 2 > 0 such that (3.41) is a quadratic Lyapunov function for the H 2 norm and by (3.24), k 1 and k 2 should satisfy

k 2 1 ≤ λ 2 (0)f 2 (0) λ 1 (0)f 1 (0) and k 2 2 < λ 1 (L)f 1 (L) λ 2 (L)f 2 (L) φ 2 (L). (4.4)
Moreover,

λ 1 f 1 > 0, λ 2 f 2 > 0, (λ 1 f 1 ) < 0, (λ 2 f 2 ) > 0, -(λ 1 f 1 ) (λ 2 f 2 ) > (af 1 + bf 2 ) 2 . ( 4.5)
Define now (see also [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2× 2 hyperbolic systems over a bounded interval[END_REF])

w(x) := λ 2 f 2 λ 1 f 1 . (4.6)
One can check from (4.5) that

w > a λ 1 + b λ 2 w 2 . ( 4.7)
As

w(0) = λ 2 (0)f 2 (0) λ 1 (0)f 1 (0) , one gets from (4.4) that k 2 1 ≤w 2 (0) and k 2 2 < φ 2 (L) w 2 (L) . (4.8)
This, together with (4.2) gives η(0) < w(0) and η(L) > w(L). (4.9)

Besides, from Lemma 3.1,

η = a λ 1 + b λ 2 η 2 , η(0) = λ 2 (0) λ 1 (0) . (4.10)
Using that η(0) < w(0), and a comparison for ODEs, one has from (4.7) and (4.10) that

η(L) < w(L), (4.11)
which contradicts with (4.9).

Proof of Theorem 2.3. Suppose that there exists a steady-state (H * , V * ) on [0, L] for any length L > 0 and there exists ε > 0 such that for any length L > 0, there exists a basic quadratic Lyapunov function for the H 2 norm with

B 2 (H * (L)) ∈ ε - ∂ H P (H * (L), L) V * (L) , -ε - V * (L) H * (L) . ( 4

.12)

We can then use the same change of variables (3.10), as in Section 3. The system (2.1)-(2.2), (2.7) becomes (3.35) with boundary conditions (3.37) and (3.38). From (4.12), we have

k 2 2 > φ 2 (L) η 2 (L) , (4.13)
where φ is defined by (3.16) and η by (3.17), k 1 , k 2 are defined by (3.38) and a and b are defined by (4.3). The difference with the proof of Theorem 2.6 given later on is that k 1 here is arbitrary, in fact we could even have k 1 = 0. As there exists a basic quadratic Lyapunov function for the H 2 norm, thus from [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2× 2 hyperbolic systems over a bounded interval[END_REF] (see also [START_REF] Hayat | On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C 1 norm[END_REF]Theorem 4.5]), noticing (3.24) and (4.13), there exists a function

η 2 ∈ C 1 ([0, L]) such that η 2 = a λ 1 + b λ 2 η 2 2 (4.14)
on [0, L] and there exists ε 1 > 0 depending only on ε such that

η 2 (L) ≤ η(L) -ε 1 . ( 4
∈ 1 2φ 2 ∞ , 2 φ 2 ∞ , (4.17)
where φ ∞ = lim x→+∞ φ(x). Thus,

η 2 = |b| λ 2 η 2 2 - a λ 1 , (4.18)
which implies that, using the estimates of [START_REF] Hayat | On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C 1 norm[END_REF]Section 4], there exist C 1 > 0 and x 1 > 0 such that for any x ≥ x 1 ,

η 2 ≥ C 1 x η 2 2 , (4.19)
hence

η 2 ≥ 1 1 η2(x1) -C 1 ln(x/x 1 ) . ( 4

.20)

But we know that η 2 exists and is positive on [x 1 , +∞), hence the contradiction. Thus η 2 converges to a limit η 2,∞ when x goes to infinity. Besides, using (4.3), (4.14) and (4.17),

η 2 = δ 1 λ 1 φ(x) φ 2 (x) - λ 1 |γ 2 | λ 2 δ 1 η 2 2 . (4.21) From [35, Section 5], lim x→+∞ λ 1 |γ 2 | λ 2 δ 1 = 1,
assuming by contradiction that η 2,∞ = φ ∞ , then there exist C 2 > 0 and x 2 > 0 such that for all x > x 2 ,

