E L Maati

Abdelaziz Rhandi
email: arhandi@unisa.it

KERNEL AND EIGENFUNCTION ESTIMATES FOR SOME SECOND ORDER ELLIPTIC OPERATORS

Keywords: 2000 Mathematics Subject Classification. Primary 47D07; Secondary 35J10, 35K05, 35K10 Heat kernels, Schrödinger operators, eigenfunctions, log-Sobolev inequality

For a potential V such that V (x) ≥ |x| α with α > 2 we prove that the heat kernel k t (x, y) associated to the uniformly elliptic operator A = -

for large x, y ∈ R n and all t > 0. Here 0 < θ ≤ 1 is an appropriate constant, b > α+2 α-2 and µ 0 is the first eigenvalue of A. We also obtain an estimate for large |x| of the eigenfunctions of A.

Introduction

In this paper we consider operators of the type

Au = - n k,j=1 ∂ k (a kj ∂ j u) + V u,
with potentials V that grow more than a polynomial. We study the behaviour for large |x| of the associated heat kernel and associated eigenfunctions. We make the following assumptions in which α > 2, η and Λ are positive constants:

(H)    a kj = a jk ∈ W 1,∞ loc (R n , R), ∂ j a kj = o(|x| α
2 ) as |x| → ∞, η|ξ| 2 ≤ n j,k=1 a kj (x)ξ k ξ j ≤ Λ|ξ| 2 for all ξ ∈ R n , V ∈ L 1 loc (R n ) such that V (x) ≥ |x| α . We define the quadratic forms

a(u, u) := n k,j=1 R n a kj ∂ k u∂ j u + R n V |u| 2 b(u, u) := R n |∇u| 2 + R n V |u| 2 for u ∈ D(a) = D(b) = {u ∈ W 1,2 (R n ); R n V |u| 2 < ∞}.
It is easy to see that a and b are closed symmetric forms. Their associated self-adjoint operators are A and B, respectively.

Formally,

Au = - n k,j=1
∂ k (a kj ∂ j u) + V u and Bu = -∆u + V u.

Since lim |x|→+∞ V (x) = +∞ it follows that A and B have compact resolvents. Therefore their spectra are discrete σ(A) = {µ i ; i = 0, 1, . . .} σ(B) = {λ i ; i = 0, 1, . . .}.

We shall denote by (ψ i ) i≥0 and (ϕ i ) i≥0 the corresponding normalized eigenfunctions of A and B, respectively. In particular, ψ 0 and ϕ 0 will denote the normalized eigenfunctions corresponding to the first eigenvalues µ 0 for A and λ 0 for B.

Since V is non-negative, the semigroup generated by -A is dominated by the Gaussian semigroup. Similarly, the semigroup of -B is dominated by the semigroup corresponding to the operator with V = 0. Therefore, A and B have heat kernels k t (x, y) and p t (x, y) which satisfy Gaussian upper bounds

p t (x, y) ≤ 1 (4πt) n/2 e -|x-y| 2 4t , k t (x, y) ≤ Ct -n/2 e -c |x-y| 2 t
for t > 0 and some constants c, C > 0 (see [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF], [START_REF] Ouhabaz | Analysis of Heat Equations on Domains[END_REF]). Better estimates for large time t are proved in [START_REF] Ouhabaz | Sharp Gaussian bounds and L p -growth of semigroups associated with elliptic and Schrödinger operators[END_REF]. Indeed, it is shown there that

p t (x, y) ≤ C t n/2 e -λ 0 t e -|x-y| 2 4t 1 + λ 0 t + |x -y| 2 t n 2
and the same arguments give

k t (x, y) ≤ C t n/2 e -µ 0 t e -ρ 2 (x,y) 4t 1 + µ 0 t + ρ 2 (x, y) t n 2 , t > 0 where ρ(x, y) := sup{φ(x) -φ(y) : φ ∈ C ∞ c (R n ), n k,j=1 a kj ∂ k φ∂ j φ ≤ 1 a.e. on R n }
is the metric associated to coefficients a kj . By uniform ellipticity, ρ is equivalent to the Euclidean distance. These estimates are sharp with respect to t but they do not reflect the behaviour of the heat kernel for large x and y. For the operator B, it is known that

p t (x, y) ≤ Ce ct -b ϕ 0 (x)ϕ 0 (y) (1.1) for all x, y ∈ R n , 0 < t ≤ 1,
|x| α 4 + n-1 2 e -|x| γ γ ≤ ϕ 0 (x) ≤ c 2 1 |x| α 4 + n-1 2 e -|x| γ γ (1.2)
for some positive constants c 1 , c 2 and γ := 1 + α 2 ; see [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF]Corollary 4.5.8]. From this and (1.1) one has for large x and y

p t (x, y) ≤ Ce ct -b 1 (|x||y|) α 4 + n-1 2 e -|x| γ γ e -|y| γ γ . (1.3)
Our contribution in this note is to show a similar estimate for the heat kernel k t (x, y) of the operator A. It is not clear how to adapt the proof in [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] to obtain (1.1) for k t (x, y) with ψ 0 instead of ϕ 0 . What we shall do is to prove that the estimate (1.3) carries over from p t (x, y) to k t (x, y). The idea is to use a log-Sobolev inequality for B from [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] and then by ellipticity the the same inequality holds for A. In order to conclude we will need an L ∞ estimate for a "weighted" semigroup ϕ -1 e -tA ϕ with an appropriate function ϕ. This estimate will be obtained by using the Beurling-Deny criterion. As a result we prove that if Λ < 1, then for large |x| and |y|

k t (x, y) ≤ Ce -µ 0 t e ct -b 1 (|x||y|) α 4 + n-1 2 e -|x| γ γ - |y| γ γ
for all t > 0. We also mention that some related results on heat kernel bounds for Schrödinger operators on manifolds are proved in [START_REF] Sikora | On-diagonal estimates on Schrödinger semigroup kernels and reduced heat kernels[END_REF].

As a consequence of our result we obtain the following estimate for the eigenfunctions ψ j , j ≥ 0 for an appropriate constant c > 0.

|ψ j (x)| ≤ C 1 |x| α 4 + n-1

Heat kernel estimates

Before we state the main results of this section, let us recall that ψ 0 is strictly positive.

Lemma 2.1. The ground state ψ 0 of A is strictly positive, i.e.

ψ 0 (x) > 0, a.e. x ∈ R n .
Proof. By the Beurling-Deny criterion the semigroup (e -tA ) is positive. This follows from the fact u ∈ D(a) ⇒ u + ∈ D(a) and a(u + , u -) = 0. Since the Gaussian semigroup (e t∆ ) is irreducible (as a convolution with the Gaussian function) we have from [9, Theorem 2.10] that for every measurable subset It is now a classical fact that this implies that µ 0 is a simple eigenvalue with

Ω of R n such that χ Ω u ∈ W 1,2 (R n ), ∀u ∈ W 1,2 (R n ), then |Ω| = 0 or |R n \ Ω| = 0,
ψ 0 > 0 (cf [2, Proposition 1.4.3]).
The following is the first main result of this paper.

Theorem 2.2. Assume that (H) is satisfied with some constant Λ < 1. Then the heat kernel k t (x, y) satisfies

k t (x, y) ≤ Ce -µ 0 t e ct -b 1 (|x||y|) β e -|x| γ γ - |y| γ γ for large x, y ∈ R n and all t > 0. Here C, c > 0, b > α+2 α-2 , β = α 4 + n-1 2 and γ = 1 + α 2 .
Proof. We note first that, by domination (cf. [9, Theorem 2.24]), one can assume that By considering the unitary operator

V (x) = |x| α . Denote by ϕ a positive and C ∞ -function on R n such that ϕ(x) = |x| -β e -|x| γ γ for |x| ≥ R large enough. By normalization, one can assume that ϕ L 2 = 1. Define on L 2 ϕ := L 2 (R n , ϕ 2 (x)dx
U ϕ : L 2 (R n , ϕ 2 dx) -→ L 2 (R n , dx) f -→ ϕf
we see that the associated operators A, B (with a and b, respectively) are

A = U ϕ -1 AU ϕ , B = U ϕ -1 BU ϕ . Hence, e -t e A = ϕ -1 e -tA ϕ, e -t e B = ϕ -1 e -tB ϕ
and their associated kernels are

k t (x, y) = k t (x, y) ϕ(x)ϕ(y) and p t (x, y) = p t (x, y) ϕ(x)ϕ(y) .
We need the following lemmas.

Lemma 2.3. For every non-negative f ∈ D( a) we have

1 ∧ f := inf{1, f } ∈ D( a).
Proof. Since (ϕ -1 0 e -tB ϕ 0 )1 = ϕ -1 0 e -λ 0 t ϕ 0 = e -λ 0 t ≤ 1, the semigroup (ϕ -1 0 e -tB ϕ 0 ) is L ∞contractive. Therefore, we can apply the Beurling-Deny criterion (see e.g. [9, Corollary 2.17]) and obtain

1 ∧ f := inf{1, f } ∈ D(b ) for all 0 ≤ f ∈ D(b ), (2.1)
where (b , D(b )) is the form associated to the operator ϕ -1 0 Bϕ 0 on L 2 (R n , ϕ 2 0 dx) which is (minus) the generator of ϕ -1 0 e -tB ϕ 0 . We claim now that D(b ) = D( b). By (1.2), ϕ and ϕ 0 are equivalent and hence

L 2 (R n , ϕ 2 dx) = L 2 (R n , ϕ 2 0 dx). In addition, R n V (ϕu) 2 dx < ∞ ⇔ R n V (ϕ 0 u) 2 dx < ∞.
Using the fact that for large |x| we have

a 1 |∇ϕ(x)| ≤ |∇ϕ 0 (x)| ≤ a 2 |∇ϕ(x)| (2.2)
where a 1 and a 2 are positive constants (see [6, Lemma 6.1], [8, Theorem 2.1, Chapter 6]), we obtain

∇ϕ 0 (x) ϕ 0 (x) 2 ≤ C 1 |x| α + C 2 , x ∈ R n ,
where C 1 and C 2 are two positive constants. Suppose now that u ∈ D( b).

