H Frankowska

G Moscariello

LONG TIME BEHAVIOUR OF SOLUTIONS TO AN EVOLUTION PDE WITH NONSTANDARD GROWTH

Keywords: Evolution PDE, time dependence of solutions at infinity, generalized Gronwall lemma. Mathematics Subject Classification. 35K51, 35K59, 35B40

In this paper we prove time estimates for solutions to a general nonhomogeneous parabolic problem whose operator satisfies nonstandard growth conditions. We also study the asymptotic behaviour of solutons to an anisotropic problem.

Introduction.

In this paper we investigate the time dependence of solutions for a wide class of nonuniformly parabolic equations whose model case is the following one:

(1.1)

   u t -div G (|∇u|) |∇u| ∇u = 0 in Ω T u(x, t) = 0 on ∂Ω × (0, T ) u(x, 0) = u 0 (x)
in Ω, where for some N ≥ 2, Ω is a bounded domain in R N with Lipschitz boundary ∂Ω, T is a positive number and Ω T = Ω × (0, T ). We assume that u 0 ∈ L 2 (Ω) and G : R + → R + is a C 1 convex function satisfying, for some k > 0 and p ≥ 2 the following conditions

(1.2)    (j) G(0) = 0; G(2s) ≤ kG(s) ∀s > 0 (jj) G(s)
s p is increasing on (0, ∞) We refer to [START_REF] Krasnosel | Convex functions and Orlicz spaces[END_REF][START_REF] Rao | Theory of Orlicz spaces[END_REF] for the properties of G that follow from assumptions (1.2). The functions listed below do satisfy the above assumptions for s ≥ 0 :

1) G(s) = s p p , p ≥ 2; 2) G(s) = s p log(1 + s), p ≥ 2; 3) G(s) = s p L k (s), p ≥ 2, L i (s) = log(1+L i-1 (s)), i = 1, ...., k, L 0 (s) = log(1+s) 4) G(s) = s 0 g(ρ)dρ where g : R + → R + is a C 1 function satisfying p -1 ≤ sg (s) g(s) ≤ q -1, ∀s > 0 with 2 ≤ p ≤ q < ∞ such that lim s→+∞ g(s) s = +∞ , see [START_REF] Baroni | The Cauchy-Dirichlet problem for a general class of parabolic equations[END_REF] where the properties of such a function G were investigated.

Date: December 3, 2019. If u ∈ L ∞ loc ((0, ∞); L2 (Ω))∩L 1 loc ((0, ∞); W 1,1 0 (Ω)) and the above holds true for all T > 0, then u is called a weak solution to Problem (1.1) on (0, ∞) × Ω.

In [START_REF] Cai | Existence and uniqueness of weak solutions for a non-uniformly parabolic equation[END_REF] the authors have shown that under (1.2) and some additional structural assumptions, there exists a unique weak solution u ∈ C([0, T ]; L 2 (Ω)) to problem (1.1). When G is as in example 4), the existence and the uniqueness of a solution u ∈ C(Ω T ) to a Cauchy-Dirichlet problem for evolution equation in (1.1) have been investigated in [START_REF] Baroni | The Cauchy-Dirichlet problem for a general class of parabolic equations[END_REF], again under several additional hypothesis.

In the elliptic framework equation in (1.1) is the Euler equation of the energy functional

Ω G(|∇u|)dx,
where the convex integrand may have nonstandard growth conditions. Note that the function G in 2) satisfies for every > 0 the growth condition

s p ≤ G(s) ≤ L (1 + s) p+ .
whenever s is sufficiently large and where L > 0 is a constant depending only on and G.

Starting with the pioneering papers by P. Marcellini [START_REF] Marcellini | Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions[END_REF][START_REF] Marcellini | Regularity and existence of solutions of elliptic equations with p, q-growth conditions[END_REF], the theory in the stationary case, especially the regularity theory, has been extensively studied ([2, 10, 14, 15, 18, 24, 22, 23]). For an almost complete treatment see the survey [START_REF] Mingione | Regularity of minima: an invitation to the dark side of the calculus of variations[END_REF] and the references therein. See also the recent paper [START_REF] Cupini | Nonuniformly elliptic energy integrals with p, q-growth[END_REF]. Following the elliptic scheme, in [START_REF] Bögelein | Parabolic systems with p,q-growth: a variational approach[END_REF] the authors recently studied variational solutions in the sense of [START_REF] Lichnewsky | Pseudosolutions of the time-dependent minimal surface problem[END_REF] to the Cauchy Dirichlet problem (1.1). The existence of such solution has been proved in [START_REF] Bögelein | Existence of evolutionary variational solutions via the calculus of variations[END_REF].

The objective of the present work is to obtain time estimates on weak solutions to (1.1) in terms of the G function. More precisely, we get the following result: Theorem 1.1. Assume (1.2) and let u : Ω T → R be a weak solution to problem (1.1).

Then for any

t ∈ [0, T ], u(•, t) 2 L 2 (Ω) ≤ 2|Ω|x(t) where x(•) is the unique solution of the problem (1.3) x (t) = -c G( x(t)) a.e. x(0) = 1 2|Ω| u 0
Due to the nature of the problem, obtaining such estimates on solutions to (1.1) is not a trivial fact. As far as we know, Theorem 1.1 is the first result in this direction. Moreover, Theorem 1.1 implies time estimates also for evolution problems related to operators having standard growth conditions, as, for instance, for parabolic p-Laplace equation. Indeed Theorem 1.1 improves the results on the behaviour of a solution contained in [START_REF] Moscariello | Quantitative asymptotic estimates for evolution problems[END_REF] and in [START_REF] Farroni | A nonlinear parabolic equation with drift term[END_REF] (see Remark 4.2 below).

The main ingredient to achieve our result is a version of the Gronwall Lemma that has an interest by itself (see Lemma 3.1 and its two Corollaries in Section 3.). An energy balance equality for solutions to (1.1) (see Proposition 2.1) is also fundamental.

More generally, in this paper we prove time estimates, similar to those of Theorem 1.1 for solutions to a nonhomogeneous parabolic problem of the type

   u t -div A(x, t, ∇u) = -div f in Ω T u(x, t) = 0 on ∂Ω × (0, T ) u(x, 0) = u 0 (x)
in Ω,

where A : Ω T ×R N → R N is a Carathéodory function satisfying some nonstandard growth conditions and f : Ω T → R N satisfies the integrability condition (2.6) below. Another application of our techniques is given in Section 5, where we study the asymptotic behaviour of solutions to an anisotropic problem.

The plan of the paper is as follows: In Section 2 we provide notations and preliminary results. In Section 3 we prove a generalized Gronwall Lemma. In Sections 4 and 5 we state and prove our main results.

Notation and preliminary results

Let us consider the parabolic problem (2.1)

   u t -div A(x, t, ∇u) = -div f in Ω T u(x, t) = 0 on ∂Ω × (0, T ) u(x, 0) = u 0 (x) in Ω,
where Ω is a Lipschitz bounded domain of R N , N ≥ 2, Ω T = Ω × (0, T ) and u 0 ∈ L 2 (Ω). We assume that the field A : Ω T × R N → R N is a Carathéodory function such that for a.e. (x, t) ∈ Ω T and for any ξ ∈ R N

(2.2)

   (i) A(x, t, ξ), ξ ≥ νG(|ξ|), 0 < ν ≤ 1 (ii) |A(x, t, ξ)| ≤ µ G(|ξ|) |ξ| ξ = 0, 1 ≤ µ (iii) A(x, t, 0) = 0, where G : R + → R + is a non zero convex function satisfying (1.2).
Let us remark that (2.2) could be imposed, equivalently, for 0 < ν < µ.

If G is a C 1 non-trivial convex function satisfying (j) in (1.2), then the field A(x, t, ξ) = G (|ξ|)
|ξ| ξ, ξ = 0, considered in the model case of Section 1 satisfies conditions (2.2). Indeed, in this case, since G(0) = 0. The convexity of G implies for all ξ ∈ R N , ξ = 0,

G(|ξ|) = 1 0 G (s|ξ|)|ξ|ds ≤ G (|ξ|) |ξ| ξ, ξ
and condition (i) follows. Moreover, by the convexity of G and from the second relation in (j), we get for all t > 0,

kG(t) ≥ G(2t) = 2t 0 G (s)ds ≥ 2t t G (s)ds ≥ tG (t)
from which (ii) follows.

The Fenchel conjugate of G is defined by G(r) = sup s≥0 (rs -G(s)).

Then the Fenchel's inequality

(2.3) sr ≤ G(s) + G(r)
holds true for every r, s ≥ 0. Note that G : R → R + , G(0) = 0 and G is increasing (in this paper we say increasing for nondecreasing). Another useful property is (see [START_REF] Rao | Theory of Orlicz spaces[END_REF] Chapt. II, (1))

(2.4) G G(r) r ≤ G(r) for every r > 0.
Observe that G is increasing and it is not difficult to realise that unless G is the zero function, (j) implies that G(s) > 0 for all s > 0. Condition (j), which is known as the ∆ 2 -condition, is equivalent to the assumption that there exists q > 1 such that G(t) t q is decreasing. On the other hand (jj) is equivalent to ∆ 2 condition for the conjugate function G of G, i.e. there exists k > 0 such that G(2r) ≤ k G(r) for every r > 0 (see [START_REF] Krasnosel | Convex functions and Orlicz spaces[END_REF] Sect. 4). Consequently, G(•) > 0 on (0, ∞).

Notice that ∆ 2 condition yields

(2.5) G(λs) ≤ λ k G(s), G(λs) ≤ λ k G(s) ∀s ≥ 0, λ ≥ 1.
For other properties of function G we refer to [START_REF] Krasnosel | Convex functions and Orlicz spaces[END_REF][START_REF] Rao | Theory of Orlicz spaces[END_REF].

In Problem (2.1) the datum f : Ω T → R N is assumed to be a measurable function such that (2.6)

T 0 Ω G(|f (x, s)|)dxds < +∞. Definition 2.1. A function u : Ω T → R is a weak solution to Problem (2.1) iff u ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 1 (0, T ; W 1,1 0 (Ω)) (2.7) T 0 Ω A(x, s, ∇u), ∇u dxds < +∞ (2.8)
and for every φ ∈ C 1 (Ω T ) with φ(•, t)| ∂Ω = 0 the following equality holds (2.9)

Ω uφ(x, t)dx- Ω u 0 φ(x, 0)dx+ t 0 Ω [-uφ s + A(x, s, ∇u), ∇φ ]dxds = t 0 Ω f, ∇φ dxds for any t ∈ [0, T ]. If u ∈ L ∞ loc ((0, ∞); L 2 (Ω))∩L 1 loc ((0, ∞); W 1,1 0 (Ω))
and the above holds true for all T > 0, then u is called a weak solution to Problem (2.1) on (0, ∞) × Ω.

Notice that assumption (2.8) is equivalent to the following condition (2.10)

T 0 Ω G(|∇u|)dxdt < +∞.
Indeed, since G is increasing, by (2.3), (ii), (2.4) and (2.5), for any (x, s) ∈ Ω T with ∇u(x, s) = 0 we have (2.11)

| A(x, s, ∇u(x, s)), ∇u(x) | ≤ |A(x, s, ∇u(x, s))| • |∇u(x)| ≤ G(|A(x, s, ∇u(x, s))|) + G(|∇u(x)|) ≤ G µ G(|∇u(x,s)|) |∇u(x,s)| + G(|∇u(x)|) ≤ µ k G G(|∇u(x,s)|) |∇u(x,s)| + G(|∇u(x)|) ≤ µ kG(|∇u(x, s)|) + G(|∇u(x)|).
The equivalence between (2.8) and (2.10) follows from the first condition in (2.2) and from (2.11).

Remark 2.1. Using conditions (2.3), (2.6) and (2.10) respectively, an argument similar to (2.11) proves that f (•), ∇φ(•) and A(•, •, ∇u(•, •)), ∇φ(•) are integrable on Ω T , for every φ ∈ C 1 (Ω T ). These considerations imply that under our assumptions, all integrals in (2.9) are well defined.

Remark 2.2. By assumptions (1.2), for any r > 0 we have

λ 1 r q ≤ G(r) ≤ λ 2 (r p + 1)
for some positive constants λ 1 , λ 2 and pp = p + p , and qq = q + q (see [START_REF] Rao | Theory of Orlicz spaces[END_REF] Chapt. 2). Then, from (2.9) we deduce that u t ∈ L q (0, T ; L q (Ω)) with 1 < q < 2, and so there exists ũ ∈ C([0, T ]; L q (Ω)) such that u(•, t) = ũ(•, t) for almost all t ∈ (0, T ). In what folllows we identify u with its continuous representant ũ.