η 2 ≥ C 2 x , (4.22)
which implies that lim x→+∞ η 2 (x) = +∞, hence the contradiction. Thus,

lim x→+∞ η 2 (x) = η 2,∞ = φ ∞ , (4.23)
moreover, as lim x→+∞ (λ 2 /λ 1 )(x) = 1 (see [START_REF] Hayat | On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C 1 norm[END_REF]), one has

lim x→+∞ η(x) = η ∞ = φ ∞ , (4.24)
which together with (4.23) indicates that the value of η 2 at x = L become arbitrarily close to the one we obtain with η when L goes to infinity and this is in contradiction with (4.15).

Proof of Theorem 2.4. We proceed similarly in the proof of Theorem 2.2. Let L > 0, and suppose that for any B 2 (H * (L)) satisfying (2.14) there exists a basic quadratic Lyapunov function for H 2 norm with

B 1 (H * (0)) ∈ R \ - ∂ H P (H * (0), 0) V * (0) , - V * (0) H * (0) . ( 4
k 2 = φ 2 (L) 2η 2 (L) + φ 2 (L) 2w 2 (L) . ( 4
B 2 (H * (L)) ∈ - ∂ H P (H * (L), L) V * (L) , - V * (L) H * (L) . (4.33)
Then using again the change of variables (3.10), the system (2. Then, using the notations (3.4), (3.5) becomes

z 1 z 2 t + λ 1 0 0 -λ 2 z 1 z 2 x + λ 1 0 0 -λ 2 ∆∆ -1 x z 1 z 2 + ∆ V * x H * x S H * + ∂ 2 HH P (H * , x)H * x S V * + V * x ∆ -1 z 1 z 2 = 0 0 , (A.1)
where λ 1 and λ 2 are given by (3.12). Let us compute the coefficient of the first part of the source term,

λ 1 0 0 -λ 2 ∆∆ -1 x = 1
k 2 1 < λ 2 (0) λ 1 (0) 2 , k 2 2 ≤ λ 1 (L) λ 2 (L) 2 . (C.3)
Thus, instead of defining η ε by (3.28) with η ε (0) assigned, we define now η ε as the solution of

η ε = δ 1 λ 1 φ + γ 2 λ 2 φ η 2 ε + ε, η ε (L) =

. 10 )

 10 Then (3.5) becomes (see Appendix A)

  ds with Ĩ defined as (3.23), as Ĩ is positive definite from (3.33), condition (3.39) is thus satisfied. Condition (3.40) is satisfied from (3.31)-(3.32) by noticing the definition of φ given in (3.16). Thus, Theorem 3.2 applies and Theorem 2.1 holds.

. 1 )

 1 Then using the change of variables (3.10), the system (2.1)-(2.2), (2.7) is equivalent to (3.35) with boundary conditions (3.37) and (3.38). Noticing relations (3.3) and (3.15), (4.1) is equivalent to

Still defining f 1 and f 2

 2 as in(3.29), then one can check that (3.32) still holds and there exists ε 3 > 0 such that for any ε ∈ (0, ε 3 ), one still has (3.31) and Ĩ defined by(3.23) is still definite positive. The rest of the proof remains similar as the proof of Proposition 3.1.

  Definition 2.1. A steady-state (H * , V * ) of the system (2.1)-(2.2), (2.7) is exponentially stable for the H 2 norm if there exist δ > 0, γ > 0 and C > 0 such that, for any (H 0

  .21) On the other hand, Condition (2.14) (resp.(2.16)) on x = L is essentially optimal if one wants to have a basic quadratic Lyapunov function for any length. To that end, assume now that (H * , V * ) is a steady-state of (2.1)-(2.2), (2.7) that exists on [0, L] for any length L > 0. Looking at (2.4)-(2.5), this holds for instance when S(H * , V * , •)+∂ x P (H * , •)/(∂ H P (H * , •)H * -V * 2 ) is bounded (we give an example later on in Subsection 2.3). We have Theorem 2.3 (Optimality of Conditions (2.14) and (2.16)). For any ε > 0, there exists L > 0 such that, if there exists a basic quadratic Lyapunov function for the H 2 norm, then

  thus Condition (2.12) is satisfied and we have the following theorem Theorem 2.7. Any steady-state (A * , V * ) of the general Saint-Venant equations (2.30), (2.32) with boundary conditions (2.7) with A instead of H, is exponentially stable for the H 2 norm provided that