Then, v := ϕu ∈ W 1,2 (R n ), R n |x| α |v(x)| 2 dx < ∞ and g := ϕ 0 ϕ satisfies |∇g(x)| 2 ≤ C 1 |x| α + C 2 , x ∈ R n . (2.3) Since C ∞ c (R n )
) is a core for B (cf. [START_REF] Kato | Schrödinger operators with singular potentials[END_REF]) and hence a core for b (cf. [9, Lemma 1.25]), it follows that there is a sequence

(v n ) in C ∞ c (R n )) such that v n → v, |∇v n | → |∇v| and |x| α 2 v n → |x| α 2 v in L 2 (R n ). Take now h ∈ R n . Then for every x g(x + h)v n (x + h) -g(x)v n (x) = (g(x + h) -g(x))v n (x) + g(x + h)(v n (x + h) -v n (x)) = v n (x) 1 0 h • ∇g(x + th) dt + g(x + h) 1 0 h • ∇v n (x + th) dt.
Hence, using (2.3), we deduce that there are positive constants

C 1 , C 2 , C 3 such that τ h (gv n ) -(gv n ) L 2 ≤ |h| C 1 R n |x| α |v n (x)| 2 dx 1 2 + C 2 v n L 2 + C 3 ∇v n L 2
holds for all n ∈ N. Passing to the limit, we obtain Proof. Starting from the expression of a we obtain

τ h (gv) -(gv) L 2 ≤ |h| C 1 R n |x| α |v(x)| 2 dx 1 2 + C 2 v L 2 + C 3 ∇v L 2 . Therefore, ϕ 0 u = gv ∈ W 1,2 (R n ) (cf.
a(u, v) = n j,k=1 R n a kj ∂ k (ϕu)∂ j (ϕv) dx + R n V uvϕ 2 dx = n j,k=1 R n a kj ∂ k u∂ j vϕ 2 dx + R n W a uvϕ 2 dx with W a := V -n j,k=1 ∂ j a kj ∂ k ϕ ϕ -n j,k=1 a kj ∂ k ∂ j ϕ ϕ .
For |x| large, we have

W a (x) = V (x) + n j,k=1 ∂ j a kj (x)x k β|x| -2 + |x| γ-2 - n j,k=1 a kj (x)x j x k (β|x| -2 + |x| γ-2 ) 2 + 2β|x| -4 -(γ -2)|x| γ-4 + n k=1 a kk (x) β|x| -2 + |x| γ-2 ≥ |x| α - k,j |∂ j a kj | β|x| -1 + |x| γ-1 - k a kk ∞ β|x| -2 + |x| γ-2 -Λ |x| 2 (β|x| -2 + |x| γ-2 ) 2 + 2β|x| -2 + η(γ -2)|x| γ-2 = (1 -Λ)|x| α - k,j |∂ j a kj | β|x| -1 + |x| γ-1 + η(γ -2)|x| γ-2 -Λ β 2 |x| -2 + 2β|x| γ-2 + 2β|x| -2 - k a kk ∞ β|x| -2 + |x| γ-2 .
Then, the assumptions Λ < 1 and

∂ j a kj = o(|x| α 2 ) yield W a (x) ≥ 0 for |x| large .
Hence, there is λ a ∈ R such that

W a (x) ≥ -λ a for a.e. x ∈ R n . (2.4)
By the previous lemma and the obvious equality D( a) = D( b) we have 1 ∧ u ∈ D( a) for all non-negative u ∈ D( a). Applying now (2.4) we obtain

a(1 ∧ u, (u -1) + ) = R n W a (1 ∧ u)(u -1) + ϕ 2 dx ≥ -λ a R n (1 ∧ u)(u -1) + ϕ 2 dx.
Applying the Beurling-Deny criterion (cf. [9, Corollary 2.17]), we obtain the lemma.

We continue the proof of the theorem. Similarly to Lemma 2.4 we have

b(u, v) = R n ∇u • ∇vϕ 2 dx + R n W b uvϕ 2 dx,
where

W b = V -∆ϕ ϕ ≥ -|x| -2 β(β + 2 -n) for large |x|.
Hence, as above, we obtain

e -t e B L(L ∞ ) ≤ e λ b t , t ≥ 0, (2.5)
for some constant λ b ∈ R.

On the other hand, let b > α+2 α-2 . Then, by (1.2) and [2, Corollary 4.5.5], we have

p t (x, y) = p t (x, y) ϕ(x)ϕ(y) ≤ C p t (x, y) ϕ 0 (x)ϕ 0 (y) ≤ M e ct -b , 0 < t ≤ 1.
For t ≥ 1 we recall first that, by the semigroup law and the symmetry of p t , p s+r (x, y) = R n p s (x, z)p r (y, z) dz, s, r > 0.

(

Hence,

p t (x, x) = R n |p t 2 (x, y)| 2 dy = R n e -( t 2 -1 2 )B p 1 2 (x, •)(y) 2 dy = e -( t 2 -1 2 )B p 1 2 (x, •) 2 L 2 ≤ e -2λ 0( t 2 -1 2 ) p 1 2 (x, •) 2 L 2 ≤ e -λ 0 t p 1 (x, x)
≤ M e -λ 0 t ϕ 2 (x).

Applying the inequality

p t (x, y) ≤ p t (x, x) p t (y, y), x, y ∈ R n ,
which is a consequence of (2.6) and Hölder's inequality, we obtain the estimate p t (x, y) ≤ M e -λ 0 t e ct -b , t > 0.

Thus, by (2.5), the semigroup e -t e B is ultracontractive. By [2, Theorem 2.2.3] we obtain that b satisfies a log-Sobolev inequality A is ultracontractive and

R n u 2 log u ϕ 2 dx ≤ ε b(u, u) + β(ε) u 2 L 2 ϕ + u 2 L 2 ϕ log u L 2 ϕ (2.7) for all ε > 0 and 0 ≤ u ∈ L 1 ∩ L ∞ ∩ D( b), with β(ε) = b 1 -λ 0 ε + cε -b ,
k t (x, y) = k t (x, y) ϕ(x)ϕ(y) ≤ Ce ct -b , 0 < t ≤ 1. (2.8)
For t ≥ 1 we proceed as above and obtain

k t (x, y) ≤ Ce -µ 0 t e ct -b ϕ(x)ϕ(y), x, y ∈ R n , t > 0.
This proves the theorem. for large |x| and a constant C > 0, where β = α 4 + n-1 2 and γ = 1 + α 2 . Proof. For each j ≥ 0 we have

|ψ j (x)|e -µ j t = |e -tA ψ j (x)| = | R n k t (x, y)ψ j (y) dy| ≤ R n k t (x, y) 2 dy 1/2 ψ j 2 = (k 2t (x, x)) 1/2 .
We apply the previous theorem and obtain the estimate of the corollary. for all large |x| and some constants a 1 , a 2 > 0.

Theorem 2.7. Assume that (H) is satisfied. Then for θ > 0 such that θΛ < 1 we have

k t (x, y) ≤ Ce -µ 0 t e ct -b 1 (|x||y|) β e - √ θ γ |x| γ e - √ θ γ |y| γ for large x, y ∈ R n and all t > 0. Here C, c > 0, b > α+2 α-2 , β = α 4 + n-1 2 and γ = 1 + α 2 .
Proof. We proceed as in the proof of Theorem 2.2. We consider the function ϕ defined above and the form h associated to H instead of b. We have a(u, v) = The assumption θΛ < 1 gives W a (x) ≥ 0 for large |x| and hence W a (x) ≥ -λ for some λ ∈ R and all x ∈ R n . By Lemmas 2.3 and 2.4 we obtain e -t e A L(L ∞ ) ≤ e λt , t ≥ 0. We conclude exactly as in the proof of Theorem 2.2.

As a consequence we obtain an estimate for the eigenfunctions ψ j of A. for large |x| and some constant C > 0, where β = α 4 + n-1 2 and γ = 1 + α 2 . Remark 2.9. Take θ > 0 such that θΛ < 1. Using Lyapunov functions techniques it is proved in [START_REF] Laidoune | Global properties of transition kernels associated with second-order elliptic operators[END_REF]Corollary 3.4] that if the assumption (H) is satisfied and the coefficients a jk ∈ C 1+η (R n ) with some 0 < η < 1 then b = α+2 α-2 can be taken. More precisely one has the following estimate:

k t (x, y) ≤ Ce ct -α+2 α-2 e -2 √ θ α+2 |x| 1+ α 2 e -2 √ θ α+2 |y| 1+ α 2
for all x, y ∈ R n , 0 < t ≤ 1.

2 e

 2 -|x| γ γ for large |x|. If the condition Λ < 1 is not satisfied we prove similar estimates with e -|x| γ γ replaced by e -c |x| γ γ

  where | • | denotes the Lebesque measure. From this and the definition of D(a), we see that χ Ω u ∈ D(a), ∀u ∈ D(a), then |Ω| = 0 or |R n \ Ω| = 0, where χ Ω (x) = 1 if x ∈ Ω, 0 otherwise denotes the characteristic function of the set Ω. By [9, Corollary 2.11], (e -tA ) is irreducible.

  ) the forms a(u, v) := a(ϕu, ϕv), b(u, v) := b(ϕu, ϕv) D( a) = D( b) = {u ∈ L 2 (R n , ϕ 2 dx); ϕu ∈ D(a) = D(b)}.

[ 1 ,Lemma 2 . 4 .

 124 Proposition 9.3]) and hence D( b) ⊂ D(b ). The reverse inclusion can be proved similarly. Using now D(b ) = D( b), the lemma follows from (2.1). There exists a constant λ a ∈ R such that e -t e A L(L ∞ ) ≤ e λat for all t ≥ 0.

where b 1

 1 , c are positive constants. Since, by ellipticity (see (H)), b(u, u) ≤ max{ 1 η , 1} a(u, u) for all u ∈ D( a) = D( b), it follows that a satisfies the same log-Sobolev inequality (2.7). Using Lemma 2.4 and [2, Corollary 2.2.8 and Example 2.3.4] we conclude that e -t e

Corollary 2 . 5 .

 25 If the assumptions of Theorem 2.2 hold, then all the eigenfunctions ψ j , j = 0, 1,• • • of A satisfy |ψ j (x)| ≤ C|x| -β e -|x| γ γ

Remark 2 . 6 .

 26 For the first eigenfunction ψ 0 we do not know if the reverse inequalityψ 0 (x) ≥ C|x| -β e -|x| γ γ holds for large |x| and some constant C > 0. ≤ φ 0 (x) ≤ c 2 |x| -β e - |x|. As in the proof of Theorem 2.2 we consider a positive and smooth function ϕ (with ϕ L 2 = 1) such that ϕ(x) = |x| -β e - √ θ γ |x| γ for large |x|. By the same computation as in [6, Lemma 6.1], we obtain a 1 |∇ϕ(x)| ≤ |∇φ 0 (x)| ≤ a 2 |∇ϕ(x)| (2.10)

  R n a kj ∂ k (ϕu)∂ j (ϕv) dx + R n |x| α uvϕ 2 dx = n j,k=1 R n a kj ∂ k u∂ j vϕ 2 dx + R n W a uvϕ 2 dx,and for large |x|, a direct computation yields W a (x) ≥ (1 -θΛ)|x| α + o(|x| α ).

Corollary 2 . 8 .

 28 Assume that (H) holds. Then for θ > 0 such that θΛ < 1 we have for all j ≥ 0|ψ j (x)| ≤ C|x| -β e -

We consider now the general case where the condition Λ < 1 is not necessary satisfied. To this purpose we consider the operator H := -∆ + θ|x| α on L 2 (R n ) with 0 < θ < 1 and denotes by h t (x, y) its corresponding heat kernel. By [2, Theorem 4.5.4] and as in the above proof, we deduce that


Antonin Delpeuch
email: antonin.delpeuch@ens.fr

Anne Preller
email: preller@lirmm.fr

From Natural Language to RDF Graphs with Pregroups

We define an algorithm translating natural language sentences to the formal syntax of RDF, an existential conjunctive logic widely used on the Semantic Web. Our translation is based on pregroup grammars, an efficient type-logical grammatical framework with a transparent syntax-semantics interface. We introduce a restricted notion of side effects in the semantic category of finitely generated free semimodules over {0, 1} to that end. The translation gives an intensional counterpart to previous extensional models. We establish a one-to-one correspondence between extensional models and RDF models such that satisfaction is preserved. Our translation encompasses the expressivity of the target language and supports complex linguistic constructions like relative clauses and unbounded dependencies.

Introduction

There is a general agreement that Natural Language Processing has two aspects. One is syntax, rules how words are put together to form a grammatical string. The other is semantics, rules how meanings of strings are computed from the meanings of words. To this we add a third aspect, namely that semantics must include rules of logic how to compute the truth of the whole string from the truth of its parts.