Lemma 2.1. Let u ∈ L ∞ (0, T ; L 2 (Ω))∩L 1 (0, T ; W 1,1 0 (Ω)), be a solution to Problem (2.1). Then, u ∈ C w ([0, T ]; L 2 (Ω)). Proof. Let t ∈ [0, T ] be fixed. Let {t n } be a sequence in [0, T ] with t n → t as n → +∞ such that u(•, t n ) ∈ L 2 (Ω) ∀n ∈ N. Since u ∈ L ∞ (0, T ; L 2 (Ω)), the sequence {u(•, t n )} is bounded in L 2 (Ω). Then, there exists a subsequence {u(•, t kn )} and ξ ∈ L 2 (Ω) such that u(•, t kn ) → ξ weakly in L 2 (Ω) as n → +∞.
On the other hand, by Remark 2.2 u ∈ C([0, T ]; L q (Ω)). Therefore

u(•, t kn ) → u(•, t) strongy in L q (Ω) as n → +∞. This implies that ξ = u(•, t) and u ∈ C w ([0, T ]; L 2 (Ω)).
We also recall the following useful result.

Lemma 2.2. Let G : R + → R + be a function satisfying (1.2). Then there exists a convex function F : R + → R + satisfying (j) such that (2.12)

F (t p ) ≤ G(t) ≤ kF (t p ).
where k and p are as in (1.2).

Proof. Since G(s 1 p )/s is increasing, it is not difficult to realize that the function

F (t) = t 0 G(s 1 p ) s ds
is well defined on (0, ∞) and can be extended by continuity to zero by setting F (0) = 0. Its first derivative is a continuous increasing function (0, ∞). Hence F is convex on R + . Moreover, by (jj), the integrand is increasing and therefore

F (t) ≤ t 0 G(t 1
p ) t ds = G(t 1 p )
implying the first inequality. We show next that F satisfies (j). Indeed, since G satisfies (j),

F (2t) = t 0 G((2s) 1 p ) s ds ≤ t 0 G(2s 1 p ) s ds ≤ k t 0 G(s 1 p ) s ds = kF (t).
The second inequality in (2.12) follows from the following estimates

G(t 1 p ) = 2t t G(t 1 p ) t ds ≤ 2t t G(s 1 p ) s ds ≤ F (2t) ≤ kF (t).
In this paper we do not investigate the existence of a solution to (2.1) and refer to [START_REF] Baroni | The Cauchy-Dirichlet problem for a general class of parabolic equations[END_REF][START_REF] Bögelein | Existence of evolutionary variational solutions via the calculus of variations[END_REF][START_REF] Cai | Existence and uniqueness of weak solutions for a non-uniformly parabolic equation[END_REF] for some existence results. Instead, we establish some properties of weak solutions. We first show that every such solution satisfies an energy balance equality. Proposition 2.1. Assume (2.2) and (2.6) and let u : Ω T → R be a weak solution to (2.1). Then for any 0

< t ≤ T 1 2 u(t) 2 L 2 (Ω) + t 0 Ω A(x, s, ∇u), ∇u dxds = 1 2 u 0 2 L 2 (Ω) + t 0 Ω f (x, s), ∇u dxds.
where

u(t) 2 L 2 (Ω) := Ω |u(x, t)| 2 dx.
Proof. Let u : Ω T → R be a weak solution to (2.1). We first extend solution u(x, t) by the initial value u 0 (x) when t < 0. As ∂Ω is Lipschitz, we can cover ∂Ω by a finite open family {U i } m i=1 and we can find corresponding positive numbers λ i and vectors p i such that for > 0 the ball B(x + λ i p i , ) ⊂ Ω for all x ∈ U i ∩ Ω (cf. [START_REF] Evans | Partial Differential Equations[END_REF]).

Choose an open set U 0 ⊂⊂ Ω such that {U i } m i=0 forms a covering of Ω and let {η i } m i=0 be a smooth partition of unity corresponding to this covering. For

x ∈ U i ∩ Ω, i = 1, ..., m, denote u i (x, t) = u(x + λ i p i , t) and let u i (•, t) be its mollification in U i ∩ Ω. Finally we mollify u(x, t) in U 0 to get u 0 . The functions u : Ω T → R defined as u = m i=0 η i u i , are C ∞ 0 (Ω) and converge to u in W 1,1 0 (Ω) when → 0 + for a.e. t ∈ [0, T ]. Consider the mollifier in time of u (x, •) defined for any natural n ∈ N such that 1 n < , as: [u ] n (x, t) = e -nt u (x, t 0 ) + n t 0 e n(s-t) u (x, s)ds, where t 0 ∈ (0, T ). Since u ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 1 (0, T ; W 1,1 0 (Ω)) and u (x, t 0 ) ∈ L 2 (Ω), we get that (2.13) [u ] n → u in L 2 (Ω T ) as n → ∞. [u ] n (x, 0) = u (x, t 0 ) ∀n ∈ N. (2.14) [u ] n ∈ L 1 [0, T ]; W 1,1 0 (Ω) (2.15) ∇ [u ] n (•, t) = [∇u ] n (•, t) [∇u ] n → ∇u in L 1 (Ω T ) when n → ∞.
Properties (2.13), (2.14) and (2.15) can be deduced from Lemma B2 in [START_REF] Bögelein | Parabolic systems with p,q-growth: a variational approach[END_REF]. See also [START_REF] Bögelein | Giuseppe Degenerate problems with irregular obstacles[END_REF]. Now we define for any > 0, n ∈ N, 0 < h << 1 such that 0

< 1 n < < h < t 0 , the Steklov average of [u ] n φ n ,h (x, t) := ([u ] n ) h = 1 2h t+h t-h [u ] n (x, τ )dτ
where we assume that [u ] n (x, t) = u (x, t 0 ) for any t ≤ 0. First let us assume that u is bounded. Then the function φ n ,h ∈ C 1 ( ΩT ) and φ n ,h (•, t) | ∂Ω = 0 for every t ∈ [0, T ], and so we can use it as a test function in (2.9) to get that for any

t ∈ [0, T ], Ω uφ n ,h dx t 0 + t 0 Ω [-u(φ n ,h ) s + A(x, s, ∇u), ∇φ n ,h ]dxds = t 0 Ω f, ∇φ n ,h dxds. Set I ,h 1,n := Ω uφ n ,h (x, t)dx - Ω u 0 φ n ,h (x, 0)dx - t 0 Ω u(φ n ,h ) s dxds.
For 0 < h < t, by definition of φ n ,h

I ,h 1,n = Ω u(x, t) 1 2h t+h t-h [u ] n (x, τ )dτ dx - Ω u 0 (x) 1 2h h -h [u ] n (x, τ )dτ dx - 1 2h t 0 Ω u(x, τ ) ([u ] n (x, τ + h) -[u ] n (x, τ -h)) dxdτ. = J ,h 1,n + J ,h 2,n + J ,h 3,n .
Observe that

J ,h 2,n = - Ω u 0 (x) 1 2h 0 -h u (x, t 0 ) dτ + 1 2h h 0 [u ] n (x, τ ) dτ dx J ,h 3,n = - 1 2h t+h t Ω u(x, τ -h)[u ] n (x, τ ) dx dτ - 1 2h t h Ω u(x, τ -h)[u ] n (x, τ ) dx dτ + 1 2h h 0 Ω u (x, t 0 )u(x, τ ) dx dτ + 1 2h t h Ω [u ] n (x, τ -h)u(x, τ ) dx dτ
From (2.13), if we pass to the limit first as n → +∞ and then as → 0 + we get

(2.16) lim →0+ lim n→+∞ I ,h 1,n = Ω u(x, t) 1 2h t+h t-h u(x, τ )dτ dx -1 2 Ω u 0 (x)u(x, t 0 ) dx -1 2h Ω u 0 (x) h 0 u(x, τ ) dτ dx -1 2h t+h t Ω u(x, τ -h)u(x, τ ) dx dτ + 1 2h h 0 Ω u(x, t 0 )u(x, τ ) dx dτ.
Then by Lemma 2.1, passing to the limit in (2.16) as h → 0 + , we get for any

t ∈ [0, T ] lim h→0 lim →0 lim n→+∞ I ,h 1,n = 1 2 Ω |u(x, t)| 2 dx - 1 2 Ω |u 0 (x)| 2 dx. Define I ,h 2,n := t 0 Ω A(x, s, ∇u), ∇φ n ,h dxds, I ,h 3,n := t 0 Ω f, ∇φ n ,h dxds.
Then,

I ,h 2,n = m i=0 t 0 Ω A(x, s, ∇u), ( u i n ) h (x, s)∇η i dxds + m i=0 t 0 Ω A(x, s, ∇u), η i (x)( ∇u i n ) h (x, s) dxds.
Observe that

| A(x, s, ∇u), η i (x)( ∇u i n ) h (x, s) | ≤ G(|A(x, s, ∇u)|) + G(η i (x)|([∇u i ] n ) h |)(x, s) ≤ µ kG(|∇u(x, s)|) + G(η i (x)|([∇u i ] n ) h |)(x, s), (2.17)
where we used condition (ii) and inequalities (2.3), (2.4), (2.5).

Let 0 < h ≤ s. By the Fubini theorem

| ∇u i n h |(x, s) ≤ 1 2h s+h s-h e -nr |∇u i (x, t 0 )| + n r 0 e n(τ -r) |∇u i (x, τ )|dτ dr = - 1 2hn [e -n(s+h) -e -n(s-h) ]|∇u i (x, t 0 )| + n 2h s+h s-h s+h τ e n(τ -r) |∇u i (x, τ )|dr dτ = 1 2hn [e -n(s-h) -e -n(s+h) ]|∇u i (x, t 0 )| + 1 2h s+h s-h 1 -e -n((s+h)-τ ) |∇u i (x, τ )|dτ ≤ 1 2hn |∇u i (x, t 0 )| + 1 2h s+h s-h |∇u i (x, τ )|dτ.
(2.18) Inequality (2.18), the monotonicity of G and the Jensen inequality imply that for (x, s)

∈ Ω × (h, t) G(η i | ∇u i n ) h | (x, s) ≤ G(|( ∇u i n ) h |)(x, s) = G 1 2hn |∇u i (x, t 0 )| + 1 2h s+h s-h |∇u i (x, τ )|dτ ≤ 1 2 G 1 hn |∇u i (x, t 0 )| + 1 2 G 2 1 2h s+h s-h |∇u i (x, τ )|dτ ≤ 1 2 G 1 hn |∇u i (x, t 0 )| + 2 k-1 1 2h s+h s-h G |∇u i (x, τ )| dτ.
(2.19) When 0 < s < h, again by the Fubini theorem, we get

| ∇u i n h |(x, s) ≤ 1 2h 0 s-h |∇u i (x, t 0 )|dr + 1 2h s+h 0 e -nr |∇u i (x, t 0 )| + n r 0 e n(τ -r) |∇u i (x, τ )|dτ dr = h -s 2h |∇u i (x, t 0 )| + 1 2hn [1 -e -n(s+h) ]|∇u i (x, t 0 )| + n 2h s+h 0 s+h τ e n(τ -r) |∇u i (x, τ )|dr dτ ≤ 1 2 + 1 2hn |∇u i (x, t 0 )| + 1 2h s+h 0 1 -e -n((s+h)-τ ) |∇u i (x, τ )|dτ ≤ 1 2 + 1 2hn |∇u i (x, t 0 )| + 1 2h s+h s-h |∇u i (x, τ )|dτ (2.20)
Arguing as in (2.19), from (2.20) we get, for (x, s)

∈ Ω × (0, h) G(η i | ∇u i n ) h | (x, s) ≤ 1 2 G 1 + 1 hn |∇u i (x, t 0 )| + 2 k-1 1 2h s+h s-h G |∇u i (x, τ )| dτ. (2.21)
Moreover, by the definition of u i ,h , the Fubini theorem and the Jensen inequality, we know that

(2.22) G(|∇u i |) h (x, s) ≤ [G(|∇u|)] ,h (x, s)
Combining (2.17), (2. [START_REF] Marcellini | Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions[END_REF]) and (2.22), for almost every (x, s) ∈ Ω × (h, t),

| A(x, s, ∇u(x, s)), ( ∇u i n ) h (x, s)η i (x) | ≤ µ kG(|∇u(x, s))|) + 1 2 G 1 hn |∇u i (x, t 0 )| + 2 k-1 [G(|∇u|)] ,h (x, s)
On the other hand, combining (2.17), (2.21) and (2.22), for almost every (x, s) ∈ Ω × (0, h),

| A(x, s, ∇u(x, s)), η i (x)( ∇u i n ) h (x, s) | ≤ µ kG(|∇u(x, s))|) + 1 2 G 1 + 1 hn |∇u i (x, t 0 )| + 2 k-1 [G(|∇u|)] ,h (x, s)
Last inequalities, the continuity of G and the Lebesgue dominated convergence theorem imply that lim

h→0+ lim →0+ lim n→∞ t h Ω A(x, s, ∇u(x, s)), η i (x)( ∇u i n ) h (x, s) dxds = t 0 Ω
A(x, s, ∇u(x, s)), η i (x)∇u(x, s) dxds and lim

h→0+ lim →0+ lim n→∞ h 0 Ω A(x, s, ∇u(x, s)), η i (x)( ∇u i n ) h (x, s) dxds = 0.
Finally, observe that

m i=0 t 0 Ω A(x, s, ∇u(x, s)), ∇u(x, s)η i (x) dxds = t 0 Ω
A(x, s, ∇u(x, s)), ∇u(x, s) dxds.