  .40) then the null steady-state of the system (3.35)-(3.38) is exponentially stable for the H 2 norm. Remark 3.2. This theorem actually shows the existence of a Lyapunov function for the H 2 norm of the form

4

  Optimality of the boundary conditions on the controlIn this section we prove Theorems 2.2-2.4 and Theorem 2.6, i.e. the optimality of the boundary conditions (2.13)-(2.14) in the two boundary controls case and the optimality of the boundary condition (2.26) in the single boundary control case. To avoid tedious statements, we only give here the proof for V * ≡ 0, the case V * ≡ 0 follows exactly similarly. One can observe from the proof that the optimality of these boundary conditions does not depend on the interior condition (2.12). of Theorem 2.2. We prove by contradiction argument. If there exists a quadratic Lyapunov function for the H 2 norm for system (2.1)-(2.2), (2.7) with

	Proof

  exists a basic quadratic Lyapunov function for the H 2 norm. Thus, we can again define w on [0, L] as in the proof of Theorem 2.2 such that

	such that there w >	a λ 1	+	b λ 2	w 2 ,	(4.28)
	and	k 2 1 < w 2 (0) and k 2 2 <	φ 2 (L) w 2 (L)	.	(4.29)
	From (4.10), (4.28), (4.27), the first inequality in (4.29), and by a comparison for ODEs, we obtain w(L) >
	η(L), thus
		1 w 2 (L)	<	1 η 2 (L)	.	(4.30)
	Recall that k 2 is arbitrary so far and only has to satisfy (4.26), noticing (4.30), we can choose
							.25)
	Then using again the change of variable (3.10), the system (2.1)-(2.2), (2.7) becomes (3.35) with boundary
	conditions (3.37) and (3.38), our assumption is equivalent to saying that for any length L > 0 and any
		k 2 2 <		φ 2 (L) η 2 (L)	,	(4.26)
	there exists k 1 satisfying
		k 2 1 > η 2 (0),	(4.27)

  is defined by(3.16) and η by(3.17). Thus, we can still define w on [0, L] as in the proof of Theorem 2.2 such that Using (4.10), (4.35) and classical result on the comparison for ODEs, one obtains thatη(L) ≤ w(L),which is in contradiction with (4.34) and(4.36). This ends the proof of Theorem 2.6.A Derivation of γ 1 , γ 2 , δ 1 and δ 2 ∂ H P (H * ,x)

	Looking at (3.10), we denote by
	∆ =	  -∂ H P (H * ,x) ∂ H P (H * ,x) H * H *	1 1	  , ∆ -1 =	1 2	H -1	H * ∂ H P (H * ,x) 1
								1)-(2.2), (2.25) is equivalent to (3.35) with
	boundary conditions (3.37) and (3.38). From (2.25) and noticing (3.3) and (3.15), one has
		k 2 1 = η 2 (0) and k 2 2 ≥	φ 2 (L) η 2 (L)	,	(4.34)
	where φ w >	a λ 1	+	b λ 2	w 2	(4.35)
	and	k 2 1 ≤w 2 (0) and k 2 2 <	φ 2 (L) w 2 (L)	.	(4.36)

*

  H P (H * ,x))∂ H P (H * ,x) ) -λ 2 ( H P (H * ,x)) ∂ H P (H * ,x) )and γ 2 , δ 1 and δ 2 can be found similarly.Differentiating λ 2 φ/λ 1 using (3.12), (3.13) and (3.16), we have(λ 1 λ 2 -λ 1 λ 2 + (λ 2 γ 1 + λ 1 δ 2 )) , x) + 2(λ 2 -λ 1 )S H * H * ∂ H P (H * , x) + 2(λ 2 + λ 1 )S V * ∂ H P (H * , x) ∂ 2 Hx P (H * , x) + H * x V * ∂ H P (H * , x) HNote that the key point of the proof of Theorem 2.1 is Proposition 3.1. In the proof of Proposition 3.1, under condition (2.17), (3.8) becomes c 1 ∈ -∂