The Resource Description Framework (RDF) [START_REF] Hayes | Rdf semantics[END_REF]) is an artificial language that takes the intuitive form of knowledge graphs. Its semantics has the expressive power of the fragment of multisorted first order logic that uses conjunction and existential quantification only. This restricted expressive power limits the statements of natural language that can be interpreted as RDF graphs. Typically, statements with negation words are excluded.

We use pregroup grammars as the linguistic and mathematical tool to recognise and interpret natural language strings in RDF. Pregroup Calculus, also known as Compact Bilinear Logic [START_REF] Lambek | Substructural Logics, chapter From categorial grammar to bilinear logic[END_REF]) [START_REF] Lambek | Type grammar revisited[END_REF], is a simplification of Lambek's Syntactic Calculus, [START_REF] Lambek | The mathematics of sentence structure[END_REF]. Pregroup grammars are based on a lexicon like all categorial grammars. Syntactical analysis consists in the construction of a proof (derivation) in the calculus.

All semantics proposed so far use functors from the lexical category of [START_REF] Preller | Free compact 2-categories[END_REF] into some symmetric compact closed category. They include the compositional distributional vector space models of [START_REF] Clark | A compositional distributional model of meaning[END_REF], [START_REF] Kartsaklis | Reasoning about Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras[END_REF] based on context and the functional logical models of [START_REF] Preller | From Sentence to Concept: Predicate Logic and Quantum Logic in Compact Closed Categories[END_REF].

We proceed as follows. Recalling that words and grammatical strings recognised by the grammar are represented by meaning-morphisms in the lexical category, we propose a "syntactic" functor from the latter into a symmetric monoidal category C S of 'morphisms with side effects' over the category C of finite dimensional semimodules over the lattice B = {0, 1}. The elements of the semimodule S identify with RDF graphs. The value of the syntactic functor at a statement is the RDF graph of the statement. The extensional models of logic are recast as "semantic" functors from the lexical category to C. We associate to any semantic functor an RDF interpretation and show that a statement is true in the semantic functor if and only if the corresponding RDF graph is true in the RDF interpretation.

Preliminaries

2.1 RDF graphs RDF is a widely-adopted framework introduced in [START_REF] Hayes | Rdf semantics[END_REF] as a standard for linked information representation on the Web. Informally, an RDF graph is a set of labeled links between objects, which represent statements concerning the linked objects.

Throughout this article we will simply consider that they are represented by strings of characters without spaces, written in a mono-spaced font: the entity John is denoted by the string John.

A link between two objects is a triple, made of one entity (the subject of the predicate), a property (the type of the link, also represented by a string) and a second entity (the object of the predicate). Graphically, we represent a triple as a directed property from the subject to the object, labeled by the predicate.

RDF allows to use nodes without labels, called blank nodes. Concretely, this means that we can always pick a fresh node, named blank-n where n is some number, such that this node is not involved in any triple yet.

An example

We can represent the sentence John owns a big boat using a fresh blank node for a big boat: Recall that our goal is to translate natural language sentences to graphs. Graphs can indeed be seen as semantic representations of sentences, in the sense that they can be used to assign a truth value to a sentence, trough the notion of entailment [START_REF] Hayes | Rdf semantics[END_REF].

A graph H is an instance of G when H can be obtained from G by assigning labels to some blank nodes (possibly none of them). A graph G 0 entails another graph G if an instance of G is a subgraph of G 0 . With these definitions, we can define true RDF graphs as the graphs entailed by some reference graph, storing all the true facts.

Pregroup grammars

A pregroup grammar [START_REF] Lambek | Type grammar revisited[END_REF], consists of the free pregroup C(B) generated by a partially ordered set B and a lexicon D B . By 'free pregroup' we mean the free compact closed category C(B) generated by B following the version given in [START_REF] Preller | Free compact 2-categories[END_REF] 1 . The lexicon D B introduces the monoidal category L B , freely generated by the inequalities and the lexical morphisms given by the lexicon. Lexical entries have "formal meanings" in the lexical category, the free compact closed category generated by B and the morphisms introduced by the words. They are analogue to the lambda-terms intervening in categorial grammars, [START_REF] Steedman | Surface Structure and Interpretation, volume 30 of Linguistic Inquiry Monograph[END_REF].

The working of pregroup grammars can be described without explicit use of category theory. The main result of this section however, the decomposition lemma, invokes properties of the graphs proved in [START_REF] Preller | Free compact 2-categories[END_REF]. 1 Semantics requires more than one morphism between objects, whereas the partial preorder of the free pregroup of [START_REF] Lambek | Type grammar revisited[END_REF]) confounds all morphisms with identical domain and codomain.

We start with the formal definition of a monoidal category followed by the condition it must satisfy to be compact closed.

Definition 1. A category C is 1. monoidal if there is a bifunctor ⊗ : C × C → C, a distinguished object I, the unit of ⊗, satisfying (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C), (f ⊗ g) ⊗ h = f ⊗ (g ⊗ h) 2 2.
compact closed if it is monoidal and there are contravariant endofunctors ( ) r and ( ) ℓ , called right adjoint and left adjoint, such that for every object A and every morphism f : A → B there are morphisms η f : I → A r ⊗ A, the name of f , and

ǫ f : A ⊗ B r → I, the coname of f , satisfying for any g : B → C (A ⊗ B) r = B r ⊗ A r , (A ⊗ B) ℓ = B ℓ ⊗ A ℓ (ǫ f ⊗ 1 C ) • (1 A ⊗ η g ) = g • f (1 A r ⊗ ǫ g ) • (η f ⊗ 1 C r ) = (f • g) r .
3. symmetric if it is monoidal and there is a natural isomorphism σ AB :

A ⊗ B → B ⊗ A such that σ -1 AB = σ BA .
Recall that ⊗ is a bifunctor if and only if the following equalities are satisfied for all objects A, B and all morphisms f i :

A i → B i , g i : B i → C i , i = 1, 2 1 A ⊗ 1 B = 1 A⊗B Interchange Law (f 2 • f 1 ) ⊗ (g 2 • g 1 ) = (f 2 ⊗ g 2 ) • (f 1 ⊗ g 1 ) (1)
The morphisms of C(B) and L B identify with graphs, which we now describe without invoking category theory.

The objects of C(B) are called types. They include the elements of B, called basic types. For instance, the set B may include the basic types s for statements, d for determiners, n for noun phrases, o for direct objects with corresponding types in relatives clauses r, n and ô. There is only one strict inequality n < o. Assimilating B to a category, we write i ab : a → b instead of a ≤ b and 1 a for i aa . A simple type is an iterated left adjoint or right adjoint of a basic type . . . , a ℓℓ = a (-2) , a ℓ = a (-1) , a = a (0) , a r = a (1) , a rr = a (2) , . . . . The iterator of t = a (z) is the integer z. An arbitrary type is a string of simple types separated by the symbol ⊗. In particular, the empty string is a type, denoted I.

The lexicon of a pregroup grammar consists of a set of pairs word : T where the type T ∈ C(B) captures the different grammatical roles a word may play. For instance

proper name : n transitive verb : n r ⊗ s ⊗ o ℓ transitive verb : nr ⊗ r ⊗ o ℓ transitive verb : n r ⊗ ôr ⊗ r determiner : d adjective : d r ⊗ d countnoun : d r ⊗ n relpronoun : n r ⊗ n ⊗ r ℓ ⊗ n relpronoun : n r ⊗ n ⊗ r ℓ ⊗ ô
A pregroup grammars analyses a string of words by constructing a graph: choose an entry w i : T i for each word w i and align the types in the order of the words. Every simple type t j of T 1 ⊗. . .⊗T n = t 1 ⊗. . .⊗t m is (the label of) a node of the graph. Place non-crossing oriented underlinks such that the tail of an underlink is a basic type with an even number of iterations, say t = a (z) , where z = ±2n. If the head is to the left of the tail then it is the left adjoint b (z)ℓ = b (z-1) , for some basic type b ≥ a. If it is to the right then its a right adjoint b (z)r = b (z+1) . Complete the graph by repeating the nodes that are not head or tail of an underlink in a line below and adding a vertical link between corresponding nodes.

The string is said to be grammatical if exactly one simple type remains that is not the tail or head of an underlink and if it is a basic type. The resulting graph is called a reduction and denotes a unique morphism r :

T 1 ⊗ . . . ⊗ T n → b of C(B).
More generally, graphs standing for morphisms align the domain above and the codomain below.

For instance, the graph below exhibits a reduction to the sentence type s

(n) ⊗ (n r ⊗ s ⊗ o ℓ ) ⊗ (d) ⊗ (d r ⊗ d) ⊗ (d r ⊗ n) john owns a red boat Y Y s g g Y Y Y Y
The following two graphs are reductions to the noun phrase type n. The first graph corresponds to the case where the relative pronoun is the object of the verb in the relative clause whereas it is the subject in the second graph

d ⊗ d r ⊗ n ⊗ n r ⊗ n ⊗ r ℓ ⊗ ô ⊗ n ⊗ n r ⊗ ôr ⊗ r a cat that bob hates c c c c n X X c c d ⊗ d r ⊗ n ⊗ n r ⊗ n ⊗ r ℓ ⊗ n ⊗ nr ⊗ r ⊗ o ℓ ⊗ n a cat that bob hates c c c c n d d c c
The meanings of words are also represented by graphs that correspond to morphisms of the lexical category L B . In fact, every entry w : T in the lexicon gives rise to a meaning morphism w T : I → T and a lexical morphism w T represented by the following oriented labelled graphs

w b = w b = w b T = a r ⊗ b w T = a r ⊗ b 4 4 w w T = a w b T = a r ⊗ b ⊗ c ℓ w T = a r ⊗ b ⊗ c ℓ w T w T = a ⊗ c b T T T jj j w T T = a r ⊗ c r ⊗ b w T = a r ⊗ c r ⊗ b w 1 1 w T = c ⊗ a b T T T jj j w T
If T has two factors that are basic types, there is besides the main lexical morphism w T an auxiliary lexical morphism j T .

T = n r ⊗ n ⊗ r ℓ ⊗ d, d = n, ô that T = n r ⊗ n ⊗ r ℓ ⊗ d j d 4 4 ~that r n that n d j d
We omit the subscript T if this does not lead to confusion.

The nodes of the meaning graphs are the simple types of T . They form the lower line of the graph, the upper line is the empty string I. The corresponding morphism has domain I and codomain T . An overlink may have several tails but only one head. The tails of overlinks are right or left adjoints of basic types, the head is a basic type3 .

The lexical category L B generated by the lexicon D B is the compact closed category freely generated by B, the symmetry morphisms σ ab and the lexical morphisms introduced by D B .

Strings of words also have meaning(s) in the lexical category. The lexical meaning of a grammatical string word 1 . . . word n recognised by the reduction r : T 1 . . . T n → b is, by definition, the composite of the tensor product of the word meanings and the chosen reduction.

r • (word 1T 1 ⊗ . . . ⊗ word nT n ) : I → b .
The meaning of a composite morphism g • f can be simplified graphically. Stack the graph of f above the graph of g and walk the paths of the graph starting at a node that is not the head of a link until you arrive at a node that is not the tail of a link. Compose the labels in the order they are encountered along the path. Replace the whole path by a single link labelling it by the composite label of the path. The resulting graph represents the morphism g • f , [START_REF] Selinger | A survey of graphical languages for monoidal categories[END_REF].