Consequently, (2.23)

lim h→0+ lim →0+ lim n→∞ m i=0 t 0 Ω A(x, s, ∇u(x, s)), η i (x)([∇u i ] n ) h (x, s)(x) dxds = t 0 Ω A(x, s, ∇u(x, s)), ∇u(x, s) dxds.
On the other hand, since G is increasing, by the Fenchel inequality, (ii), (2.4), (2.5), (2.13) and the boundedness of u, for some c 0 > 0, for any i ∈ {0, 1, ..., m} and every (x, s) ∈ Ω T with ∇u(x, s) = 0 we have (2.24) A(x, s, ∇u), ∇u dxds.

| A(x, s, ∇u(x, s)), ([u i ] n ) h (x, s)∇η i | ≤ |A(x, s, ∇u(x, s))| • |([u i ] n ) h (x, s)∇η i | ≤ G(|A(x, s, ∇u(x, s))|) + G(|([u i ] n ) h (x, s)∇η i |) ≤ G µ G(|∇u(x,s)|) |∇u(x,s)| + c 0 ≤ µ k G G(|∇u(x,s)|) |∇u(x,s)| + c 0 ≤ µ kG(|∇u(x, s)|) + c 0 . Hence (2.10) implies that | A(•, •, ∇u(•, •)), ([u i ] n (•, •)) h ∇η i (x) | is
In the same way, by using (2.3), (2.5) and (2.6), we can prove that lim

h→0+ lim →0+ lim n→∞ I ,h 3,n = t 0 Ω f, ∇u dxds.
If u is not bounded, define for any j ∈ N, u j = 1 2 (|u + j| -|u -j|). Then,

u j ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 1 (0, T ; W 1,1 0 (Ω)) ∩ L ∞ (Ω T ).
We can constuct (u j ) as before and we can apply the above arguments by using (φ j ) n ,h as test function in (2.9) Finally we can pass to the limit as j → +∞ since |u j | ≤ |u| a.e. A(x, s, ∇u) -A(x, s, ∇v), ∇w dxds = 0.

Hence, (2.26) yields u = v a.e. in Ω T .

A Version of the Gronwall Lemma

For arbitrary non-empty closed sets in D , D ⊂ R n , define

d D (D ) := inf{β > 0 | D ⊂ D + βB} .
where B denotes the closed unit ball in R n . Consider reals 0 ≤ t 0 < T and a multifunction P : [t 0 , T ] ; R n with non-empty closed values. P is called left absolutely continuous if the following condition is satisfied: given any > 0 there exists δ > 0 such that, for any finite partition

t 0 ≤ s 1 < τ 1 ≤ s 2 < τ 2 ≤ . . . ≤ s m < τ m ≤ T satisfying m i=1 (τ i -s i ) < δ, we have m i=1 d P (τ i ) (P (s i )) < .
For any y ∈ P (t) the contingent derivative DP (t, y)(1) is the set of all v ∈ R n such that there exist h i → 0+ and v i → v satisfying y + h i v i ∈ P (t + h i ).

A function γ : [t 0 , T ] → R + is called upper semicontinuous from the right, if for every t ∈ [t 0 , T ) we have lim sup s→t+ γ(s) ≤ γ(t). Similarly, it is called lower semicontinuous from the left, if for every t ∈ (t 0 , T ] we have lim inf s→t-γ(s) ≥ γ(t). Lemma 3.1. Consider a Carathéodory function ψ : [t 0 , T ] × R → R + such that for every r > 0 there exists k r ∈ L 1 (t 0 , T ; R + ) satisfying for a.e. t ∈ [t 0 , T ] sup |x|≤r ψ(t, x) ≤ k r (t).

Let g ∈ L 1 (t 0 , T ; R) and γ : [t 0 , T ] → R + be measurable, bounded and satisfy

(3.1) γ(t 2 ) -γ(t 1 ) + t 2 t 1 ψ(t, γ(t))dt ≤ t 2 t 1 g(t)dt t 0 ≤ t 1 ≤ t 2 ≤ T.
Then there exists a solution x(•) ∈ W 1,1 ([t 0 , T ]) of the Cauchy problem

(3.2) x (t) = -ψ(t, x(t)) + g(t), a.e. x(t 0 ) = γ(t 0 ) such that γ(t) ≤ x(t) for all t ∈ [t 0 , T ]. Furthermore, if g ∈ L 1 (t 0 , ∞; R), ψ is defined on [t 0 , +∞) × R, γ : [t 0 , +∞) → R + is measurable
and locally bounded and for every T > t 0 the above assumptions hold true, then there exists a solution x to (3.2) defined on [t 0 , ∞) such that γ ≤ x. In particular,

lim sup t→∞ γ(t) ≤ lim sup t→∞ x(t).
Proof. By (3.1), γ is upper semicontinuous from the right and lower semicontinuous from the left. Define the set-valued map P : [t 0 , T ] ; R by

P (t) = γ(t) + R + t ∈ [t 0 , T ) R t ≥ T.
By our assumptions for all t 0 ≤ t i < τ i < T γ(τ i ) -

τ i t i g(t)dt ≤ γ(t i ).
This implies that P (•) is left absolutely continuous. Let t ∈ (t 0 , T ) be a Lebesgue point of ψ(•, γ(•)) and of g(•). The set A of all such points is of full Lebesgue measure in [t 0 , T ]. By the assumptions of our lemma for every t ∈ A,

Dγ(t) := lim inf ε→0+ γ(t + ε) -γ(t) ε ≤ -ψ(t, γ(t)) + g(t).
This and the definition of P (•) imply that -ψ(t, γ(t)) + g(t) ∈ DP (t, γ(t))(1). Since γ is upper semicontinuous from the right, it enjoys the following property: if y > γ(t), then for every r ∈ R and all small ε > 0 we have y + εr > γ(t + ε). Therefore, DP (t, y)(1) = R for any y > γ(t). Consequently for all t ∈ A and y ∈ P (t) we have -ψ(t, y) + g(t) ∈ DP (t, y) [START_REF] Adams | Anisotropic Sobolev inequalities[END_REF]. By [13, Theorem 4.2] and our assumptions, for some S ∈ (t 0 , T ] there exists a solution of (3.2) satisfying x(t) ∈ P (t) on [t 0 , S). Therefore 0 ≤ γ(t) ≤ x(t) for all t ∈ [t 0 , S). Furthermore, x(t) ≤ x(t 0 ) + To prove the last statement assume that g ∈ L 1 (t 0 , ∞; R), ψ is defined on [t 0 , +∞) × R, γ is locally bounded on [t 0 , +∞) and for every T > t 0 the assumptions of Lemma hold true. Then the same construction allows one to get x(•) defined on [t 0 , +∞) satisfying for every t > τ ≥ t 0 ,

0 ≤ γ(t) ≤ x(t) ≤ x(τ ) + t τ g(s)ds.
Since lim τ →∞ ∞ τ g(s)ds = 0 the proof follows. 2 Remark 3.1. Indeed, assume that the solution x : [t 0 , T ] → R + of (3.2) is unique. Then for γ = x we have

γ(t 2 ) -γ(t 1 ) + t 2 t 1 ψ(t, γ(t))dt = t 2 t 1 g(t)dt t 0 ≤ t 1 ≤ t 2 ≤ T.
That is (3.1) holds true. The estimate γ ≤ x obtained in Lemma 3.1 can not be further improved.

Corollary 3.1. Under all the assumptions of Lemma 3.1 suppose that ψ(t, a) = 0 for all a ≤ 0, g(•) ≥ 0, ψ(t, •) is increasing for a.e. t ∈ [t 0 , T ] and that for any R > r > 0 there exists kR,r ∈ L 1 (t 0 , T ; R + ) satisfying for a.e. t ∈ [t 0 , T ]

|ψ(t, x) -ψ(t, y)| ≤ k R,r (t)|x -y| ∀ x, y ∈ [r, R].
Then the solution z(•) of

(3.3) z (t) = -ψ(t, z(t)), a.e. z(t 0 ) = γ(t 0 )
is unique and well defined on [t 0 , T ], z(•) ≥ 0 and γ(t) ≤ x(t) ≤ z(t) + t t 0 g(s)ds for all t ∈ [t 0 , T ].

Proof. Clearly, if z(•) is a solution of (3.3), then it is decreasing on its interval of existence. Moreover, if for some t ≥ t 0 , z(t) = 0, then z(s) = 0 for any t ≤ s ≤ T. This and the Lipschitz continuity assumption imply the existence and uniqueness on the whole time interval [t 0 , T ]. Consider any solution x(•) of (3.2) defined on [t 0 , T ]. Then it is not difficult to realize that x(•) ≥ 0.

We claim that z ≤ x on [t 0 , T ]. Indeed otherwise we can find

t 0 ≤ s 0 < t 1 < T such that z(s 0 ) = x(s 0 ) and z(s) > x(s) on [s 0 , t 1 ]. Then -ψ(t, z(t)) ≤ -ψ(t, x(t)) for a.e. t ∈ [s 0 , t 1 ]. Thus x (s) -z (s) ≥ g(s) ≥ 0 a.e. in [s 0 , t 1 ]. This implies that x(t) ≥ z(t) for t ∈ [s 0 , t 1 ]
leading to a contradiction. Let x(•) be as in Lemma 3.1. Observe that for a.e. t ∈ [t 0 , T ] we have

x (t) = -ψ(t, x(t)) + g(t) ≤ -ψ(t, z(t)) + g(t) = z (t) + g(t).
Integrating we obtain that x(t) ≤ z(t) + t t 0 g(s)ds for all t ∈ [t 0 , T ]. This and Lemma 3.

1 yield γ(t) ≤ x(t) ≤ z(t) + t t 0 g(s)ds . 2 Corollary 3.2. Consider g ∈ L 1 (t 0 , ∞; R + ), a Carathéodory function ψ : [t 0 , ∞) × R → R + ,
a measurable locally bounded γ : [t 0 , ∞) → R + and let assumptions of Corollary 3.1 hold true on [t 0 , T ] for every T > t 0 . If for every r > 0 there exists α r > 0 such that inf x≥r ψ(t, x) ≥ α r for a.e. t ≥ t 0 , then lim t→∞ γ(t) = 0.

Proof. Fix any sequence t i → ∞ and consider the solutions z i to (3.3) with t 0 replaced by t i . By Corollary 3.1 applied with t 0 replaced by t i , we know that z i ≥ 0. Notice that z i is decreasing for every i. Observe next that if for some i we have lim t→∞ z i (t) =: r > 0, then, z i (t) ≥ r for every t ≥ t i and

0 ≤ z i (t) = z i (t i ) - t t i ψ(s, z i (s))ds ≤ z i (t i ) -(t -t i )α r .
When t is sufficiently large, the right-hand side of the above expression becomes negative. Thus implies that lim t→∞ z i (t) = 0 for all i. This and Corollary 3.1 imply that for every i, every ε > 0 and all large t we have

γ(t) ≤ ε 2 + t t i g(s)ds.
Taking i sufficiently large, we deduce that for every ε > 0 and all large t we have γ(t) ≤ ε completing the proof. 2

In [START_REF] Moscariello | Quantitative asymptotic estimates for evolution problems[END_REF] a particular instance of ψ(t, x) was considered, namely ψ(t, x) = M |x| 1+ν , where M and ν are non negative constants. It was shown then that for any continuous γ : [t 0 , T ] → R + satisfying the inequality (3.1) we have

(3.4) γ(t) ≤ y(t) := γ(t 0 ) [1 + νM γ(t 0 ) ν (t -t 0 )] 1 ν + Γ(t) ∀ t ∈ [t 0 , T ]
for every choice of continuous function Γ : [t 0 , T ] → R + satisfying

(3.5) Γ(t 1 ) ≤ Γ(t 2 ) + M t 2 t 1 Γ(t) 1+ν dt - t 2 t 1 g(t)dt t 0 ≤ t 1 ≤ t 2 ≤ T,
where g ∈ L 1 ([t 0 , T ], R). Let Γ, M, ν be as above and x be the solution of

(3.6) x = -M |x| 1+ν + g(t), a.e. t > t 0 x(t 0 ) = γ(t 0 )
We already know that x(t) ≥ γ(t) ≥ 0 for all t ∈ [t 0 , T ]. We claim that x(t) ≤ y(t), which implies that our result provides a better estimate on the time behavior of γ than the one from [START_REF] Moscariello | Quantitative asymptotic estimates for evolution problems[END_REF]. Indeed, since y(t 0 ) = γ(t 0 ) + Γ(t 0 ) ≥ x(t 0 ), if for some t 1 ∈ (t 0 , T ] we have y(t 1 ) < x(t 1 ), then we can find s 0 ∈ [t 0 , t 1 ] such that x(s 0 ) = y(s 0 ) and y(s) < x(s) for all s ∈ (s 0 , t 1 ].