	λ 2 λ 1	φ =	φ λ 2 1
	4 HH P (H * , x)H *   λ 1 ( H * x H * -∂ 2 H (B.1) H * x H * - . (A.2)  2 ) -3 1 -λ 2 (V + = φ λ 2 1 1 4 (λ 2 H * x H * -= φ 1 λ 2 V *
	The coefficient of the second part of the source term is Noticing the notations (3.4) and from (2.4), (B.1) becomes
	∆ 1 2   ∂ 2 HH P H * x which, together with (3.26) gives V * x = λ 2 λ 1 φ = H Thus,	H * λ 2 φ 1	x ∂ , x) ∆ -1 λ 2 λ 1 φ = φδ 1 λ 1 + γ 2 λ 2 φ 2 λ 2 λ 1 φ .	S H * ,	(B.2) (B.3)
	γ 1 =	1 4	λ 1	H * x H * -	∂ 2 HH P H * x + ∂ 2 Hx P ∂ H P	+ 2 ∂ 2 HH P H * x	H * ∂ H P	+ S H *	H * ∂ H P	+ H * x	∂ H P H * + 2V * x + S V *
	=	1 4	2S V * + 2S H *	H * ∂ H P	-λ 1	∂ 2 Hx P ∂ H P
		+ (V ∂ 2 HH P H * x ∂ H P	+ 2∂ 2 HH P H * x	H * ∂ H P	+ 2H * x	∂ H P H * + 4V * x	(C.1)
	= and (3.9) becomes 1 4 2S V * + 2S H *	H * ∂ H P	+ 3λ 2	H * x H * -λ 1	∂ 2 Hx P ∂ H P	+ λ 2	∂ 2 HH P H * x ∂ H P	,
												c 1 < 0, c 2 ≥ 0.	(C.2)
	Correspondingly, (3.30) now becomes

d dx (∂ H P (H * ,x)) ∂ H P (H * ,x) ) -λ 1 ( H * x H * -d dx (∂ H P (H * ,x)) ∂ H P (H * ,x) ) λ 2 ( H * x H * -d dx (∂ d dx (∂ x S H * + ∂ 2 HH P (H * , x)H * x S V * + V * * ∂ H P +S H * H * ∂ H P +H * x ∂ H P H * +2V * x +S V * -∂ 2 HH P H * x H * ∂ H P -S H * H * ∂ H P +H * x ∂ H P H * +S V * ∂ 2 HH P H * x H * ∂ H P +S H * H * ∂ H P -H * x ∂ H P H * +S V * -∂ 2 HH P H * x H * ∂ H P -S H * H * ∂ H P -H * x ∂ H P H * +2V * x +S V *   . * + ∂ H P H * ) H * x H * -B Proof of Lemma 3.1 * + ∂ H P (H * , x)H * ) -V * x + (∂ H P (H * , x)H * ) 2 ∂ H P (H * , x)H * -( ∂ H P (H * , x)H * -V * ) V * x + (∂ H P (H * , x)H * ) 2 ∂ H P (H * , x)H * x ∂ H P (H * * * + V * x ∂ H P (H * , x)H * -V * S

H * H * ∂ H P (H * , x) + ∂ H P (H * , x)H * S V * . H P (H * , x)H * S V * -V H * ∂ H P (H * C Adapting the proof of Theorem 2.1 to prove Remark 2.2 H P (H * (0), 0) V * (0) , -V * (0) H * (0) , c 2 ∈ R \ -∂ H P (H * (L), L) V * (L) , -V * (L) H * (L) ,

Acknowledgements

The authors would like to thank Jean-Michel Coron for his advice and constant support. They would also like to thank the National Natural Science Foundation of China (No. 11771336 and 11971119), the ANR project Finite4SoS (No.ANR 15-CE23-0007), the LIASFMA and the French Corps des IPEF for their financial support.