For instance, the grammatical strings mentioned above above simplify to

(n) ⊗ (n r ⊗ s ⊗ o ℓ ) ⊗ (d) ⊗ (d r ⊗ d) ⊗ (d r ⊗ n) john big boat Y Y s g g Y Y Y Y own a ) ) ) ) = n ⊗ o john boat•red•a s T T T jj j own = I s own•(john⊗(boat•red•a)) T = nr ⊗ r ⊗ o ℓ d ⊗ d r ⊗ n ⊗ n r ⊗ n ⊗ r ℓ ⊗ n ⊗ nr ⊗ r ⊗ o ℓ ⊗ n a cat that bob hate T c c c c n d d c c d d ) ) 3 3 Ò Ò that • hate T • ((cat • a) ⊗ bob)] : I ⊗ I → n T ′ = n r ⊗ ôr ⊗ r d ⊗ d r ⊗ n ⊗ n r ⊗ n ⊗ r ℓ ⊗ ô ⊗ n ⊗ n r ⊗ ôr ⊗ r a cat that bob hate T ′ c c c c n d d X X c c ) ) 3 3 Ò Ò ) ) = that • hate T ′ • ((cat • a) ⊗ bob)] : I ⊗ I → n
All unlabelled links in the graph above correspond to inequalities of basic types. Inserting the strict inequalities at the appropriate place and applying the interchange law, we decompose the label into minimal building blocks that correspond one-to-one to the resources occurring in the RDF graph associated to the statement. For example,

own • (john ⊗ (boat • big • a)) = own • (1 n ⊗ in) • (1 n ⊗ boat) • (1 n ⊗ big)) •(john ⊗ a) .
(2) The expression after the equality symbol above is a composite of tensor products the factors of which are either inequalities between basic objects or lexical morphisms. Only the rightmost tensor product contains more than one lexical morphism. In fact, all factors are lexical morphisms with domain I.

The translation of statements to RDF graphs rests on the existence of this decomposition. Borrowing the terminology of RDF graphs, call any lexical morphism with domain I a node word and any other lexical morphism a property word.

Lemma 1 (Decomposition). Let word 1 . . . word n be a grammatical string with lexical morphisms word 1 , . . . , word n and a reduction r : T 1 . . . T n → b. Then there is an enumeration of the node words word i1 : I → b 1 , . . . , word im : I → b m such that the lexical meaning of the string satisfies

r • (word 1 ⊗ . . . ⊗ word n ) = p 1 • • • • • p m ′ • (word i1 ⊗ . . . ⊗ word im ) ,
where each p k is either a tensor product of inequalities of basic types or a tensor product of inequalities and one property word

word j k . Moreover, k < k ′ implies j k < j k ′ .
In particular, the meaning of the string belongs to the monoidal category generated by the lexicon This is a straightforward consequence of the characterisation of morphisms as normal graphs in the free category, [START_REF] Preller | Free compact 2-categories[END_REF] and the interchange law.

We map lexical morphisms to "unfinished" RDF triples

The Translation

Let B = {0, 1}, +, • be the commutative semiring in which the addition, +, is the lattice operator ∨ and the multiplication, •, the lattice operator ∧ on {0, 1}.

The semantic category which hosts the RDF graphs and our models of grammatical strings is the category C of finitely generated semimodules over B. It is compact closed satisfying A ℓ = A = A r for each object A.

Every object A has a'canonical' basis (e i ) i such that every element v ∈ A can be written uniquely as v = i α i e i , where α i ∈ B. We refer to elements ofA as vectors. Morphisms of C are maps that commute with addition and scalar multiplication.

We interpret RDF graphs as elements of a semimodule S determined by the RDF vocabulary, see below. The translation of grammatical strings is given by a functor from the monoidal category generated by the lexical morphisms into a category of morphisms with side effects mapping the decomposition of a grammatical string to a vector of S.

Let L be a set of labels that includes the property words and a 'very big', but finite number n 0 of labels blank i , for i = 1, . . . n 04 . The other elements of L are node names and property names of an RDF vocabulary.

Define N as the semimodule over the semi-ring B freely generated by L and denote e label the basis vector of N corresponding to label ∈ L. We present RDF triples by basis vectors of S = N ⊗ N ⊗ N and RDF graphs by sums of basis vectors of S, for instance e bob ⊗e hate ⊗e blank +e blank ⊗e rdf-type ⊗e cat . Hence, the vector sum models the union of RDF graphs.

We want to interpret the lexical morphisms such that they construct a triple when applied to an argument and add it to the vector of S already constructed. Composition of the category C alone is not powerful enough to achieve this. We define a new category C S in which the entity a big boat will be denoted by the pair (e blank-1 , e blank-1 ⊗e rdf:type ⊗e boat +e blank-1 ⊗e is-big ⊗e true ).

Therefore we switch from C to the monoidal category C S below in which arrows have two components. The first component creates the triple and the second component adds the new triple to the graph as a 'side effect'.

Definition 2 (Category with Side Effects). Let {a 1 , . . . , a m } and {b 1 , . . . , b n } be the basis5 of A and B. For any p ∈ C(A, S), q ∈ C(B, S), define p ⊗ + q ∈ C(A ⊗ B, S) as the unique linear map satisfying for an arbitrary basis vector

a i ⊗ b j of A ⊗ B (p ⊗ + q) : a i ⊗ b j → p(a i ) + q(b j ) .
The category C S of morphisms with side effects in S has:

• objects as in C • morphisms (f, p) : A → B where f ∈ C(A, B), Ker(f ) = {0} and p ∈ C(A, S).

• arrows 1 A = (1 A , 0). • an operation • defined by (f, p) • (h, q) = (f • h, p • h + q).
• an operation ⊗ on objects defined as in C • an operation ⊗ on arrows defined by (f, p) ⊗ (h, q) = (f ⊗ h, p ⊗ + q). Examples of morphisms in the first component are the symmetry σ :

N ⊗N → N ⊗N , π 1 , π 2 : N ⊗N → N defined by σ(a i ⊗ b j ) = b j ⊗ a i , π 1 (a i ⊗ b j ) = a i and π 2 (a i ⊗ b j ) = b j .
The new operation ⊗ + introduced above concerns the second component only. The morphism p ⊗ + q computes at a i ⊗ b j the union of the RDF graphs p(a i ) and q(b j ), computed separately by p and q.

As an illustration, consider the following morphisms of C S m john = (e john , 0), m blanki = (e blanki , 0)

The arrow of a proper noun consists of the node representing this entity, e.g. e john , paired with the empty graph represented by 0 ∈ S. Choosing the empty graph means that nothing is said about this node. A similar remark holds for determiners.

m big = (1 N , 1 N ⊗ e is-big ⊗ e true ), m boat = (1 N , 1 N ⊗ e rdf-type ⊗ e boat )
An adjective or a noun is a morphism that maps a node word to itself and to the empty slot in the corresponding triple. As a side effect, it adds this triple to the second component.

m own = (1 N ⊗ e own ⊗ 1 N , 1 N ⊗ e own ⊗ 1 N )
A transitive verb is a morphism that maps an ordered pair of nodes to a triple making the first the subject and the second the object of the triple.

Compose the morphisms

m big • m blanki = (e blanki , t 1 ) t 1 = (e blanki ⊗ e is-big ⊗ e true ) m boat • m big • m blanki = (e blanki , t 2 + t 1 ) t 2 = (e blanki ⊗ e rdf-type ⊗ e boat ) m own • (m john ⊗ (m boat • m big • m blanki )) = m own • (e john ⊗ e blanki , 0 + t 2 + t 1 ) = (e john ⊗ e own ⊗ e blanki , t 3 + t 2 + t 1 )
t 3 = e john ⊗ e own ⊗ e blanki (4) The effect of composition is to create a new triple on the left of the comma and to store it on the right.

Proposition 1. The category with side effects C S is a monoidal category.

The proof of this proposition is given in appendix A. Define a monoidal structure preserving functor F from the monoidal category generated by the lexical morphisms to

C S thus • F(s) = S = N ⊗ N ⊗ N • F(a) = N if a = s • F(1 a ) = F(i a b ) = F(j a b ) = 1 F (a), for all basic types a, b ∈ B • F(that : r → n) = (1, 0) • F(name : I → d) = (e name , 0) • F(determiner : I → d) = (e blanki , 0) • F(word : d r ⊗ n) = (1, 1 ⊗ e rdf-type ⊗ e word ) • F(word : d r ⊗ d) = (1, 1 ⊗ e is-word ⊗ e true ) • F(word n r ⊗s⊗o l ) = (1 ⊗ e word ⊗ 1, 1 ⊗ e word ⊗ 1) F(word nr ⊗r⊗o l ) = (π 1 , 1 ⊗ e word ⊗ 1) F(word n r ⊗ ôr ⊗r ) = (π 2 , (1 ⊗ e word ⊗ 1) • σ).
Note that the morphisms in the example above satisfy m word = F(word). Computation (4) shows that F maps the lexical meaning of john owns a big boat to the three RDF triples t 1 , t 2 , t 3 . Lemma 2. Let word 1 . . . word n be a statement with corresponding lexical entries word 1 : T 1 . . . word n : T n and r : T 1 . . . T n → s a reduction to the sentence type. Then second component of

F(r • (word 1 ⊗ . . . ⊗ word n )) = (f, t 1 + • • • + t m ) ∈ C S (I, S)
is a sum of RDF triples t i ∈ S, for k = 1, . . . , m. Moreover, t k has the form (3) determined by the property word word j k such that the unlabelled nodes are filled by node words of the statement.

The proof is given in Appendix B.

⊗ (boat • big • a)) = m own • (m john ⊗ (m boat • m big • m blanki )) .
The translation algorithm from text to RDF graphs starts with a parsing algorithm of pregroup grammars that chooses a type for each word of the statement and provides a reduction to the sentence type. Next it computes the decomposition of the formal meaning in the lexical category by 'yanking'. Finally it computes the RDF graph by applying the translation functor F to the decomposition. The parsing algorithm is cubic polynomial in the number of words. Decomposition is linear, because the number of links is proportional to the number of words. Finally, translation again is linear, because the sum of the number of property words and of the number of node words in the decomposition is proportional to the number of words.

C-Models and RDF Interpretations

In this section, we establish the connection between the extensional models of meaning of [START_REF] Preller | From Sentence to Concept: Predicate Logic and Quantum Logic in Compact Closed Categories[END_REF] with the RDF interpretations of [START_REF] Hayes | Rdf semantics[END_REF] via the translation F from natural language to RDF graphs. We show that a statement is true in an extensional model if and only if the RDF graph computed by F is true in the RDF interpretation associated to the extensional model.

Choose an object U of C, the 'universe of discourse' of the fragment of natural language. The basis vectors of U stand for individuals or concepts. Properties are represented by maps that test (n-tuples of) entities and let (part of) them pass if the test succeeds.

Let

1 ≤ i 1 < • • • < i m ≤ n be a strictly increasing sequence. A linear map q : U 1 ⊗ . . . ⊗ U n → U i1 ⊗ . . . ⊗ U im is said to be a selector if for any basis vector e 1 ⊗ . . . ⊗ e n ∈ U 1 ⊗ . . . ⊗ U n q(e 1 ⊗. . .⊗e n ) = e i1 ⊗. . .⊗e im or q(e 1 ⊗. . .⊗e n ) = 0 .
We say that q selects the i-th factor if m = 1 and i = i 1 and that it is a projector if m = n. If the latter is the case then q is an idempotent and self-adjoint endomorphism, hence a 'property' in quantum logic, see [START_REF] Abramsky | A categorical semantics of quantum protocols[END_REF].