Note that -y(•) is continuous on [t 0 , T ). By the same arguments as in the proof of Lemma 3.1, we check that t ; -Γ(t) + R + is left absolutely continuous on [t 0 , T ] and that the map

t → γ(t 0 ) [1 + νM γ(t 0 ) ν (t -t 0 )] 1 ν
is Lipschitz. Therefore t ; P (t) = -y(t) + R + is left absolutely continuous on [t 0 , T ]. Furthermore, for a.e. s ∈ (s 0 , t 1 ) we have

D(-Γ)(s) ≤ M Γ(s) 1+ν -g(s).
Hence, for a.e. s ∈ (s 0 , t 1 ),

D(-y)(s) ≤ M γ(t 0 ) 1+ν [1 + νM γ(t 0 ) ν (t -t 0 )] 1+ν ν + M Γ(s) 1+ν -g(s) ≤ M y(s) 1+ν -g(s)
= M y(s) 1+ν -M x(s) 1+ν -x (s) ≤ -x (s). Therefore, for a.e. s ∈ [s 0 , t 1 ] and all z ∈ P (s) we have -x (s) ∈ DP (s, z) [START_REF] Adams | Anisotropic Sobolev inequalities[END_REF]. From [START_REF] Frankowska | Measurable viability theorems and Hamilton-Jacobi-Bellman equation[END_REF]Theorem 4.2] applied to the single-valued map s → {-x (s)} we deduce that -x(t) ∈ P (t) for all t ∈ [s 0 , t 1 ]. This yields x(t) ≤ y(t) on [s 0 , t 1 ] contradicting the choice of t 1 .

It was observed in [START_REF] Moscariello | Quantitative asymptotic estimates for evolution problems[END_REF] that if g ≥ 0, then Γ(t) = t t 0 g(s)ds verifies (3.5) and that for any continuous γ : [t 0 , T ] → R + satisfying the inequality (3.1), the estimate (3.4) holds true for this choice of Γ. In particular, if g ∈ L 1 (t 0 , ∞; R + ), then (3.4) is verified with this Γ and t 0 replaced by any τ > t 0 and T > τ . This implies that lim t→∞ γ(t) = 0.

Define ψ(t, x) = M |x| 1+ν for x ≥ 0 and ψ(t, x) = 0 for x < 0 and observe that it satisfies the assumptions of Corollary 3.2. Thus the equality lim t→∞ γ(t) = 0 is a consequence of Corollary 3.2.

The main results

In this section we assume that the function G : R + → R + in (2.2) is continuous, convex and satisfies (1.2) for some k > 0 and p ≥ 2.

We state next the main result of this paper.

   ẋ(t) = -c G( x(t) ) + g(t) x(0) = 1 2|Ω| u 0 2 L 2 (Ω)
for some c > 0 depending only on p, k, N, |Ω|, ν and g(t

) = 1 |Ω| G( 2 ν |f (•, t)|) L 1 (Ω) . If f : Ω → R is a integrable function on Ω, we set Ω |f |dx = 1 |Ω| Ω |f |dx
where |Ω| > 0 denotes the Lebesgue measure of Ω.

Proof. For 0 < t

G 1 2 Ω |u(x, t)| 2 dx 1 2 ≤ G Ω |u(x, t)| 2 dx 1 2 ≤ C Ω G(|∇u|)dx. (4.5)
Going back to (4.2), we deduce from (4.5) the following inequality

1 2 Ω |u(x, t 2 )| 2 dx - 1 2 Ω |u(x, t 1 )| 2 dx + ν 2C t 2 t 1 G 1 2 Ω |u(x, t)| 2 dx 1/2 dt ≤ 1 |Ω| t 2 t 1 Ω G 2 ν |f (x, s)| dxds. (4.6)
Now we are in position to apply Lemma 3.1 with

γ(t) =: 1 2 Ω |u(x, t)| 2 dx and ψ(t, x) =: ν 2C G( √ x ).
By the measurable viability theorem, cf. [START_REF] Frankowska | Measurable viability theorems and Hamilton-Jacobi-Bellman equation[END_REF], there exists a solution x(•) : R + → R + of (4.1). We claim that it is unique. Indeed, for positive reals z ≥ y, we have √ z ≥ √ y and ψ(t, z) ≥ ψ(t, y). Hence (ψ(t, z) -ψ(t, y))(z -y) ≥ 0. Similarly, it can be verified that if z ≤ y, the same inequality holds true.

Consider any solution y : [t 0 , T ] → R + of (4.1). Then

d dt 1 2 |x -y| 2 (t) = -(ψ(t, x(t)) -ψ(t, y(t)))(x(t) -y(t)) ≤ 0
and therefore x = y on [t 0 , T ]. Setting ψ(t, a) = 0 for a < 0, we may apply Lemma 3.1 and so to get the desired estimate.

Corollaries 3.1 and 3.2 yield the following result. Theorem 4.2. In Theorem 4.1 assume that g ∈ L 1 ([0, ∞)). Let u : Ω × [0, ∞) → R be a weak solution to (2.1). Then, for any t 0 ≥ 0 and t > t 0 ,

u(t, •) 2 L 2 (Ω) ≤ 2|Ω|x(t) ≤ 2|Ω| z(t) + t t 0 g(s)ds
where z(•) is the unique solution of the problem

ż(t) = -c G( z(t) ) z(t 0 ) = 1 2|Ω| u(•, t 0 ) L 2 (Ω)
for some c > 0 depending only on p, k, N, |Ω|, ν.

Furthermore, lim t→∞ u(t, •) [START_REF] Moscariello | Quantitative asymptotic estimates for evolution problems[END_REF] and the reference therein). For istance, when p = 2, Theorem 4.1 implies that for any t ∈ [0, T ]

u(t) 2 L 2 (Ω) ≤ e -c 2 t u 0 2 L 2 (Ω) + 2 t 0 e -c 2 (t-s) f (•, s) 2 L 2 (Ω) ds
where c is the constant in Problem (4.1).

We conclude this section by an example. Let us consider for simplicity problem (1.1) with

G(s) = (s 2 + 1) log(1 + s 2 ).
Then, if u is the weak solution of problem (1.1), we get that

(4.7) u(•, t) 2 L 2 (Ω ≤ 2|Ω| exp log(1 + 1 2|Ω| u 0 L 2 (Ω) )e -ct - 1
for some c > 0 and all t ≥ 0. Estimate (4.7) is new and cannot be achieved by the arguments of [START_REF] Moscariello | Quantitative asymptotic estimates for evolution problems[END_REF].

Solutions of an anisotropic problem

In this section we consider the following anisotropic problem (5.1)

           u t -N i=1 ∂ ∂x i (|u x i | q i -2 u x i ) = 0 in Ω T , u(x, t) = 0 on ∂Ω × (0, T ), u(x, 0) = u 0 (x) in Ω,
where u 0 ∈ L 2 (Ω) and 1 < q i , i ∈ {1, ..., N }.

Here, for any ξ ∈ R N the vector field is A(ξ) =: N i=1 |ξ i | q i -2 ξ i . In [START_REF] Cai | Existence and uniqueness of weak solutions for a non-uniformly parabolic equation[END_REF] the authors prove that there exists a unique function u ∈ C ([0, T ], L 2 (Ω)) ∩ L 2 0, T, W

|u x i | q i dxdt = 1 2 u 0 2 L 2 (Ω) .
The proof of (5.2) is similar to the one of Proposition 2.1. In this case the mollifier in time of test functions is not necessary and we can argue directly on the convex vector field A(ξ). (See also [START_REF] Cai | Existence and uniqueness of weak solutions for a non-uniformly parabolic equation[END_REF]).

Set 1 q =: 1 N N i=1 1 q i , q =: max{q i , i = 1, ..., N } p =: min{q i , i = 1, ..., N } and B(t) =: max{1, u x i (t) L q i (Ω), i=1,...,N }.

Theorem 5.1. Assume that 2 ≤ q and 2N N +2 < q < N . Let u : Ω T → R be a weak solution to problem (5.1). Then for any t ∈ [0, T ], u(•, t) 2 L 2 (Ω) ≤ 2x(t) where x(t) is the unique solution of the problem (5.3) x (t) = -c B(t) p-q x(t) q 2

x(0) = 1 2 u 0 2 L 2 (Ω) . for some c > 0 depending only on |Ω|, q and N . Proof. By (5.2) for any 0 ≤ t 1 < t 2 ≤ T (5.4)

1 2 u(t 2 ) 2 L 2 (Ω) - 1 2 u(t 1 ) 2 L 2 (Ω) + t 2 t 1 Ω N i=1 |u x i | q i dxdt = 0
We claim that, for a.e. t ∈ (0, T ), (5.5)

Ω |u| q * dx q q * ≤ cB(t) q-p Ω N i=1 |u x i | q i dx,
where c depends on N, |Ω|, q.

In fact, by the definition of B(t), since q i ≥ p and B(t) ≥ 1, for any i = 1, . . . , N we have u x i q L q i (Ω) ≤ B(t) q-q i u x i q i L q i (Ω) ≤ B(t) q-p u x i q i L q i (Ω)

The convexity of the real function t → t q implies 1 N N i=1

u x i L q i (Ω) q ≤ 1 N N i=1 u x i q L q i (Ω)
Then, since 2N N +2 < q < N , we can apply the anisotropic Sobolev-type inequality [START_REF] Troisi | Teoremi di inclusione per spazi di Sobolev non isotropi[END_REF] to obtain

u q L q * (Ω) ≤ c q 0 1 N N i=1 u x i L q i (Ω) q ≤ c q 0 N -1 B(t) q-p N i=1 u x i q i L q i (Ω)
where c 0 depends on N, |Ω|, q. Then, (5.5) follows.

The elliptic version of relation (5.5) has been proved in [START_REF] Boccardo | L ∞ -regularity for variational problems with sharp non-standard growth conditions[END_REF]. Now, from (5.5) we get (5.6) B(t) p-q

Ω |u| 2 dx q 2 ≤ B(t) p-q |Ω| q 2 -q q * Ω |u| q * dx q q * ≤ c 1 Ω N i=1 |u x i | q i dx
where c 1 depends on N, |Ω|, q. Integrating (5.5) on (t 1 , t 2 ), from the relations (5.4) and (5. dt ≤ 0 for some c > 0 depending only on N, |Ω|, q, q. Now if γ(t) = 1 2 u(t) 2 L 2 (Ω) we can rewrite (5.7) as γ(t 2 ) -γ(t 1 ) + t 2 t 1 ψ(t, γ(t)) dt ≤ 0 where ψ(t, x) = cB(t) p-q x q 2 ∀ x ≥ 0, ψ(t, x) = 0 ∀ , x < 0.

We apply Corollary 3.1 to achieve the announced result. Indeed the above proof can be applied as well.

Definition 1 . 1 .

 11 A function u : Ω T → R is called a weak solution to Problem (1.1) iff u ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 1 (0, T ; W 1)dxdt < +∞and for every φ ∈ C 1 (Ω T ) with φ(•, t)| ∂Ω = 0 the following equality holds Ω uφ(x, t)dx -Ω u 0 φ(x, 0)dx + t 0 Ω -uφ s + G (|∇u|) |∇u| ∇u, ∇φ dxds = 0 for any t ∈ [0, T ].

Theorem 4 . 1 .

 41 Assume (2.2) and (2.6). Let u : Ω T → R be a weak solution to (2.1). Then, for any t ∈ [0, T ], u(•, t) 2 L 2 (Ω) ≤ 2|Ω|x(t) where x(•) is the unique solution of the problem (4.1)

Remark 5 . 1 .

 51 Let β : [0, T ] → (1, +∞) be a measurable function. The statement of Theorem 5.1 still holds if we replace the function B(t), t ∈ [0, T ], with the function B(t) = max{ u x i (t) q i , β(t) : i = 1, . . . , N }.

  is bounded. Hence, by our assumptions and since x(•) is a solution of (3.2), it can be extended, by continuity, on [t 0 , S]. Furthermore, γ(S) ≤ x(S) because γ is lower semicontinuous from the left. Applying the Zorn lemma and [13, Theorem 4.2], we extend this solution x(•) to the time interval [t 0 , T ].

t t 0 g(s)ds for all t ∈ [t 0 , S) implying that x(•)

  1 < t 2 < T , applying Proposition 2.1 at times t 1 and t

	By (i) and Fenchel's inequality (2.3) we get
	(4.2)
	1 2	u(t 2 ) 2 L 2 (Ω) -	1 2	u(t 1 ) 2 L 2 (Ω) +	ν 2	t 2 t 1		Ω	G(|∇u|)dxds ≤	t 2 t 1	Ω	G 2 ν	|f (x, s)| dxds.
	Applying Hölder inequality we obtain, since p ≥ 2,
										p
	(4.3)					|u| 2 dx	2	≤ c	|∇u| p dx
						Ω					Ω
	where c > 0 depends only on p, N, |Ω|.
	Now, let F be the convex function as in the claim of Lemma 2.2. Since F is positive
	and F (0) = 0, it is nondecreasing on R + . From (4.3) we deduce that
									p
	(4.4)			F	|u| 2 dx	2			≤ F c
					Ω
													2 and subtracting
	the results we get
	1 2	u(t 2 ) 2 L 2 (Ω) -	1 2	u(t 1 ) 2 L 2 (Ω) +	t 2 t 1	Ω	A(x, s, ∇u), ∇u dxds =	t 2 t 1	Ω	f, ∇u dxds.

Ω |∇u| p dx .

In what follows C denotes a positive constant depending only on N, p, k, |Ω|, which may vary from line to line. From (2.5), (2.12), (4.4) and the Hölder and Jensen inequalities we deduce that for any t ∈ [0, T ],

  2 L 2 (Ω) = 0. Remark 4.1. Theorem 1.1 follows trivially from Theorem 4.1 and Theorem 4.2. Remark 4.2. If G(s) = s p p , p ≥ 2,, as observed in Section 3, the statement of Theorem 4.1 improves the behaviour in time of a solution to Problem (2.1) for a p-Laplace type operator (see

L 2 (Ω)

Acknowledgments

The authors are grateful to the referee for the constructive comments that helped to improve the presentation of the results. The research was partially supported by CNRS when G. Moscariello were visiting the center. Gioconda Moscariello was partially supported by Ministry of Education, University and Research PRIN 2017 and she is member of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of INdAM.


Notions of Utility: Construction of the B2 foreign language level of competence in Europe and France

Laura M. Hartwell

For the last several decades, the Council of Europe has encouraged foreign language learning. Beginning in the 1980s, the Council of Europe's stated political objectives link language teaching and learning to greater respect, tolerance, mutual understanding, as well as independence of thought, judgment, and action. Today, [START_REF] Räisänen | ESP teaching and learning in Western European higher education[END_REF] comment that the Common European Framework established the definition of language competence to include linguistic, sociolinguistic, and pragmatic aspects within the four language activities of reception, production, interaction, and mediation, and this within personal, occupational, and educational domains.

Although the Council of Europe's language objectives extend beyond questions of language competence, an essential outcome is the ubiquitous Common European Framework of Reference (CEFR). The CEFR has become a source of reference for numerous institutions in primary, secondary, and higher education across Europe. The six levels of competence described in the CEFR were created by inventorying evaluation criteria gathered from teachers from several European countries and have in turn influenced new language teaching and assessment projects. [START_REF] Jones | European Language Policy: Assessment, Learning, and the CEFR[END_REF] underscore the impact of the movement over the last decade:

Certainly the publication of the CEFR and the subsequent preliminary pilot version of a manual for relating language examinations to it (Council of Europe, 2003) has stimulated a great deal of work by language testers. It has led to increasing insistence by governments and other test users on proof of alignment, and a rush on the part of testers to provide it. This influence is set to increase. (Jones and Saville, 2009, p. 53) The CEFR model divides foreign language knowledge and use into six categories ranging from basic (level A1) to near-native proficiency (level C2). The focus of this study is the construction, across Europe, of an ensemble of descriptors and characteristics of the fourth or B2-level of language competence or above. In France, the CEFR has been integrated into the programs of the Éducation nationale and has served as a reference for the creation and implementation of the Certificat en langues de l'enseignement superieur (CLES). Future primary and secondary school teachers in France will be required to obtain this certificate at the B2-level before attaining the status of a civil servant (cf. JORF n°0123 du 31 mai 2010).

The aim of this paper is, given the stated political objectives of the Council of Europe, how do the criteria of European CEFR-based assessment projects and the CLES fulfill Mill's objectives of Utility. According to Mill's theory of life, as described in his seminal work Utilitarianism, pleasure and the absence of pain are the only desirable ends of any action. Desirable things are those that lead, either directly or indirectly, to pleasure. This, with the absence of pain, is at the heart of his view of moral standards:

The creed which accepts as the foundation of morals, Utility, or the Greatest Happiness Principle, holds that actions are right in proportion as they tend to promote happiness, wrong as they tend to produce the reverse of happiness. By happiness is intended pleasure, and the absence of pain; by unhappiness, pain, and the privation of pleasure. To give a clear view of the moral standard set up by the theory, much more requires to be said; in particular, what things it includes in the ideas of pain and pleasure; and to what extent this is left an open question. But these supplementary explanations do not affect the theory of life on which this theory of morality is grounded-namely, that pleasure, and freedom from pain, are the only things desirable as ends; and that all desirable things (which are as numerous in the utilitarian as in any other scheme) are desirable either for the pleasure inherent in themselves, or as means to the promotion of pleasure and the prevention of pain. (Mill, 1863) In this paper, we will first describe the Council of Europe's stated political objectives and examine how they correspond to tenets of Utility. Second, we will examine how CEFR-based descriptors of foreign language competence sidestep the questions of justice or tolerance that are found both in Mill's theory of life and the Council of Europe objectives. Then, we will consider how the CLES occasionally addresses issues of justice or tolerance, but does not incorporate literary aspects of language, which are integral to pleasure and hence Utility. The aim of this study is not the validity of language teaching of assessment based on the CEFR, but the theory of life or moral standards, as Mill evoked, that transpires through its descriptors and in turn impacts teaching and assessment. In other words, the very question of what one does or is expected to do with foreign languages.

Pleasure within Mill's definition of Utility, or the Greatest Happiness Principle

In 1828, the Webster's Revised Unabridged Dictionary defined utility as "Usefulness; production of good; profitableness to some valuable end; as the utility of manures upon land; the utility of the sciences; the utility of medicines" (Webster's Revised Unabridged Dictionary, 2011). This definition highlights the immediate and long-range positive aspects of utility.

However, the examples given within the domains of agriculture and the sciences do not foretell the notion of morals and justice that Mill would link to this term in his work Utilitarianism (Mill, 1863). For Mill, this definition is inadequate in that the notion of pleasure is absent:

Having caught up the word utilitarian, while knowing nothing whatever about it but its sound, they habitually express by it the rejection, or the neglect, of pleasure in some of its forms; of beauty, of ornament, or of amusement. Nor is the term thus ignorantly misapplied solely in disparagement, but occasionally in compliment; as though it implied superiority to frivolity and the mere pleasures of the moment. And this perverted use is the only one in which the word is popularly known, and the one from which the new generation are acquiring their sole notion of its meaning. (Mill, 1863) Mill's notion of Utility is broad as beauty, ornament, and amusement are integrated and not superfluous components of Utility. The popular conception of the word utility associated with the practical advantages of manure or medicine is a restrictive one. Mill conceives of pleasure to be a desirable objective, for his notion of Utility is also that of the Principle of Greatest Happiness.

The Council of Europe's objectives

The 2001 book, Common European Framework of Reference for Languages: Learning, Teaching, Assessment clearly states that the Common European Framework of Reference is not set out to be a "normalization" of language teaching. The Council of Europe objectives are more far-ranging. There is a political agenda of cohesion behind the recommendation of teaching foreign languages. During a First Summit of Heads of State, xenophobia and ultranationalism were singled out as threats to European mobility, integration, stability, and democracy. So, the promotion of greater respect, tolerance, and mutual understanding became stated objectives as necessary to the construction of Europe. During a later Summit, Heads of State continued to link foreign language teaching to the construction of Europe by giving priority to promoting independence of thought, judgment and action, as well as social skills, through modern language teaching.

Along with multilingualism and plurilingualism, the Council of Europe recommends that national and regional literatures be protected and developed. It lists the many educational purposes to teach these literatures, including aesthetic, but also intellectual, moral, emotional, linguistic, and cultural. Aesthetic uses of oral or written language include a range of activities, such as singing pop songs, rewriting stories, performing scripted plays, and listening to opera.

The Council of Europe's seminal book recognizes that national and regional literatures have traditionally been a major aspect of modern language studies in upper secondary and higher education and considers them "a valuable common resource to be protected and developed".

The respect of identities and cultural diversity is equally a stated objective (Council of Europe, 2001, p. 56).

As we will see, a commonly mentioned objective is "to equip all Europeans for the challenges of intensified international mobility and closer co-operation not only in education, culture and science but also in trade and industry". This objective is of particular concern in higher education where professional objectives are of immediate interest for many institutions and students.

European conception of the B2-level

New European language assessments are generally based on a six-level framework. The beginning level is A1, followed by A2. The highest level approaches a native level and is called C2. In France, teachers and counselors will be required to be certified at a B2 level, which is also the level that students are expected to attain when leaving lycée. The levels are based on a collection of various assessment projects found in Europe and are not based on any given theory of language acquisition. According to Fulcher and Davidson (2009, p. 126), the CEFR is one of the most well-known models and it "attempts to be encyclopaedic, by adding details of possible contexts for communication and sometimes performance conditions". Today, many language assessment organizations claim that their certificates are benchmarked on this Common European Framework, such as the Test of English for International Communication (TOEIC) (Chauncey Groupe Europe, 2001) or the The International English Language Testing System (IELTS) (Cambridge University, British Council, and IELTS Australia, 2011) David North has compiled the descriptors of high-level language performance (B2+ to C2) from 13 European projects or guidelines of language assessment (Appendix A). Some are from specific countries, including France, but other projects are the result of the participation of members from multiple countries. Certain projects are called Portfolios and are collections of language artifacts and summaries of performances. Among the 13 projects we find a total of 304 descriptors, such as "Can follow the essentials of lectures, talks and reports and other forms of academic / professional presentation which are propositionally and linguistically complex" Hartwell, L. M. (2010). Notions of utility: Construction of the B2-foreign language level of competence in Europe and France. Dans Baillé, J., Compeyron, A. et al. (dirs), Du mot au concept : . Grenoble: Presses Universitaires de Grenoble.

or "Good grammatical control; occasional "slips" or non-systematic errors and minor flaws in sentence structure may still occur, but they are rare and can often be corrected in retrospect" (CEFR).

The majority of the descriptors are clearly identified in categories such as oral or written comprehension and oral or written expression. As can be seen in Table 1, oral interaction is more frequently mentioned than written interaction, but in general, there is a fairly equal attention given to both oral and written language skills. This approach of separating language into competencies has become a very standard approach to language assessment, especially task-based assessment (see [START_REF] De Ketele | De la nécessité et de la relativité de la notion de seuil en évaluation[END_REF].

Descriptors Subtotals

Oral In order to evaluate language skills, evidence is most often collected from candidates following a task or a set of tasks. A score or label is then determined so that others, such as university admissions offices or human resource services, can easily compare or categorize language skills. The definition of an ability and how the score is derived is the construct. According to Bachman and Palmer (1996, p. 21), the term construct validity refers to "the extent to which we can interpret a given test score as an indicator of the ability(ies), or construct(s), we want to measure." Selected tasks linked to the language competencies mentioned in The empirical validation in language testing is to ensure the defensibility and fairness of interpretations based on test performance.

[…] Validation of the evidential basis of testscore interpretation involves thinking about the logic of the test, particularly its design and its intentions, and also looking at the empirical evidencethe hard factsemerging from data from test trials or operational administrations. (McNamara, 2000: 33) The issue raised by this paper is how the "design and its intentions" of certain European language descriptors and the CLES correspond to stated European political objectives and Mill's Theory of Utility.