If the domain V and the codomain U i1 ⊗. . .⊗U im are fixed then the selectors are in a one-to-one correspondence with the 'truth-value' functions p : A → {⊤, ⊥} on the set A of basis vectors of V related by the condition p(a) = ⊤ if and only if q(a) = 0 for all a ∈ A . Let v = k l=1 a j l be an arbitrary vector of V . A selector q : V → W is said to be true at v if q(a j l ) = 0, for l = 1, . . . , k.

Lemma 3. Selectors are closed under composition and the tensor product. Every identity is a selector.

Let p : V → W and q : W → X be selectors and v ∈ V . Then q • p is true at v if and only if p is true at v and q is true at w = p(v).

Proof. The first assertion is straight forward. To show the second, assume that a is a basis vector for which (q • p)(a) = 0. Then p(a) = 0 because q(0) = 0. Hence p(a) is a basis vector of W selected by p. The property now follows for an arbitrary vector from the definitions. The last condition guarantees that the interpretations of a transitive verb in a statement and in a relative clause are equal if the relative pronoun is the subject of the verb in the relative clause and that they differ only by the symmetry isomorphism if the relative pronoun is the object. Definition 5. A statement word 1 . . . word n with reduction r :

Definition 4. A compact closed structure preserving functor M : L B → C is a C-model with universe U if it satisfies • M(s) = U ⊗ U • M(a) = U for any basic type a = s • M(1 a ) = M(i a b ) = M(j a b ) = 1 M (a)
T 1 ⊗ . . . ⊗ T n → b is true in M if M(r • (word 1 ⊗ . . . ⊗ word n ) = 0.
Truth in a model can be reformulated in terms of selector truth with the help of Lemma 1.

Lemma 4. Let p 1 •• • •• p m ′ •(word i1 ⊗. . .⊗word im )
be the decomposition of the formal meaning of the statement word 1 . . . word n . Then M(p l ) is a selector and M(word i k ) a vector in U, for

1 ≤ l ≤ m ′ , 1 ≤ k ≤ m.
Moreover, the statement is true in M if and only if the selector M(p l ) is true at

M(p l+1 )•• • ••M(p m ′ )• (M(word i1 ) ⊗ . . . ⊗ M(word im )) for 1 ≤ l ≤ m ′ .
Proof. M maps the meaning of the string to M(p 1 ) •

• • • • M(p m ′ ) • (M(word i1 ) ⊗ . . . ⊗ M(word im )).
Note that for k = 1, . . . , m ′ any factor of M(p k ) is the identity of U unless it is M(word j k ) = q k . Thus M(p k ) is a tensor product of selectors. Hence both assertions follow from Lemma 3.

Conclusion

The modelling of natural language by RDF graphs remains limited by the expressivity of RDF. The latter goes way beyond the few examples presented here. So far, we have only considered the extensional branch of a word. Future work will take advantage of the distinction between a property and its extension in RDF to introduce the conceptual branch of a plural, which does not refer to an extension, e.g Tom likes books, or capture the difference between Eve and John own a boat and Eve and John are athletes.

A Proof of proposition 1

Note first that composition and the tensor are well defined.

Assume (f, p) : A → B and (g, q) : B → C. Then q • f : A → S and p : A → S, so q

• f + p : A → S. Therefore (g, q) • (f, p) = (g • f, q • f + p) : A → C is well defined. Similarly, if (f i , p i ) : A i → B i for i = 1, 2 then p 1 ⊗ + p 2 : A 1 ⊗ A 2 → S and therefore (f 1 ⊗ f 2 , p 1 ⊗ + p 2 ) : A 1 ⊗ A 2 → B 1 ⊗ B 2 as required.
The operation ⊗ + is associative on arrows. Indeed, let (a i ) i , (b j ) j and (c k ) k be the basis of A, B and C, respectively. Then for r :

C → S ((p ⊗ + q) ⊗ + r)(a i ⊗ b j ⊗ c k ) =(p ⊗ + q)(a i ⊗ b j ) + r(c k ) =p(a i ) + q(b j ) + r(c k ) =(p ⊗ + (q ⊗ + r))(a i ⊗ b j ⊗ c k ) .
Hence the tensor product on arrows of C S is associative.

To show the interchange law (1), we need a lemma:

Lemma 5. Let p : C → S, q : D → S and f : A → C, g : B → D with Ker(f ) = Ker(g) = {0}. Then (p ⊗ + q) • (f ⊗ g) = (p • f ) ⊗ + (q • g) Indeed, let (a i ) i , (b i ) i , (c i ) i and (d i )
i be the bases of A, B, C and D respectively. For all i and j, we decompose f (a i ) on the basis (c i ) i and similarly for g(b j ) on (d i ) i :

f (a i ) = r c ir and g(b j ) = s d is Each family of indices (i r ) r and (i s ) s is non empty, because Ker(f ) = Ker(g) = {0}. Hence, ((p ⊗ + q) • (f ⊗ g))(a i ⊗ b j ) =(p ⊗ + q)(f (a i ) ⊗ g(b j )) =(p ⊗ + q)(( r c ir ) ⊗ ( s d is )) =(p ⊗ + q)( r,s c ir ⊗ d is ) = r,s p(c ir ) + q(d is ) = r p(c ir ) + s q(d is ) =p(f (a i )) + q(g(b j )) =((p • f ) ⊗ + (q • g))(a i ⊗ b j )
The fifth equality uses the fact that 1 + 1 = 1 and the sum is non empty. This terminates the proof of the lemma. Now let (f 1 , p 1 ) : A → C, (f 2 , p 2 ) : C → E, (g 1 , q 1 ) : B → D and (g 2 , q 2 ) : D → F . We show first the following equality (f 1 ⊗ + g 1 )+(f 2 ⊗ + g 2 ) = (f 1 +f 2 )⊗ + (g 1 +g 2 ). (5) Indeed, let (e i ) i and (f j ) j be the bases of A and B respectively. Then, for all i and j, ((f 1 ⊗ + g 1 ) + (f 2 ⊗ + g 2 ))(e i ⊗ f j ) =(f 1 ⊗ + g 1 )(e i ⊗ f j ) + (f 2 ⊗ + g 2 )(e i ⊗ f j ) =f 1 (e i ) + g 1 (f j ) + f 2 (e i ) + g 2 (f j ) =((f 1 + f 2 ) ⊗ + (g 1 + g 2 ))(e i ⊗ f j ) .

Finally, the Interchange Law follows from (5) and the definitions thus ((f 2 , p 2 ) ⊗ (g 2 , q 2 )) • ((f 1 , p 1 ) ⊗ (g 1 , q 1 )) =(f 2 ⊗ g 2 , p 2 ⊗ + q 2 ) • (f 1 ⊗ g 1 , p 1 ⊗ + q 1 ) =((f 2 ⊗ g 2 ) • (f 1 ⊗ g 1 ), (p 2 ⊗ + q 2 ) • (f 1 ⊗ g 1 ) + (p 1 ⊗ + q 1 )) =((f 2 • f 1 ) ⊗ (g 2 • g 1 ), (p 2 • f 1 ) ⊗ + (q 2 • g 1 ) + (p 1 ⊗ + q 1 )) =((f 2 • f 1 ) ⊗ (g 2 • g 1 ), (p 2 • f 1 + p 1 ) ⊗ + (q 2 • g 1 + q 1 )) =(f 2 • f 1 , p 2 • f 1 + p 1 ) ⊗ (g 2 • g 1 , q 2 • g 1 + q 1 ) =((f 2 , p 2 ) • (f 1 , p 1 )) ⊗ ((g 2 , q 2 ) • (g 1 , q 1 )) .

Therefore C S is a monoidal category.

B Proof of Lemma 2 and Proposition 2

Proof. Note that both F and M map inequalities of basic types to identities, hence also any atom without a lexical morphism to an identity. Let word j l be the property word occurring in p l , say as the k l -th factor, q l = M(word j l ) and m l = F(word j l ), for l = 1, . . . , m. Then M(p l ) = 1 U ⊗ . . . q l . . . ⊗ 1 U and F(p l ) = 1 N ⊗ . . . m l . . . ⊗ 1 N . Suppose that e i is a basis vector of U and e nodei a basis vector of N satisfying e i = I(e nodei ).

Use induction on n-m-l and assume that e Let t l be the triple created when composing F(p l ) with (e node1 , 0) ⊗ . . . ⊗ (e noder l , 0). We want the show that t l is true in I M if and only if M(p l ) is true at e 1 ⊗ . . . ⊗ e r l .

Consider the case where word j l : d → d. The other cases are shown similarly. Recall that m l • (e node k l , 0) = (e node k l , t l ), with t l = e node k l ⊗ e is-wordj l ⊗ e true . Hence, F(p l )((e node1 , 0) ⊗ . . . ⊗ (e noder , 0)) = ((e node1 , 0) ⊗ . . . (e node k l , t l ) . . . ⊗ (e noder , 0) = (e node1 ⊗ . . . ⊗ e noder , t l ). Then t l is true in I M if and only if (I(e node k l ), true) ∈ IEXT(I(e is-word )) if and only if q l is true at e k l , by definition of I. On the other hand, 1 U ⊗. . . q l . . .⊗1 U is true at e 1 ⊗. . . e k l . . .⊗e r if and only if q l is true at e k l . If that is the case then M(p l )(e 1 ⊗ . . . e k l . . . ⊗ e r ) = e 1 ⊗ . . . e k l . . . ⊗ e r .

  lexical morphism RDF triple noun : d → n ? rdf : type noun adjective : d → d ? is-adjective true verb : a ⊗ b → c ? verb ? determiner : I → d blank ? ?, ? ? blank propername : I → n propername ? ?, ? ? propername (3) The question marks designate unoccupied positions in the triple. Node words occupy either the subject or the object position, unary property words leave only the subject position open, binary property words occupy the centre position leaving subject and object position unoccupied.

  , for all basic types a, b ∈ B • M(that) = 1 U • M(word a r ⊗b ) and M(word n r ⊗s⊗o l ) are projectors • M(word n⊗r⊗o l ) = π 1 • M(word n r ⊗s⊗o l ) M(word n r ⊗ ôr ⊗r ) = π 2 • M(word n r ⊗s⊗o l ) • σ • M maps determiners and proper names in the singular to basis vectors.

  Any C-model M defines an RDF interpretation. The vocabulary consists of the basis vectors e label ∈ N , see previous section. The set of property symbols is given by P = {e is-adjective : adjective ∈ D} ∪ {rdf:type} ∪ {e verb : verb ∈ C} Define an RDF interpretation I M as follows • set of properties IP = {M(word) : e word ∈ P } ∪ {rdf:type} • set of resources IR = IP ∪ U ∪ {true} • mapping IS and its extension to blank nodes IS(e word ) = M(word), IS(e blanki ) = M(determiner i ) • mapping IEXT from IP into the power set of IR × IR IEXT(rdf-type) = { e, M(noun) : M(noun) is true at e} IEXT(IS(e is-adjective )) = { e, true : M(adjective) is true at e} IEXT(IS(e verb )) = { e 1 , e 2 : M(verb) is true at e 1 ⊗ e 2 } . Proposition 2. A statement word 1 . . . word n is true in a C-model M if and only if every triple of its RDF translation G is true in the RDF interpretation I M The proof is given in Appendix B.