Language assessment is comprised of several characteristics that [START_REF] Bachman | Language testing in practice[END_REF] regroup under the concept of usefulness. They posit that the notion of overall usefulness should be maximized over the test's individual interrelated qualities. These qualities, including reliability, construct validity, authenticity, interactiveness, practicality, and impact, should not be evaluated independently, but should be determined related to the specific testing situation.

Although these authors concentrate more specifically on how a specific testing context may impact students, they posit that testing impacts societal and educational value systems. The focus of this paper is the impact of how Mill's theory of Utility and the Council of Europe's value systems are manifested within the descriptors of thirteen European projects and the French CLES.

It is important to examine testing criteria as the washback on teaching may include the impact on teaching method, curriculum, and materials, and remains "a far more complex and thorny" issue then simply the effect of testing on teaching (Bachman and Palmer, 1996, p. 31)

The Construction of context within European descriptors

We have already seen that the descriptors of high-level competencies from the thirteen evaluation projects are fairly evenly distributed between oral and written competencies.

However, these descriptors also refer to the context or type of language use targeted. In order to examine the context of how language is used, keywords related to context were pulled from the 304 descriptors (Appendix 2). Some descriptors contained more than one keyword, such as "I can readily appreciate most narratives and modern literary texts (e.g., novels, short stories, poems, plays)" (CERCLES).

Keywords or expressions found in the descriptors and the descriptor titles were analyzed in order to identify the context of purpose of language use. The descriptors were then classified in one of four categories: academia and the professions, literature and culture, pragmatic, and finally aesthetics. Some descriptors include more than one keyword. If a keyword was found more than once within one descriptor and its title, it was only included in the analysis only one time. Certain keywords or expressions, such as style or keeping the floor (i.e. maintain attention when speaking) could be listed in more than one category, as appropriate style can be, for example, an advantage within the professional realm. In this case, the keyword or expression was coded in the most applicable category.

Table 2 displays the number of descriptors per category, the number of keywords per category, the number of times keywords are mentioned per category, and the top five keywords per category in order of frequency. We see that academia or the professions are represented by the greatest number of keywords. That is to say that the seventeen keywords are cited 134 times within 86 descriptors. Second, literature and cultural are evoked in greatest frequency, these descriptors are of high-level language performance, so among the forty-two keywords, we find references to idioms or colloquialisms that are not present at a B2 level or lower. The language descriptors studied here include a limited range of keywords related to literary realms, such as "short story" cited three times and "novel", cited only once (Appendix 2).

The descriptors of the thirteen evaluation guidelines catalogued by North include some 67 references to twenty keywords related to pragmatic issues in a wide range of contexts and this category includes the greatest range of different keywords, including announcements, health, legal, pain, show visitors, and useful, which is cited five times. This word selection directly reflects the objective of the Council of Europe to develop the ability of Europeans to communicate with each other across linguistic boundaries. Finally, the smallest category "aesthetics" includes notions of style or humor, especially as it relates to oral or written expression. However, the capacity to converse does not imply the capacity to respect others or to be tolerant.

Domain

Number of descriptors

Number of keywords

Number of citations

Top five keywords in order of frequency

The stated European objectives of respect, tolerance or mutual understanding are not reflected within these descriptors. These themes mayor may notbe found in the materials chosen by the instructor, the educational institution, or the certification establishments. However, it should be noted that these themes are not reflected in the assessment criteria of the descriptors of the thirteen European projects. Although descriptors evoke the necessity to adopt an appropriate or flexible manner, these remain limited to more pragmatic or undefined circumstances:

Can write a letter dealing with financial matters in an appropriate manner which will encourage a positive outcome, e. Mill posits that the "entire history of social improvement" is an evolution in increments, for which an awareness of injustice is a mandatory first step:

All persons are deemed to have a right to equality of treatment, except when some recognised social expediency requires the reverse. And hence all social inequalities which have ceased to be considered expedient, assume the character not of simple inexpediency, but of injustice, and appear so tyrannical, that people are apt to wonder how they ever could have been tolerated; forgetful that they themselves perhaps tolerate other inequalities under an equally mistaken notion of expediency, the correction of which would make that which they approve seem quite as monstrous as what they have at last learnt to condemn. The entire history of social improvement has been a series of transitions, by which one custom or institution after another, from being a supposed primary necessity of social existence, has passed into the rank of a universally stigmatised injustice and tyranny. So it has been with the distinctions of slaves and freemen, nobles and serfs, patricians and plebeians; and so it will be, and in part already is, with the aristocracies of colour, race, and sex. (Mill, 1963) Hartwell, L. M. ( 2010). Notions of utility: Construction of the B2-foreign language level of competence in Europe and France. Dans Baillé, J., Compeyron, A. et al. (dirs), Du mot au concept : .

Grenoble: Presses Universitaires de Grenoble.

146

In other words, an attention to social inequalities is necessary before their eradication. Other notions related to the stated linguistic and political objectives of the Council of Europe could be added to our assessment criteria, such as:

Can write a letter dealing with social matters in an appropriate manner which will encourage a positive outcome, e.g. write a letter to a newspaper complaining forcefully yet tactfully about an issue of inequity.

I can present respect and tolerance in a very flexible manner in order to give emphasis, to differentiate and to eliminate ambiguity.

A French interpretation of European guidelines: The Certificat en Langues de l'Enseignement Supérieur

Based on the CEFR, the CLES is not based on any language acquisition theory, but encompasses a range of languages, including, at the B2-level: English, Spanish, German, Italian, and Portuguese. Despite a massive increase in the number of candidates passing the CLES at the B2-level, English remains the language chosen by over three-quarters of the candidates (Table 3). The CLES exam covers four language competencies: listening, reading, writing and oral interaction. Each section is eliminatory, which makes the certificate especially difficult.

English

Nationally, less than half the candidates obtain the certificate. However, there are differences according to language chosen, year of study, and academic program (see [START_REF] Olive | CLES : Bilan statistiques[END_REF]2010).

The CLES includes general or academic topics, such as electronic games, traffic signs, or waste management and requires the candidates to defend a point of view based on information from oral and written sources about the given subject (Official CLES website, 2010 andHartwell, 2010). These topics are not directly linked to reflecting on greater respect, tolerance, mutual Hartwell, L. M. (2010). Notions of utility: Construction of the B2-foreign language level of competence in Europe and France. Dans Baillé, J., Compeyron, A. et al. (dirs), Du mot au concept : . Grenoble: Presses Universitaires de Grenoble.

understanding as evoked by the Council of Europe's stated political objectives. However, the CLES subjects, such as the Glass Ceiling or Homophobia in Schools, touch upon notions of respect and tolerance. The CLES requires students to "use information from the documents [they] have been studying to discuss, negotiate, and reach a compromise according to the role assigned" in ten minutes (Official CLES website, 2010). The CLES topics are questions of society that have yet to be resolved despite years or perhaps decades of public debate. A candidate who feels strongly about the topic may find it very difficult to negotiate a compromise or defend a point of view that is contrary to their own. Before the introduction of the CLES, the competitive exam to become a state-employed primary school teacher included a language exam for which a low score could be compensated by a high score in another subject, such as mathematics or history. That is no longer the case. The first effect of introducing a language certificate is the transfer of the cost of the exam from the Éducation National to the university-funded CLES center or, in some cases to the candidates.

Second, it has the capacity to reduce the number of permanent primary school teachers, as candidates may pass the state competitive exam, but without obtaining the necessary CLES certificate. Furthermore, although the former competitive exam included a section on language didactics, this is not present in the CLES. Therefore, students preparing to become primary school teachers may privilege the language capacities necessary to pass the CLES, before studying foreign language didactics or foreign language and literature for young learners.

All notions of beauty, ornament, amusement, common qualities of literary works and also components of Mill's Theory of Utility, are absent from the CLES. Furthermore, Mill offers a definition of a "cultured mind" that comes from "being taught". He posits that teaching leads to an endless intellectual attraction to many subjects, from natural phenomena to historical events:

A cultivated mind -I do not mean that of a philosopher, but any mind to which the fountains of knowledge have been opened, and which has been taught, in any tolerable degree, to exercise its faculties-finds sources of inexhaustible interest in all that surrounds it; in the objects of nature, the achievements of art, the imaginations of poetry, the incidents of history, the ways of mankind, past and present, and their prospects in the future. It is possible, indeed, to become indifferent to all this, and that too without having exhausted a thousandth part of it; but only when one has had from the beginning no moral or human interest in these things, and has sought in them only the gratification of curiosity. (Mill, 1863) The CLES culminates with a task constructed on negotiating a compromise, which is a social language-related skill encouraged by the Council of Europe's stated political objectives for Modern Language teaching. But primary school teachers would be better equipped by foreign language classes that are oriented toward "fountains of knowledge", including the "imagination of poetry" and the "achievements of the arts" as Mill evoked. Yet, literary or fictional texts are completely absent from the CLES assessment project Approximately 80% of candidates passing the CLES at the B2-level chose English. So, in practice, general or communication skills in English centered on general topics are likely to come to the forefront in many language learning settings for future teachers. The transversal nature of CLES subjects are intended to be accessible by candidates emanating from a range of academic fields, but primary school teachers also need to be introduced to literary works in a foreign language. Literature and fiction have traditionally offered a commentary on current issues. Incorporating literary works into the CLES would broaden the scope to include beauty and ornament, elements of Utility.

The capacities to synthesize and negotiate in a foreign language are important skills and reflect stated Council of Europe objectives. However, these capacities are not those most needed by future primary school teachers whose mission should include "cultivating minds" as Mill insists, and, as part of foreign language teaching and learning, greater respect, tolerance, mutual understanding, and independence of thought, judgment, and action as recommended by the Council of Europe.

Conclusion

According to Mill, the creed of Utility, or the Principle of Greatest Happiness, deems that actions are right if they tend to promote happiness and prevent pain. Mill links this not with greater professional efficiency, but equality of treatment, justice, and a cultivated mind. The stated political objectives of the Council of Europe also point to these objectives. In practice, though, there is a discordance between stated Council of Europe objectives and the manifestations found in both the language descriptors found across Europe and the French Certificat en Langues de l'Enseignement Superieur.

Foreign language assessment projects are and should remain under constant scrutiny in order to correspond to evolving stakeholder and societal needs and objectives. Notions of tolerance, respect, and justice could be added as a criterion to CEFR-based teaching or assessment projects, alongside existing pragmatic, occupational, and educational considerations. Hartwell, L. M. (2010). Notions of utility: Construction of the B2-foreign language level of competence in Europe and France. Dans Baillé, J., Compeyron, A. et al. (dirs)

  g. write a letter to a bank complaining forcefully yet tactfully about a delay in transferring money. (C2-level, ALTE, 2003) I can present ideas and viewpoints in a very flexible manner in order to give emphasis, to differentiate and to eliminate ambiguity. (ELP: Switzerland: European Language Portfolio. Version for Young People and Adults (15+), 2000).

Table 1 . B2+ to C2 Language descriptors found in 13 European projects.

 1

	comprehension	45
	Oral interaction	55	156
	Oral expression	56
	Written comprehension	59
	Written interaction	17	127
	Written expression	51
	General	21	21

Table 1

 1

	can be

  Hartwell, L. M. (2010). Notions of utility: Construction of the B2-foreign language level of competence in Europe and France. Dans Baillé, J.,Compeyron, A. et al. (dirs), Du mot au concept :.

Grenoble: Presses Universitaires de Grenoble. 144

Table 2 . Descriptors and keywords related to language use context.

 2 Hartwell, L. M. (2010). Notions of utility: Construction of the B2-foreign language level of competence in Europe and France. Dans Baillé, J.,Compeyron, A. et al. (dirs), Du mot au concept :. Grenoble: Presses Universitaires de Grenoble.145Although there are domains as heteroclite as tenancy or exports, keywords such as respect, tolerance, or identity are absent. The linguistic and cultural complexity of Europe is alluded to in the thrice-mentioned term socio-inter-cultural. Issues of social, personal, cultural, or intercultural are evoked, as in the following descriptors:

	Can converse comfortably and appropriately, unhampered by any linguistic limitations in
	conducting a full social and personal life. (CEFR, 2001)
	I can participate effectively in extended discussions and debates on subjects of personal,
	cultural, intercultural or social interest (Ireland: European Language Portfolio. Model for
	learners in post-primary education, 2001).

Table 3 . Number of CLES2 (B2-level) candidates by language and success rate.

 3

,

  . Grenoble: Presses Universitaires de Grenoble.