  1 ⊗. . .⊗ e r l = M(p l+1 )•• • ••M(p n-m )•(M(node i1 )⊗. . .⊗ M(node im )) and (e node1 , 0) ⊗ . . . ⊗ (e noder l , 0) = F(p l+1 ) • • • • • F(p n-m ) • (F(e nodei 1 ) ⊗ . . . ⊗ F(e nodei m )). 6

  Definition 3 (RDF Translation). The RDF graph translating the statement word 1 . . . word n with reduction r : T1 . . . T n → s is the second component of F(r • (word 1 ⊗ . . . ⊗ word n )).For instance, the translation of the statement John owns a big boat is e john ⊗ e own ⊗ e blanki + e blanki ⊗ e is-big ⊗ e true + e blanki ⊗ e rdf-type ⊗ e boat ,

	because
	F(own • (john

Strictly speaking, these equalities hold up to natural isomorphisms, but the coherence theorem of[START_REF] Mac | Categories for the Working Mathematician[END_REF]) makes it possible to replace the isomorphisms by equalities without loss of generality.

"basic type" is replaced by simple type with an even iterator in the general case

it suffices that n0 exceeds the number of occurrences of determiners in the set of digitalised documents

In C, every object has a unique basis: the canonical one.

0 can be replaced by any vector of S without changing the proof.

Acknowledgments

This work was supported by the École Normale Supérieure and the LIRMM. The first author wishes to thank David Naccache, Alain Lecomte, Antoine Amarilli, Hugo Vanneuville and both authors the members of the TEXTE group at the LIRMM for their interest in the project.


Gérard Scorletti
email: gerard.scorletti@ec-lyon.fr

Vincent Fromion
email: vincent.fromion@jouy.inra.fr

Safta De Hillerin

Toward nonlinear tracking and rejection using LPV control

Quasi) LPV control methods and more generally L2 gain control methods, referred to as nonlinear H∞ control methods, are usually applied in order to ensure reference tracking and disturbance rejection. In this paper, we exhibit a counterexample that reveals that these specifications can not be ensured by these methods. We then propose a new LPV based approach in order to a priori ensure these specifications by combining the LPV method with the incremental L2 gain analysis of nonlinear performance. Its benefit is illustrated on the counterexample.

I. INTRODUCTION

The gain-scheduling approach is a very classical and widespread but heuristic nonlinear control approach. The underlying idea of the traditional gain-scheduling is to design at one or more operating points LTI controllers using the linearised plant models associated to the operating points. The nonlinear control law is then obtained by interpolating (or scheduling) these LTI controllers as a function of the operating point, see e.g. [START_REF] Rugh | Research on gain scheduling[END_REF] and references therein. Despite its widespread application, up to recently, there was no systematic gain scheduling controller design method which a priori ensures the desired specifications to the closed loop system.

Two alternative approaches to the traditional gainscheduling approach are based on the Linear Parameter-Varying (LPV) methods. They were initially developed as an extension of the H ∞ control problem to the case of LPV systems [START_REF] Shamma | Gain scheduling: potential hazards and possible remedies[END_REF]- [START_REF] Apkarian | A convex characterization of gainscheduled H∞ controllers[END_REF], that is, linear systems whose state space matrices depend on time-varying parameters. The first alternative is actually an improvement of the traditional gain-scheduling approach where the operating points Linear Time Invariant (LTI) controllers are computed in one shot as a single LPV controller. As pointed out in [START_REF] Rugh | Research on gain scheduling[END_REF], "Typically, stability can be assured only locally and in a "slowvariation" setting, and typically there are no performance guarantees." In order to offer the potential of both stability and performance guarantees [START_REF] Rugh | Research on gain scheduling[END_REF], an alternative approach, referred to as the quasi-LPV approach, proposes to embed the nonlinear system in an LPV model. An LPV controller can be computed using convex optimization involving Linear Matrix Inequality (LMI) constraints in order to ensure an upper bound on the L 2 gain of the closed loop system, see [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF]- [START_REF] Scherer | LPV control and full block multipliers[END_REF] to cite a few. A nonlinear controller is then deduced Since the quasi-LPV approach seems seducing, it was largely applied for tracking and disturbance rejection, see e.g. [START_REF] Wu | Gain-scheduling control of LFT systems using parameter-dependent Lyapunov functions[END_REF], [START_REF] Vermeiren | Modeling, control and experimental verification on a twowheeled vehicle with free inclination: An urban transportations system[END_REF]. Nevertheless, beyond the L 2 gain performance, it is necessary to question the actual rigorous guarantees on usual tracking and rejection specifications. In section III, we reveal that there are none using an illustrative example. This example emphasizes that in contrast with the LTI closed loop systems, it is not possible to ensure tracking and rejection specifications using the L 2 gain stability and performance concepts, in the case of LPV or nonlinear closed loop systems. For these usual specifications, the quasi-LPV approach does not actually introduce performance guarantees, as in the gain-scheduling one.

The theoretical explanation of this fact can be found in our longstanding investigations on the nonlinear system performance, see e.g. [START_REF] Fromion | Asymptotic properties of incrementally stable systems[END_REF]- [START_REF] Fromion | The weighted incremental norm approach: from linear to nonlinear H∞ control[END_REF]. As the final objective is to ensure tracking and rejection specifications for a nonlinear system, the question is how to test these specifications on a nonlinear system. In [START_REF] Fromion | Nonlinear performance of a PI controlled missile: an explanation[END_REF], [START_REF] Fromion | The weighted incremental norm approach: from linear to nonlinear H∞ control[END_REF], it was pointed out that L 2 gain stability and performance fail to ensure these specifications. It was proved that these properties can be ensured by incremental L 2 gain stability and performance. As the two LPV approaches fail to ensure incremental L 2 gain stability and performance, in Section IV, we pave the way to a third LPV approach whose objective is to compute a nonlinear controller which ensures incremental L 2 gain stability and performance and which is able to ensure tracking and rejection specifications. A first solution in a special but important case is discussed here based on the results proposed in [START_REF] De Hillerin | Reduced-Complexity Controllers for LPV Systems: Towards Incremental Synthesis[END_REF].

Notations The identity matrix of R n×n is denoted In and the zero matrix of R n×m is denoted 0n×m. The subscripts are omitted when obvious from the context. For two operators A and B, diag(A, B) denotes the operator A 0 0 B . For a full-rank matrix U , U ⊥ denotes an orthogonal complement of U , i.e., U U ⊥ = 0 and U T U ⊥ is of maximal rank. For a square matrix M , M > 0 and M ≥ 0 mean respectively positive and semi-positive definiteness. The symbol ⋆ denotes the Redheffer star product [START_REF] Zhou | Robust and Optimal Control[END_REF]. L2 is the space of R n square integrable valued functions defined on R, where the norm is defined by f 2 = ( f (t) 2 dt) 1/2 . The causal truncation PT f is defined by PT f (t) = f (t) for t ≤ T and 0 otherwise. The extended space L e 2 is the space of R n valued functions defined on R whose causal truncations belong to L2. For a system G, respectively the H∞ norm, the L2 gain and the incremental norm, if they exist, are respectively denoted ||G||∞, ||G||i,2 and ||G||∆, see e.g. [START_REF] Fromion | Nonlinear performance of a PI controlled missile: an explanation[END_REF], [START_REF] Fromion | The weighted incremental norm approach: from linear to nonlinear H∞ control[END_REF].

II. ILLUSTRATIVE CONTROL PROBLEM

Let G N L be the nonlinear plant defined by y

= G N L (u) with          ẋ1 (t) = -100ϕ(x 1 (t)) -70x 2 (t) + 300u(t) ẋ2 (t) = 70x 1 (t) -14x 2 (t) y(t) = x 1 (t) x(t 0 ) = x 0 (1)
where x(t) ∈ R n is the state vector, u(t) ∈ R nu the input, y(t) ∈ R ny the measured to-be-controlled output. The function ϕ is defined as follows (see Fig. 1):

   for |x| < 2 ϕ(x) = 0.9x 3 -2|x|x + 1.2x for x ≥ 2 ϕ(x) = 2x -2.4 for x ≥ 2 ϕ(x) = 2x + 2.4
The purpose is to design an output feedback controller u = K N L (r, y) where r denotes the reference signal such that the closed loop system satisfies the following specifications:

• tracking of step reference with a null static error and a response time less than 0.1 s; • rejection of step disturbance at the plant input;

• limited control energy that is, typical control specifications.

III. NONLINEAR CONTROL USING LPV CONTROL A. LPV control

A Linear Parameter Varying (LPV) system is a linear system whose dynamics (e.g. defined by a state space representation) depend on time-varying exogenous parameters whose trajectories are a priori unknown [START_REF] Shamma | Gain scheduling: potential hazards and possible remedies[END_REF]- [START_REF] Apkarian | A convex characterization of gainscheduled H∞ controllers[END_REF], [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related problems[END_REF]. Nevertheless, some information is available, in particular the intervals the parameters belong to. An LPV system G LP V is usually defined by state-space equations of the form:

y = G LP V (u) with      ẋ(t) = A(θ(t))x(t) + B(θ(t))w(t) y(t) = C(θ(t))x(t) + D(θ(t))w(t) x(0) = x 0 , ∃θ(.) ∈ Θ (2)
where

• x(t) ∈ R n is the state vector, w(t) ∈ R nw the input, z(t) ∈ R nz the output; • Θ is a set of measurable functions from R + to R r such that for all θ(.) ∈ Θ, for all t ≥ 0, θ(t) belongs to an hyper rectangle Θ t , usually

[-1, 1] × • • • × [-1, 1]; • the matrix function A(θ) B(θ) C(θ) D(θ) (3)
is a continuous matrix of real rational functions defined from Θ t to R (n+nz)×(n+nw) . The different LPV control problem can be expressed as follows. Given the (generalized) plant P LP V defined as:

z y = P LP V w u with      ẋ(t) = A(θ(t))x(t) + Bw(θ(t))w(t) + Bu(θ(t))u(t) z(t) = Cz(θ(t))x(t) + Dzw(θ(t))w(t) + Dzu(θ(t))u(t) y(t) = Cy(θ(t))x(t) + Dyw(θ(t))w(t) x(0) = x0 ∃θ(.) ∈ Θ , (4)
where x(t) ∈ R n is the state vector, u(t) ∈ R nu the command input, y(t) ∈ R ny the measured output, z(t) ∈ R nz the controlled output, w(t) ∈ R nw the disturbance input, with the time-varying exogenous parameters θ(t) measured on-line, find an controller defined as:

u = K LP V (y) with      ẋK (t) = A K (θ(t))x K (t) + B K (θ(t))y(t) u(t) = C K (θ(t))x K (t) + D K (θ(t))y(t) x K (0) = 0 (5)
where x K ∈ R n , such that the closed-loop LPV system P LP V ⋆ K LP V is asymptotically stable (for null input) with an L 2 gain less than a given γ. In the case when the rate of variation of θ(t) is possibly unbounded, solutions to the LPV control problem are based on a (parameter independent) quadratic Lyapunov function V (x) = x T P x, see e.g. [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF], [START_REF] Apkarian | A convex characterization of gainscheduled H∞ controllers[END_REF], [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related problems[END_REF], [START_REF] Scherer | LPV control and full block multipliers[END_REF], [START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parameter uncertainty[END_REF]. The solutions usually rely on convex optimization involving Linear Matrix Inequality constraints, an important class of problems for which there exist efficient solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], [START_REF] Ben-Tal | Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, ser. Series on Optimization[END_REF].

B. Application of the LPV approach to nonlinear control

Roughly speaking, from a technical point of view, the LPV L 2 gain control problem was developed as an extension of the LTI H ∞ control problem to LPV systems. On the other hand, a strong motivation for the introduction of LPV systems was the development of computationally efficient methods for the control of nonlinear systems. Different approaches exist. In the most popular approach, referred to as quasi-LPV, the purpose is to compute a non linear controller K N L such that the closed loop system P N L ⋆ K N L has an L 2 gain less than a given γ where the (augmented) nonlinear

plant z y = P N L w u is defined by the state-space representation:            ẋ(t) = f (x(t), w(t), u(t)) z(t) = g(x(t), w(t), u(t)) y(t) = h(x(t), w(t)) x(0) = x 0 (6)
where

x(t) ∈ R n is the state vector, u(t) ∈ R nu the command input, y(t) ∈ R ny the measured output, z(t) ∈ R nz the controlled output, w(t) ∈ R nw the disturbance input.
The functions f and h are assumed uniformly Lipschitz and C 1 and such that f (x 0 , 0) = 0 and h(x 0 , 0) = 0. When P N L and K N L are LTI, this problem reduces to the H ∞ control problem.

To this purpose, an LPV augmented plant is obtained from the nonlinear plant by including the nonlinear terms in newly defined time-varying parameters. It is an embedding approach, that is, the LPV system is selected such that the trajectories of the nonlinear system are trajectories of the LPV system, that is, with the following sets defined on L 2 ,

Ω N L = x z y w u | (6) is satisfied and Ω LP V = x z y w u | (4) is satisfied , we have Ω N L ⊂ Ω LP V . (7)
In this approach, the components of the θ(t) can usually be computed from the measurement of components of the state space vector, see e.g. [START_REF] Ghaoui | Control of rational systems using Linear-Fractional Representations and Linear Matrix Inequalities[END_REF]. As a consequence, the objective of the quasi-LPV approach is to compute a nonlinear controller on the LPV model using a quadratic Lyapunov function, in order to ensure the stability of the closed loop nonlinear system and performance evaluated by an upper bound on the L 2 gain of the closed loop system.

C. Application to the illustrative control problem

The quasi LPV approach is applied to the illustrative control problem introduced section II using a quite typical approach. To this purpose, an augmented plant presented Fig. 2 which corresponds to the four-block L 2 gain criterion is proposed. This is the nonlinear counterpart of the usual four-block H ∞ criterion [START_REF] Skogestad | Multivariable feedback control analysis and design[END_REF], with a small modification in order to introduce a pure integrator in the controller. It is a well-known fact that for LTI closed loop systems, integral control ensures the tracking of step reference signals and the rejection of step disturbances, with a null static error.

The first step is to derive an LPV model G LP V associated to the plant G N L defined by [START_REF] Rugh | Research on gain scheduling[END_REF] such that the inclusion (7) holds. A simple LPV model is straightforwardly defined as:

y = G LP V (u) with      ẋ(t) = A G (θ(t))x(t) + 300 0 u(t) y(t) = 1 0 x(t) , θ(t) ∈ [0, 2]
where

θ(t) = ϕ(x 1 (t)) x 1 (t) (8)
✲ K LP V W 1 ✻ ✲ ✲ ✲ z 1 (t) ✲ ✻ ✲ ✲ r(t) y(t) u(t) ǫ(t) - + ✻ W 2 ✲ z 2 (t) W 3 ❄ G LP V ✛ Fig. 2. Augmented plant corresponding to a modified four-block L 2 gain criterion and A G (θ(t)) = 0 -70 70 -14 + θ(t) -100 0 0 0 (9)
Since the measured signal y is equal to x 1 , the parameter θ can be computed on-line using [START_REF] Wu | Gain-scheduling control of LFT systems using parameter-dependent Lyapunov functions[END_REF]. The augmented LPV plant P LP V is then obtained from Fig. 2 where the weighting functions are defined by

• W 1 = 50 in order to ensure a time response of the reference tracking less than 0.1 s; • W 3 = 0.1 in order to ensure a time response of the disturbance rejection larger than for the reference tracking one;

• W 2 (s) = 0.1

s 10 + 1 s 1000 + 1
in order to limit the controller bandwith. The weighting functions are selected following the usual H ∞ approach as in [START_REF] Apkarian | A convex characterization of gainscheduled H∞ controllers[END_REF]. The LPV controller is then computed using the approach presented in [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related problems[END_REF] in order to minimize an upper bound γ on the L 2 gain of the augmented closed loop system P LP V ⋆ K LP V . The obtained value of γ is 0.89. In order to evaluate the possible conservatism of this upper bound, θ(t) is set to a constant θ i ∈ [0, 2]: in this case, the augmented closed loop system is LTI and its L 2 gain boils down to the H ∞ norm which can be easily computed. By computing the maximum value of the H ∞ norm when θ i goes from 0 to 2 by step of 0.01, a lower bound on the actual L 2 gain is obtained: 0.85. The true value of the L 2 gain is between 0.85 and 0.89: in this example, the use of a quadratic Lyapunov function is not very conservative. Let us now evaluate the time domain performance with the following scenario: (i) tracking of square periodic reference whose mean value is 0.4 with amplitude 0.35 and frequency 0.1 Hz, see Fig. 3; (ii) rejection of a step disturbance of amplitude 1.75 at time 4s.

The first simulation is performed on P LP V ⋆ K LP V with θ(t) which is set to a constant, 0 (see Fig. 4) and 2 (see Fig. 5). In both cases, the closed loop system is LTI. We observe that the specifications are satisfied. Now, let us simulate the LPV closed loop system with θ(t) ∈ [0, 2] defined Fig. 6 (top): the output y is represented Fig. 6 (middle and bottom). Note that even if the reference input is constant, the output is oscillating and the oscillations are not damped. This result seems to be quite surprising since the closed loop system is (L 2 gain) stable. It reveals that, in contrast with the case of LTI systems, in the case of Linear Time Varying (LTV) systems, (exponential or asymptotic) stability does not ensure that for constant inputs, the internal signals and the output signals of the system tend to a constant. As a consequence, even if the closed loop system is (L 2 gain) stable, integral control is unable to ensure the tracking of step reference with a null static error. Nevertheless, if the control specifications has not satisfied, it does not mean that the closed loop system behaviour does not have nice properties. Due to asymptotic stability, with the same time function θ, zero inputs but different initial conditions, the system output converges to the same steady state, see Fig. 7.

From the LPV controller K LP V , a nonlinear controller is obtained by replacing θ(t) by ϕ(y(t)) y(t) in ( 5), thanks to [START_REF] Wu | Gain-scheduling control of LFT systems using parameter-dependent Lyapunov functions[END_REF]. Let us simulate the behavior of this controller on the nonlinear plant G N L defined by [START_REF] Rugh | Research on gain scheduling[END_REF], see Fig. 8. Note that if the disturbance is rejected, the tracking of step references depends strongly on the actual value of the step. If the tracking is satisfying for reference close to 0, it is no longer true for constant input close to 0.75. This fact is in accordance with the result proved, in [START_REF] Van Der Schaft | L 2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control[END_REF], for the zero input, the L 2 gain control solution reduces to the H ∞ control solution. As in the LTV case, in the case of nonlinear systems, in contrast with the case of LTI systems, even if the closed loop system is (L 2 gain) stable, integral control is unable to ensure the tracking of step reference with a null static error. The behaviour is actually worse than it seems. Actually, the linearisations of the closed-loop system defined by constant inputs could be not unique with some of them unstable. This is another drawback of the general L 2 gain approach. For L 2 gain stable systems, the stability of the linearisations defined by constant input is not sufficient for ensuring the good behaviour of the system: resonance due to jumps can occur, see [START_REF] Fromion | Multipliers and continuity of the input/output map[END_REF].

IV. NONLINEAR INCREMENTAL CONTROL USING LPV

In [START_REF] Fromion | Nonlinear performance of a PI controlled missile: an explanation[END_REF], [START_REF] Fromion | The weighted incremental norm approach: from linear to nonlinear H∞ control[END_REF], it was revealed that reference tracking and disturbance rejections can be achieved for a nonlinear closedloop system if the nonlinear controller is computed such that a suitable augmented plant ( 6) is incrementally L 2 stable with its incremental L 2 gain less than 1.

In order to compute such a controller, we propose a new approach, based on the LPV control. One of the key ingredients of this approach is the generalized version of the mean value theorem [START_REF] Fromion | A theoritical framework for gain scheduling[END_REF] which is now recalled.

Theorem 4.1: Let G be a dynamical system defined from L e 2 into L e 2 and let us assume that its Gâteaux derivative (time varying linearisation), D G[u 0 ] exists for any u 0 belonging to L e 2 . Then

G ∆ ≤ η if and only if D G[u 0 ] i2 ≤ η for any u 0 belonging to L 2 .
This theorem is applied as follows. For any input u 0 belonging to L e 2 , the Gâteaux derivative of P N L at u 0 exists and is defined by

z y = DP N L [w 0 , u 0 ] w u with        ẋ(t) = A(t)x(t) + B w (t)w(t) + B u (t)u(t) z(t) = C z (t)x(t) + D zw (t)w(t) + D zu (t)u(t) y(t) = C y (t)x(t) + D yw (t)w(t) + D yu (t)u(t) x(0) = 0, (10)
with

A(t) = ∂f ∂x (x 0 (t), w 0 (t), u 0 (t)), • • • D yu (t) = ∂h ∂u
(x 0 (t), w 0 (t), u 0 (t)) where x 0 (t) is the solution of ( 6)

under inputs w 0 (t), u 0 (t) and the initial condition x(0) = x 0 .

In order to compute a nonlinear controller K N L such that the closed loop system P N L ⋆ K N L has an incremental L 2 gain less than a given γ where P N L is defined by ( 6), we first compute the time varying linearisations (Gâteaux derivative) DK N L [y 0 ] of the nonlinear controller K N L such that the closed loop system DP N L [u 0 , w 0 ] ⋆ DK N L [y 0 ] has an L 2 gain less than a given γ for any w 0 ∈ L e 2 . The second step is to compute K N L such that DK N L [y 0 ] are the time varying linearisation of K N L at y 0 .

We propose to realize the first step (computation of DK N L [y 0 ] for any y 0 ) by application of the LPV control method. To this purpose, an LPV (augmented) plant ( 4) is obtained such that the trajectories of the time varying linearisations [START_REF] Fromion | Asymptotic properties of incrementally stable systems[END_REF] of the nonlinear plant ( 6) are trajectories of the LPV system, that is, with the sets of L 2 Ω DN L =

x z y w u | (10) is satisfied and

Ω LP V = x z y w u | (4) is satisfied , we have DN L ⊂ Ω LP V . ( 11
)
The LPV controller is then computed such that the L 2 gain of the closed loop LPV system is less than γ. Since the obtained LPV controller has to correspond to the linearisations of the nonlinear controller, the question is how to enforce this property during the synthesis, that is, how to compute the LPV controller such that there exists a nonlinear controller which can be obtained such that the trajectories of its time-varying linearisations are trajectories of the LPV controller. To our best knowledge, this problem is still open in the general case. In the sequel, we nevertheless exhibit a special class of nonlinear control problems, referred to as "filtered cancellation control" problems for which this problem can be solved.