	Academic and Professional	report	24
	Aestetic	style	22
	Academic and Professional	lecture	21
	Academic and Professional	field	19
	Culture and Literature	literary	17
	Culture and Literature	idiom /atic	16
	Academic and Professional	work	14
	Academic and Professional	profession/al	13
	Academic and Professional	academic	11
	Culture and Literature	colloquial	11
	Pragmatic	service	11
	Culture and Literature	cultur e/al	10
	Pragmatic	instructions	10
	Academic and Professional	technical	9
	Pragmatic	conclusion	9
	Culture and Literature	film	7
	Pragmatic	announcement	6
	Aestetic	allusion	6
	Aestetic	humour	6
	Academic and Professional	meeting	5
	Pragmatic	legal	5
	Pragmatic	useful /ness	5
	Culture and Literature	play	4
	Academic and Professional	scientific	3
	Academic and Professional	colleague	3
	Academic and Professional	vocational	3
	Culture and Literature	short stories	3
	Culture and Literature	context	3
	Culture and Literature	socio-inter-cultural	3
	Pragmatic	keep the floor	3
	Pragmatic	tenancy	3
	Aestetic	elaborate	3
	Aestetic	pun	3
	Academic and Professional	expertise

Preserving pluriculturalism could be supported by the incorporation of documents written in English in non-anglophone countries or translated documents within the English classroom. This type of integration, of course, would go against English teaching as it is known in France and as it is embodied in the CLES, the CAPES or the Agrégation. It would mean conceiving of English teaching, at least at the university level, as having a much wider perspective, such as that of Lettres Modernes, instead of the current concentration on British and US documents.

The risk is that an orientation towards professional and pragmatic topics do not necessarily lead to greater respect, tolerance, mutual understanding, and independence of thought, judgment, and action. Furthermore, a narrowing of foreign language study to English reduces the scope of cross-cultural awareness, unless English studies are conceived of as a language of communication, but also a path to accessing literary and cultural works from across Europe. It is worth taking a step back to examine the new road that we and our students are undertaking.


Vincent Andrieu
email: vincent.andrieu@gmail.com

Laurent Praly
email: laurent.praly@ensmp.fr

Alessandro Astolfi
email: a.astolfi@ic.ac.uk

‡ A Astolfi

An output

Introduction

We consider a system, with state X = (X 1 , . . . , X n ) in R n described by : Ẋ

= S X + Bu , y = X 1 , ( 1
)
where y is the output, S in R n×n is the left shift matrix defined as : S X = (X 2 , . . . , X n , 0) T , and B in R n is the vector defined by B = (0, . . . , 1) T

In the whole paper we refer to tools developed within the framework of homogeneity in the bi-limit, notion introduced and studied in (Andrieu et al., 2008, SICON). We give in Section 2 a brief summary and definition.

Selecting d 0 = 0 and arbitrary degrees 0 ≤ d ∞ < 1 n-1 , homogeneity in the bi-limit (see Definition 3 below) with degrees d 0 and d ∞ is obtained for system (1) provided the weights r 0 = (r 0,1 , . . . , r 0,n ) and r ∞ = (r ∞,1 , . . . , r ∞,n ) are :

r 0,i = 1 , r ∞,i = 1 -d ∞ (n -i) .
(2)

In (Andrieu et al., 2008, SICON), we have proposed an output feedback for system (1) given by : u = φ( X) , Ẋ = S X + B φ( X) + K(X 1 -X 1 ) ,

(3)

where X = (X 1 , . . . , Xn ) in R n , and K and φ are respectively a homogeneous in the bi-limit vector field and function with weights r 0 and r ∞ , and degrees d 0 and d ∞ (respectively 1 + d 0 and 1 + d ∞ ). Setting :

E = (e 1 . . . , e n ) T = X -X
the chain of integrator (1) with the controller (3) can be described by : Ẋ = S X + B φ( X) + K(e 1 ) Ė = S E + K(e 1 ) .

(4)

In (Andrieu et al., 2008, SICON), the design of K and φ is done recursively, following an observer / controller approach in such a way that there exists a homogeneous in the bi-limit Lyapunov function U of degree d U which time derivative along the trajectories of system (4) satisfies for some positive real number

c 1 : ˙ U ( X, E) ≤ -c 1 U ( X, E) + U ( X, E) d U +d ∞ d U .
(5)

In (Andrieu et al., 2008, IEEE-Tac), to synthesize a control law in the context of trajectory tracking, we combine this tool with dynamic scaling. To do so, we need a specific property on the Lyapunov function U . This property is a homogeneous in the bi-limit version of the one given in the seminal paper on high-gain dynamic scaling (Praly, 2003, equation (16)) or in (Krishnamurthy et al., 2003, Lemma A1) and also used in the context of observer design in (Andrieu et al., 2008, Automatica). Namely, given a diagonal matrix

D = diag(d 1 , . . . , d n ) ,
with d j > 0, the function φ and the vector field K have to be selected such that the associated Lyapunov function U satisfies (5), and also :

∂U ∂E ( X, E) D E + ∂U ∂ X ( X, E) D X ≥ c 2 U ( X, E) , ( 6
)
for some positive real number c 2 . This is summarized in the following Theorem which proof will constitute the main (only) contribution of this paper.

Theorem 1 Let d U be a positive real number satisfying d U ≥ 2 + d ∞ . There exists a homogeneous in the bi-limit function φ : R n → R with associated triples (r 0 , d 0 , φ 0 ) and (r ∞ , d ∞ , φ ∞ ) and a homogeneous in the bi-limit vector field K : R → R n , with associated triples (r 0 , d 0 , K 0 ) and (r ∞ , d ∞ , K ∞ ) and a positive definite, proper and C 1 function U : R 2n → R + , homogeneous in the bi-limit with associated triples (r 0 , d U , U 0 ) and (r ∞ , d U , U ∞ ), such that :

1. The functions U 0 and U ∞ are positive definite and proper and the functions ∂U ∂e j and ∂U ∂ X j are homogeneous in the bi-limit with approximating functions respectively ∂U 0 ∂e j , ∂U∞ ∂e j , ∂U 0 ∂ X j , and ∂U ∞ ∂ X j .

2. There exist two positive real numbers c 1 and c 2 such that ( 5) and ( 6) are satisfied.

The proof of this Theorem was omitted in (Andrieu et al., 2008, IEEE-Tac) due to space limitation and is given in Section 3. Section 2 gives some prerequisite needed to address this proof.

Some prerequisite

The proof of this Theorem needs some prerequisite. Indeed, we recall the definition of homogeneity in the bi-limit, introduced in (Andrieu et al., 2008, SICON), and give some related properties.

Given a vector r = (r 1 , . . . , r n ) in (R + /{0}) n , we define the dilation of a vector x in R n as

λ r x = (λ r 1 x 1 , . . . , λ r n x n ) T .
Definition 1 (Homogeneity in the 0-limit)

• A continuous function φ : R n → R is said homogeneous in the 0-limit with associated triple (r 0 , d 0 , φ 0 ), where r 0 in (R + /{0}) n is the weight, d 0 in R + the degree and φ 0 : R n → R the approximating function, respectively, if φ 0 is continuous and not identically zero and, for each compact set C in R n and each ε > 0, there exists λ * such that we have :

max x ∈ C φ(λ r 0 x) λ d 0 -φ 0 (x) ≤ ε ∀λ ∈ (0, λ * ]. • A vector field f = n i=1 f i ∂ ∂x i is said homogeneous in the 0-limit with associated triple (r 0 , d 0 , f 0 ), where f 0 = n i=1 f 0,i ∂ ∂x i , if, for each i in {1, . . . , n}, the function f i is homo- geneous in the 0-limit with associated triple (r 0 , d 0 + r 0,i , f 0,i ) 1 .
Definition 2 (Homogeneity in the ∞-limit)

• A continuous function φ : R n → R is said homogeneous in the ∞-limit with associ- ated triple (r ∞ , d ∞ , φ ∞ ) where r ∞ in (R + /{0}) n is the weight, d ∞ in R + the
degree and φ ∞ : R n → R the approximating function, respectively, if φ ∞ is continuous and not identically zero and, for each compact set C in R n and each ε > 0, there exists λ * such that we have :

max x ∈ C φ(λ r∞ x) λ d∞ -φ ∞ (x) ≤ ε ∀λ ∈ [λ * , +∞) . • A vector field f = n i=1 f i ∂ ∂x i is said homogeneous in the ∞-limit with associated triple (r ∞ , d ∞ , f ∞ ), with f ∞ = n i=1 f ∞,i ∂ ∂x i , if, for each i in {1, . . . , n}, the function f i is homogeneous in the ∞-limit with associated triple (r ∞ , d ∞ + r ∞,i , f ∞,i ).
Definition 3 (Homogeneity in the bi-limit) A continuous function φ : R n → R (or a vector field f ) is said homogeneous in the bi-limit if it is homogeneous in the 0-limit and homogeneous in the ∞-limit.

The following propositions are proved, or are direct consequences of results, in (Andrieu et al., 2008, SICON).

Proposition 1 Let η and µ be two continuous homogeneous in the bi-limit functions with weights r 0 and r ∞ , degrees d η,0 , d η,∞ and d µ,0 , d µ,∞ , and continuous approximating functions

η 0 , η ∞ , µ 0 , µ ∞ 1. The function x → η(x)µ(x) is homogeneous in the bi-limit with associated triples (r 0 , d η,0 + d µ,0 , η 0 µ 0 ) and (r ∞ , d η,∞ + d µ,∞ , η ∞ µ ∞ ).

If the degrees satisfy

d η,0 ≥ d µ,0 and d η,∞ ≤ d µ,∞
and the functions µ, µ 0 and µ ∞ are positive definite then there exists a positive real number c satisfying : The same result holds for the cases of homogeneity in the ∞-limit and in the bi-limit.

η(x) ≤ c µ(x) , ∀ x ∈ R n .
Proposition 3 Suppose η and µ are two functions homogeneous in the bi-limit, with weights r 0 and r ∞ , degrees d 0 and d ∞ , and such that the approximating functions, denoted η 0 and η ∞ , and, µ 0 and µ ∞ are continuous. If µ(x) ≥ 0 and

{ x ∈ R n \ {0} , µ(x) = 0 } ⇒ η(x) > 0 , { x ∈ R n \ {0} , µ 0 (x) = 0 } ⇒ η 0 (x) > 0 , { x ∈ R n \ {0} , µ ∞ (x) = 0 } ⇒ η ∞ (x) > 0 ,
then there exists a strictly positive real number k * such that, for all k ≥ k * , the functions

η(x) + k µ(x), η 0 (x) + k µ 0 (x) and η ∞ (x) + k µ ∞ (x) are positive definite.
3 Proof of Theorem 1

Main proof

The proof of this theorem is based on two other results. The first one is associated to the observer part of the output feedback controller and has already been stated in (Andrieu at al., 2008, Hal) in a more general version. The other one is associated to the state-feedback part and constitutes one of the contribution of this paper.

Theorem 2 ( (Andrieu et al., 2008, Hal))

Given d ∞ in [0, 1 n-1 ), let d W be a positive real number satisfying d W ≥ 2 + d ∞ .
There exist a vector field K : R → R n which is homogeneous in the bi-limit with associated weights r 0 and r ∞ , and a positive definite, proper and C 1 function W : R n → R + , homogeneous in the bi-limit with associated triples (r 0 , d W , W 0 ) and (r ∞ , d W , W ∞ ), such that the following holds.

1. The functions W 0 and W ∞ are positive definite and proper and, for each j in {1, . . . , n}, the function ∂W ∂e j is homogeneous in the bi-limit with approximating functions ∂W 0 ∂e j and ∂W ∞ ∂e j .

2. There exist two positive real numbers c 1 and c 2 such that we have, for all

E in R n , ∂W ∂E (E) (S E + K(e 1 )) ≤ -c 3 W (E) + W (E) d W +d ∞ d W , ( 7
) ∂W ∂E (E) D E ≥ c 4 W (E) , (8)
The proof of this Theorem has been given in (Andrieu et al., 2008, Hal).

The other Theorem needed to prove the main result can be stated as follows :

Theorem 3 Given d ∞ in [0, 1 n-1 ), let d V be a positive real number satisfying d V ≥ 2 + d ∞ .
There exist a function φ : R n → R which is homogeneous in the bi-limit with associated weights r 0 and r ∞ and degrees 1 and 1 + d ∞ , and a positive definite, proper and C 1 function V : R n → R + , homogeneous in the bi-limit with associated triples (r 0 , d V , V 0 ) and (r ∞ , d V , V ∞ ), such that the following holds.

1. The functions V 0 and V ∞ are positive definite and proper and, for each j in {1, . . . , n}, the function ∂V ∂X j is homogeneous in the bi-limit with approximating functions ∂V 0 ∂X j and ∂V∞ ∂X j .