A. Filtered cancellation control problem

Let us consider the particular class of (generalized) nonlinear plants [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related problems[END_REF] denoted in the sequel P N L and defined by the state-space representation:

       ẋ(t) = Ax(t) + B1w(t) + B2u(t) + f (x(t)) z(t) = C1x(t) + D11w(t) + D12u(t) + g(x(t)) y(t) = C2x(t) + D21w(t) + D22u(t) + h(x(t)) x(0) = x0 (12) with   f (x(t)) g(x(t)) h(x(t))   =   B 0 D 10 D 20   p(t) (13)
where

• x(t) ∈ R n is the state vector, u(t) ∈ R nu the command input, y(t) ∈ R ny the measured output, z(t) ∈ R nz the controlled output, w(t) ∈ R nw the disturbance input; • p(t)
is either measured on-line or constructed when the components of x(t), w(t) and u(t) necessary for the computation of p(t) are measured, that is, there exists a function β such that p(t) = β(x(t), w(t), u(t)). The nonlinear controller will be defined as follows:

u = K p y p with    ẋK (t) = A K x K (t) + B Ky y(t) + B Kp p(t) u(t) = C K x K (t) + D Ky y(t) + D Kp p(t) x K (0) = x 0 (14)
In the case when p(t) is computed from the measure of components of x(t), w(t) and u(t) a nonlinear state space representation is obtained:

u = K xwu     y x w     with    ẋK (t) = A K x K (t) + B Ky y(t) + B Kp β(x(t), w(t), u(t)) u(t) = C K x K (t) + D Ky y(t) + D Kp β(x(t), w(t), u(t)) x(0) = x 0 (15)
The incremental gain control problem Given γ > 0 and the generalized plant P N L defined by ( 6), find a controller of the form [START_REF] Zhou | Robust and Optimal Control[END_REF] or [START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parameter uncertainty[END_REF] such that the closed loop system has an incremental gain less than γ.

In the next subsection, a solution to this problem is proposed.

B. LPV filtered cancellation control

When the matrices of the state space representation of the (generalized) LPV plant (4) are rational functions of θ(t), the latter can be modeled by a Linear Fractional Transformation (LFT) on a parameter block diagonal structure [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related problems[END_REF]:

               ẋ(t) z(t) y(t)   =   A B0 B1 B2 C1 D10 D11 D12 C2 D20 D21 0      x(t) p(t) w(t) u(t)    p(t) = Θ(t) (I -D00Θ(t)) -1 (C0x(t) + D01w(t) + D02u(t)) (16)
where the parameter block Θ(t) is defined as Θ(t) = diag(θ 1 (t)I m1 , • • • , θ r (t)I mr ). We now assume that the signal p(t) is measured on-line, instead of the parameters θ(t). The problem of the design of a cancellation controller for an LPV (generalized) system can be defined as the following.

The LPV filtered cancellation control problem Given γ > 0 and the LPV generalized plant ( 16), find a filtered cancellation controller u = K cancel (y, p) of the form:

     ẋK (t) = A K x k (t) + B Ky y(t) + B Kp p(t) u(t) = C K x K (t) + D Kuy y(t) + D Kup p(t)
x K (0) = 0 [START_REF] Ben-Tal | Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, ser. Series on Optimization[END_REF] such that the closed loop system is asymptotically stable (for null input) with an L 2 gain less than γ.

In the papers [START_REF] Apkarian | A convex characterization of gainscheduled H∞ controllers[END_REF], [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related problems[END_REF], convex optimization approaches, involving LMI constraints, were proposed in order to compute an LPV controller of the form (5) in the case when the generalized plant is defined by an LFT representation [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. The LPV controller has also an LFT representation with the same Θ(t) than in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. In [START_REF] De Hillerin | Reduced-Complexity Controllers for LPV Systems: Towards Incremental Synthesis[END_REF], we reveal that in the case when in addition to y(t), some components of the vector p(t) are measured on-line, the size of Θ(t) block of the controller can be reduced. If these results are applied to the case when all the components of p(t) are measured then the state space matrices of the "LPV" controller can be chosen independent of θ(t), that is, the controller has a state-space representation [START_REF] Ben-Tal | Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, ser. Series on Optimization[END_REF]. In this case, a similar result can be found in [START_REF] Wu | On convexified robust control synthesis[END_REF]. This result is very nice since, in the general case (that is, no component of p(t) is measured), if a reduced size is enforced for the Θ(t) block of the controller then a non convex constraint is introduced in the optimization problem.

Theorem 4.2: The LPV filtered cancellation control problem has a solution when

Θ t = [-1, 1] × • • • × [-1, 1] if there exist matrices P = P T , Q = Q T ∈ R n×n , T ∈ R k×k such that: N T        A T P + P A P B1 C T 0 C T 1 B T 1 P -γI D T 01 D T 11 C0 C1 D01 D11 - T 0 0 γI        N < 0 (18)
M T         AQ + QA T + • • • +B0T B T 0 + γ -1 B1B T 1 QC T 0 + D00T B T 0 + • • • + γ -1 D01B T 1 C0Q + B0T D T 00 + • • • + γ -1 B1D T 01 D00T D T 00 + +γ -1 D01D T 01 + • • • - T 0 0 γI         M < 0 (19)
P I I Q > 0 and T > 0. ( 20
)
with

N = C 2 D 21 ⊥ 0 0 I nz+k and M = B T 2 D T 02 ⊥ Proof:
The proof is a direct application of Theorem 3.2 presented in [START_REF] De Hillerin | Reduced-Complexity Controllers for LPV Systems: Towards Incremental Synthesis[END_REF].

For a given γ > 0, find P , Q, T such that ( 18), ( 19) and ( 20) is a (feasibility) convex optimization problem involving LMI constraints. If this optimization problem has a solution then the state space matrices of the cancellation controller can be computed, see [START_REF] De Hillerin | Reduced-Complexity Controllers for LPV Systems: Towards Incremental Synthesis[END_REF] for the details.

V. APPLICATION TO THE NONLINEAR PERFORMANCE CONTROL

A. A filtered cancellation control solution to the incremental gain control problem

The major interest of the cancellation controller is that it can be readily obtained from its time-varying linearizations. The time-varying linearization D P N L of ( 12) are defined by: 

                ẋ(t) = Ax(t) + B 1 w(t) + B 2 ū(t) + ∂ f ∂x (x 0 (t))x(t) z(t) = C 1 x(t) + D 11 w(t) + D 12 ū(t) + ∂ g ∂x (x 0 (t))x(t) ȳ(t) = C 2 x(t) + D 21 w(t) + D 22 ū(t) + ∂ h ∂x (x 0 (t))x(t)
x(0) = 0 (21) Let us introduce the vector p(t) such as

        ∂ f ∂x (x 0 (t)) ∂ g ∂x (x 0 (t)) ∂ h ∂x (x 0 (t))         =   B 0 D 10 D 20   p(t)
The solution is obtained in several steps. 1) Compute an LPV system for the time-varying linearization of the nonlinear generalized plant ( 21) such that for any p(t), there exists a θ ∈ Θ, matrices C 0 , D 00 , D 01 and D 02 such that ∀t p(t) is defined by

Θ(t) (I -D00Θ(t)) -1 (C0 x(t) + D01 w(t) + D02 ū(t)) .
2) Solve the following (feasibility) convex optimization problem involving LMI constraints: find P , Q, T such that ( 18), [START_REF] Skogestad | Multivariable feedback control analysis and design[END_REF] and [START_REF] Van Der Schaft | L 2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control[END_REF]. 3) If the problem is feasible, compute the state space matrices of the linearization of the controller [START_REF] Zhou | Robust and Optimal Control[END_REF] using [START_REF] De Hillerin | Reduced-Complexity Controllers for LPV Systems: Towards Incremental Synthesis[END_REF], that is:

ū = D K p y 0 p 0 ȳ p with    ẋK (t) = A K xK (t) + B Ky ȳ(t) + B Kp p(t) ū(t) = C K xK (t) + D Ky ȳ(t) + D Kp p(t) xK (0) = 0
The corresponding nonlinear controller is then given by: u

= K p y p with    ẋK (t) = A K x K (t) + B Ky y(t) + B Kp p(t) u(t) = C K x K (t) + D Ky y(t) + D Kp p(t) x K (0) = 0
where p(t) is such that [START_REF] De Hillerin | Reduced-Complexity Controllers for LPV Systems: Towards Incremental Synthesis[END_REF].

B. Application to the illustrative control problem

In order to apply the previous method, we first compute an LPV system which embeds the time-varying linearization The time-varying linearization of the nonlinear generalized plant ( 21) is then obtained from Fig. 9. The resolution of the LMI optimization problem allows to obtain a controller (that is, the matrices A K , B K , C K and D K ) for γ ≈ 1.

DG N L [u 0 ]: with A G defined by (9),      ẋ(t) = A G (θ(t))x(t) + 300 0 ū(t) ȳ(t) = 1 0 x(t) , θ(t) ∈ [-0.3, 4] . K LP V W 1 ✻ ✲ ✲ ✲ z 1 (t) ✲ ✻ ✲ ✲ r(t) y(t) u(t) . . ǫ(t) - + ✻ W 2 ✲ z 2 (t) W 3 ✲ ❄ ✛ G θ ✲ ✛ ✻ p(t)
The simulation is performed on the nonlinear plant using the nonlinear controller of the form [START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parameter uncertainty[END_REF], see Fig. 10. Note that the tracking and disturbance rejection specifications are satisfied. Since for any constant input, the internal signal of an incremental L 2 gain stable system goes to a constant [START_REF] Fromion | Asymptotic properties of incrementally stable systems[END_REF] when t goes to ∞, necessarily the input of the integrator Fig. 9 goes to zero when t goes to ∞, in order to ensure that the integrator output goes to a constant. The static error is then well-defined and equal to 0. The damping can be improved by computing a controller which uses y as an extra input, see Fig 11 and

VI. CONCLUSION

In this paper, we exhibit a counterexample which reveals that a control approach whose purpose is to ensure for a 12. Simulation of the non linear closed loop system with incremental controller, ouput y(t) (top), zoom on the output y(t) (bottom) nonlinear closed loop system its L 2 gain stability with the L 2 gain less than a given γ fails to ensure usual tracking and rejection specifications and that the quasi LPV approach does not ensure such specifications, even if they are largely applied for such specifications. We emphasize that in order to ensure tracking and rejection specifications when applying LPV control, it is necessary to associate to the LPV approach a nonlinear framework in order to test such specifications on a nonlinear system. This framework is the incremental gain approach introduced in [START_REF] Fromion | The weighted incremental norm approach: from linear to nonlinear H∞ control[END_REF]. We propose a new LPV approach in order to compute nonlinear controllers with a priori guarantees on the tracking and rejection specifications for a class of nonlinear systems. We point out that for the generalization of this approach, it is necessary to solve the "integrability problem", that is, the computation of a nonlinear controller from its time-varying linearisations. To our best knowledge, such a problem is still an open problem.

K LP V W 1 ✻ ✲ ✲ ✲ z 1 (t) ✲ ✻ ✲ ✲ r(t) y(t) u(t) . . ǫ(t) - + ✻ W 2 ✲ z 2 (t) W 3 ✲ ❄ ✛ G θ ✲ ✛ ✻ p(t) ✲

Fig. 1 .

 1 Fig. 1. Plot of the function ϕ

Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 3456 Fig. 3. Square periodic reference signal r

Fig. 7 .

 7 Fig. 7. Simulation of the LPV closed loop system with time varying θ(t), zero input and different initial conditions

Fig. 8 .

 8 Fig. 8. Simulation of the non linear closed loop system, ouput y(t) (top), zoom on the output (bottom)

Fig. 9 .

 9 Fig. 9. Augmented plant corresponding to a modified four-block L 2 gain criterion

  Fig 12.

Fig. 10 .

 10 Fig.10. Simulation of the non linear closed loop system with incremental controller 2, ouput y(t) (top), zoom on the output (bottom)

Fig. 11 .

 11 Fig. 11. Augmented plant corresponding to a modified four-block L 2 gain criterion