2. There exist two positive real numbers c 1 and c 2 such that we have, for all

X in R n , ∂V ∂X (X) (S X + B φ(X)) ≤ -c 5 V (X) + V (X) d V +d ∞ d V , (9) ∂V ∂X (X) D X ≥ c 6 V (X) , ( 10
)
The proof of this Theorem is given in Section 3.2.

With these two theorems in hand the proof follows the line of the one given in (Andrieu et al., 2008, SICON) and is a direct application of Proposition 3. Indeed, consider a positive real number d ∞ in [0, 1 n-1 ), a diagonal matrix D and a sufficiently big positive real number d U . Employing Theorem 2 and 3 we get the existence of a homogeneous in the bi-limit vector field K, three homogeneous in the bi-limit functions φ, V and W and four positive real numbers c 3 , c 4 , c 5 and c 6 such that (7), ( 8), ( 9) and (10) are satisfied.

Let U : R 2n → R + be the candidate Lyapunov function for the closed loop system (4) defined as :

U ( X, E) = V ( X) + c W (E) , ( 11
)
where c is a positive real number to tuned.

Note that this function is homogeneous in the bi-limit with appropriate weights and degrees and with approximating functions

U 0 ( X, E) = V 0 ( X) + c W 0 (E) , U ∞ ( X, E) = V ∞ ( X) + c W ∞ (E) .
Moreover, due to the property of V and W exhibited in point 1) of Theorems 2 and 3, for all j in {1, . . . , n}, we get that the homogeneous approximating functions of ∂U ∂ X j are ∂U 0 ∂ X j and ∂U∞ ∂ X j , while the homogeneous approximating functions of ∂U ∂e j are ∂U 0 ∂e j and ∂U∞ ∂e j . Hence point 1) of Theorem 1 is satisfied.

Moreover, along the trajectory of the system (4), the function U satisfies :

˙ U ( X, E) = -η( X, E) -c µ(E) ( 12
)
where µ and η are two homogeneous in the bi-limit functions with weights (r 0 , r ∞ ) and degrees d U and d U + d ∞ defined as,

µ(E) = ∂W ∂E (E) (S E + K(e 1 )) , η( X, E) = ∂V ∂ X ( X) S X + B φ( X) + K(e 1 ) .
With ( 7), the function µ and its homogeneous approximations µ 0 and µ ∞ are negative definite. Furthermore, with (9), we have for all X = 0 :

η( X, 0) < 0 , η 0 ( X, 0) < 0 , η ∞ ( X, 0) < 0 .
Consequently, Proposition 3 yields the existence of a positive real number c * such that for all c ≥ c * the function ˙ U ( X, E) and its homogeneous approximation is negative definite. With Proposition 1, point 2), we conclude that there exists c 1 such that the inequality (5) is satisfied.

Finally, with ( 8) and ( 10) we get that the homogeneous in the bi-limit function

( X, E) → ∂U ∂E ( X, E) D E + ∂U ∂ X ( X, E) D X
and its homogeneous approximations are positive definite. Hence, employing Proposition (1) we get a positive real number c 2 such that (6) is satisfied.

Proof of Theorem 3

The proof we propose here is an adaptation of the one given in (Andrieu et al., 2008, SICON). It is done by induction. To do so we use notations with an index showing the value from which we stop counting. For instance X i = (X 1 , . . . , X i ) T denotes a state vector in R i and B i is a vector in R i defined as

B i = (0, . . . , 0, 1) T .
Also, S i is the left shift matrix of dimension i, i.e.

S i X i = (X 2 , . . . , X i , 0) T .
Proposition 4 Let d V be a positive real number satisfying d V ≥ 2 + d ∞ and d 1 , . . . , d i be i positive real numbers. Suppose there exist a C 1 homogeneous in the bi-limit function φ i : R i → R of degrees 1 and r ∞,i + d ∞ with approximating functions φ i,0 and φ i,∞ , and a positive definite, proper and C 1 function homogeneous in the bi-limit

V i : R i → R + , with associated triples (r 0 , d V , V i,0 ) and (r ∞ , d V , V i,∞
) such that the following holds :

1. for all j in {1, . . . , i}, the function ∂φ i X j is homogeneous in the bi-limit with approximating functions

∂φ i,0 X j and ∂φ i,∞ X j
2. the function V i,0 and V i,∞ are positive definite and proper and for all j in {1, . . . , i}, the functions ∂V i ∂X j are homogeneous in the bi-limit with approximating functions

∂V i,0 ∂X j and ∂V i,∞ ∂X j .
3. There exist positive real numbers c 5 and c 6 , such that for all X i in R i :

1 j=1 d j ∂V i ∂X j (X i ) X j ≥ c 5 V i (X i ) , ( 13
) ∂V i ∂X i (X i ) (S i X i + B i φ i (X i )) ≤ -c 6 V i (X i ) + V i (X i ) d V +d∞ d V (14)
Then, for any positive real number d i+1 , there exist a C 1 homogeneous in the bi-limit function φ i+1 : R i+1 → R, with degree 1 and r ∞,i+1 + d ∞ and a positive definite, proper and C 1 function V i+1 : R i+1 → R + homogeneous in the bi-limit with associated triples (r

0 , d V , V i+1,0 ) and (r ∞ , d V , V i+1,∞
) such that the following holds :

1. for all j in {1, . . . , i + 1}, the function ∂φ i+1 X j is homogeneous in the bi-limit with approximating functions

∂φ i+1,0 X j and ∂φ i+1,∞ X j .
2. The functions V i+1,0 and V i+1,∞ are positive definite and proper and for all j in {1, . . . , i+ 1}, the functions ∂V i+1 ∂X j are homogeneous in the bi-limit with approximating functions

∂V i+1,0 ∂X j and ∂V i+1,∞ ∂X j .
3. There exists a positive real number c5 and c6 such that for all

X i+1 in R i+1 , i+1 j=1 b j ∂V i+1 ∂X j (X i+1 ) X j ≥ c5 V i+1 (X i+1 ) . (15) ∂V i+1 ∂X i+1 (X i+1 ) (S i+1 X i+1 + B i+1 φ i+1 (X i+1 )) ≤ -c 6 V i+1 (X i+1 ) + V i+1 (X i+1 ) d V +d ∞ d V (16)
Proof : The proof is divided in three steps.

1. Construction of the Lyapunov function. Consider the candidate Lyapunov function V i+1 : R i+1 → R + defined by :

V i+1 (X i+1 ) = V i (X i ) + σ L(X i+1 ) ,
where σ is a positive real number which will be specified later and L is the function defined as :

L(X i+1 ) = |φ i (X i ) -X i+1 | d V + |φ i (X i ) -X i+1 | d V r ∞,i+1 .
The function V i+1 is positive definite and proper. Moreover, the function φ i (X i ) being C 1 and since d V ≥ 2, it yields that the function V i+1 is C 1 . Finally, for each j in {1, . . . , i + 1}, the function ∂V i+1 ∂X j is homogeneous in the bi-limit with associated triples r 0 , d V -1,

∂V i+1,0 ∂X j , r ∞ , d V -r ∞,j , ∂V i+1,∞ ∂X j
. Consequently Point 2) of Proposition 4 is satisfied.

2. Properties of the Lyapunov function. Let d i+1 be any positive real number. Let J : R i+1 → R be the function defined as :

J(X i , X i+1 ) = i+1 j=1 d j ∂L ∂X j (X i+1 ) X j .
The functions V i and J are homogeneous in the bi-limit with associated weights 1 and r ∞ , and unique degree d V . By assumption V i is positive definite and the same holds for its homogeneous approximations in the 0-limit and in the ∞-limit and we have :

J(0, X i+1 ) = d i+1 d V |X i+1 | d V + d V r ∞,i |X i+1 | d V r ∞,i , > 0 ∀X i+1 = 0 .
It follows that the assumptions of Proposition 3 are satisfied with µ = V i and η = J. Hence, with c 5 given in (13), there exists a positive real number σ such that the functions c 5 V i,0 + σJ 0 , c 5 V i,∞ + σJ ∞ and c 5 V i + σJ are continuous and positive definite in X i+1 . But then, from Proposition 1.2, there exists a positive real number c5 satisfying :

1 c5 [c 5 V i + σJ] ≥ V i+1 . Since assumption (13) gives readily, for all X i+1 in R i+1 , i+1 j=1 d j ∂V i ∂X j (X i+1 ) X j ≥ c 5 V i (X i ) + σ J(X i , X i+1 ) ,
we have established inequality (15) of Proposition 4.

3. Construction of the function φ i+1 . Given a real number σ, we define the homogeneous in the bi-limit function φ i+1 as :

φ i+1 (X i ) = -k (X i+1 -φ i (X i )) + (X i+1 -φ i (X i )) r ∞,i +d∞ r ∞,i ,
With Propositions 1 and 2 it is a homogeneous in the bi-limit function with degrees 1 and r ∞,i + d ∞ .

Moreover, with Proposition 2 and the properties of the function φ i , for all j in {1, . . . , i+1}, the function ∂φ i+1 X j is homogeneous in the bi-limit with approximating functions

∂φ i+1,0 X j and ∂φ i+1,∞ X j
. Hence point 1) of the Theorem is satisfied.

We show now that by selecting k large enough we can satisfy (16). Note that :

∂V i+1 ∂X i+1 (X i+1 ) [S i+1 X i+1 + B i+1 φ i+1 (X i+1 )] = T 1 (X i+1 ) -k T 2 (X i+1 ) ,
with the functions T 1 and T 2 defined as :

T 1 (X i+1 ) = ∂V i+1 ∂X i (X i+1 ) [S i X i + B i X i+1 )] T 2 (X i+1 ) = σ ∂L ∂X i+1 (X i+1 )φ i+1 (X i+1 ) ,
where,

∂L ∂X i+1 (X i+1 ) = d V (X i+1 -φ i (X i )) d V -1 + d V r ∞,i+1 (X i+1 -φ i (X i )) d V -r ∞,i+1 r ∞,i+1
φ i+1 (X i+1 ) .

By definition of homogeneity in the bi-limit and Proposition 2, T 1 and T 2 are homogeneous in the bi-limit with weights r 0 and r ∞ , and degrees d V . Moreover, since φ i+1 (X i+1 ) has the same sign as X i+1 -φ i (X i ), T 2 (X i+1 ) is non-negative for all X i+1 in R i+1 and as φ i+1 (X i+1 ) = 0 only if X i+1 -φ i (X i ) = 0 we get :

T 2 (X i+1 ) = 0 =⇒ X i+1 = φ i (X i ) . and,

X i+1 = φ i (X i ) =⇒ T 1 (X i+1 ) = ∂V i ∂X i (X i ) [S i X i + B i φ i (X i )] .
Consequently, equations ( 14) yield :

X i+1 ∈ R i+1 \ {0} : T 2 (X i+1 ) = 0 ⊆ X i+1 ∈ R i+1 : T 1 (X i+1 ) < 0 .
The same implication holds for the homogeneous approximations of the two functions at infinity and around the origin. Hence, by Proposition 3, there exists k * > 0 such that, for all k ≥ k * , we have for all X i+1 = 0 :

∂V i+1 ∂X i+1 (X i+1 ) [S i+1 X i+1 + B i+1 φ i+1 (X i+1 )] < 0 , ∂V i+1,0 ∂X i+1 (X i+1 ) [S i+1 X i+1 + B i+1 φ i+1,0 (X i+1 )] < 0 , ∂V i+1,∞ ∂X i+1 (X i+1 ) [S i+1 X i+1 + B i+1 φ i+1,∞ (X i+1 )] < 0 .

Proposition 2

 2 If φ : R n → R and ζ : R → R are homogeneous in the 0-limit functions, with weights r φ,0 and r ζ,0 , degrees d φ = r ζ,0 and d ζ in R + , and approximating functions φ 0 and ζ 0 , then ζ • φ is homogeneous in the 0-limit with weight r φ,0 , degree d ζ , and approximating function ζ 0 • φ 0 .

In the case of a vector field the degree d 0 can be negative as long as d 0 + r 0,i ≥ 0, for all 1 ≤ i ≤ n.

Employing Proposition 1, this implies that there exists c6 such that ( 16) is satisfied. Hence Proposition 4 is proved.

2

To construct the Lyapunov function V and the vector field φ, which prove Theorem 3, it is sufficient to iterate the construction proposed in Proposition 4 starting from

Note that with these datas, Assumptions of Proposition 4 are satisfied.

We apply this Proposition recursively for i ranging from 1 to n -1.

Conclusion

In this short paper we have given a proof to a result stated in (Andrieu et al., 2008, IEEE-Tac).