Bernhard H Haak
email: bernhard.haak@math.u-bordeaux1.fr

E L Maati Ouhabaz
email: elmaati.ouhabaz@math.u-bordeaux1.fr

EXACT OBSERVABILITY, SQUARE FUNCTIONS AND SPECTRAL THEORY

Keywords: . 1991 Mathematics Subject Classification. 93B07, 43A45, 47A60 Exact observability, admissibility, spectral theory of semigroups and groups, H ∞ functional calculus, square function estimates

. We can show that unless the spectum contains a halfplane, the BFC property occurs only in siutations where the underlying semigroup extends to a group. In a second part we present a sufficient condition for exact observability in Banach spaces that is designed for infinitedimensional output spaces and general strongly continuous semigroups. To obtain this we make use of certain weighted square function estimates. Specialising to the Hilbert space situation we obtain a result for contraction semigroups without an analyticity condition on the semigroup.

Introduction

In this article we study exact observability of linear systems (A, C) on Banach spaces of the form

   x ′ (t) + Ax(t) = 0 x(0) = x 0 y(0) = Cx(t)
We suppose throughout this article that -A is the generator of a strongly continuous semigroup T (t) t≥0 on a Banach space X. For details on semigroup theory used frequently in this article we refer to e.g. to the textbooks [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF][START_REF] Hille | Functional analysis and semi-groups[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]]. Since we deal with unbounded operators in general, we will note D(A) the domain of A and R(A) its range. Let Y be another Banach space and suppose that the observation operator C : D(A) → Y is bounded and linear when D(A) is endowed with the graph norm x D(A) = x + Ax . Here we denote by ˙ the norm of X. Since the observation operator C is generally unbounded, the concept of admissibility is introduced. It means that the output y of the system (usually measured in L 2 norm) depends continuously on the initial value x 0 .

Definition 1.1. We say that C is L 2 -admissible in time τ > 0 (for A or for T (t) t≥0 ) if there exists a constant M (τ ) > 0 such that sup

x∈D(A), x =1 τ 0 CT (t)x 2 Y dt =: M (τ ) 2 < ∞.
Definition 1.2. We say that C is exactly L 2 -observable for A (or for T (t)) in time η > 0 if there exists a constant m(η) > 0 such that inf

x∈D(A), x =1 η 0 CT (t)x 2 Y dt =: m(η) 2 > 0.
For more information the notion of admissible observation (or control) operators we refer the reader to the overview article [START_REF] Jacob | Admissibility of control and observation operators for semigroups: a survey, Current trends in operator theory and its applications[END_REF] or, both for admissibility and observability issues to the recent book [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF] and references therein. We summarise some well-known facts and notations: When there is no risk of confusion, 'admissible' means L 2 admissible in some finite time τ > 0 and 'exact observable' means exactly L 2 -observable for A in some finite time η > 0. We say that C is infinite-time admissible if M (∞) < ∞ and exactly observable in infinite time if m(∞) > 0. Finite-time admissibility does not depend on the choice of τ > 0. Nevertheless it turns out to be useful to study the (clearly non-decreasing) functions t → m(t) and t → M (t). In the 'dual' situation of a control operator B the quantity m(η) -1 is often referred to as control cost of a system. We refer e.g. to [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF][START_REF] Thomas | On uniform null controllability and blowup estimates, Control theory of partial differential equations[END_REF][START_REF] Tenenbaum | On the null-controllability of diffusion equations[END_REF] and references therein for more details.

Independence of the time τ > 0 of the notion of admissibility means that a lack of admissibility expresses either by M (τ ) = ∞ for all τ > 0 or by M (τ ) < ∞ for finite τ while M (∞) = ∞. On the other hand, a lack of exact observability expresses by m(η) = 0 for 0 < η < η 0 for η 0 ∈ (0, ∞]. We remark that Example 2.3 below satisfies m(η) = 0 for 0 < η < 2, m(2) = 1 while m(η) → +∞ for η → +∞. Since most parabolic equations like for example the heat equation are not exactly observable unless very special observations are chosen whereas exact observability appears frequently for hyperbolic systems such as the wave equation, it appears natural to study necessary spectral conditions of the generator -A that make exact observability possible or impossible. In this direction we extend and complete former results of [START_REF] Xu | Necessary conditions for the exact observability of systems on Hilbert spaces[END_REF]. We introduce the notion of backward-forward conditioning BFC-systems. These are admissible and exactly observable systems for which M (η) < m(τ ) for some η < τ . We analyse spectral properties of the generator -A of the semigroup of such systems. In particular, we prove that the approximate point spectrum of A is contained in a vertical strip. Therefore, the boundary of the spectrum is also contained in a strip. We prove in addition that if (A, C) is an admissible BFC-system such that the spectrum of A does not contain a half-plane then the semigroup actually extends to a group. Note that every bounded group with an admissible operator C is a BFC-system. Since (BFC) is a frequent property that typically is more likely to hold the more 'regular' the operators A and C are, this shows that exact observability is considerably rare outside the group context. A second part of this paper is devoted to a new sufficient criterion for exact observability. Under an assumption of square function type estimate we prove that a condition like

CA -α x Y ≥ δ x ,
implies exact observability. Here α ∈ (0, 1) and δ is a positive constant. Without any further assumption, we show that if -A is the generator of a contraction semigroup on a Hilbert space and C : D(A) → Y is such that CA -1 /2 x Y ≥ δ x , then (A, C) is exactly observable. In order to state and prove our criterion we make a heavy use of square function estimate of type

x 2 ≤ K 2 ∞ 0 (tA) -β (T (2t 2β ) -T (t 2β ))x 2 dt t ,
where K is a positive constant, β ∈ (0, 1) and T (t) denotes the semigroup generates by -A. In the case where β = 1 / 2 , this corresponds to a lower square function estimate

x 2 ≤ K 2 ∞ 0 ϕ(tA)x 2 dt t ,
where ϕ(z) := z -1 /2 (e -2z -e -z ). On Hilbert spaces, it is well known that such estimate is related to the holomorphic functional calculus of the operator A. The needed results on this functional calculus and associated square function estimates for sectorial operators will be sketched in the last two sections. As we will explain later our criterion applies for bounded analytic semigroups on Hilbert spaces whose generator admits a bounded H ∞ -calculus -but the first part of this paper reveals this to be impossible for a large class of systems unless A is bounded. One important aspect of the criterion that might also help in other situations is therefore how to avoid making use of analyticity assumption of the semigroup. We discuss at the end of this papers two examples.

BFC-systems

Let X and Y be Banach spaces with norms • and • Y , respectively. Throughout this section, (T (t)) t≥0 is a strongly continuous semigroup on X whose generator is denoted by -A.

Definition 2.1. An admissible observation operator

C for A is called zero-class admissible, if lim τ →0+ M (τ ) = 0.
Note that if the semigroup T (t) is bounded analytic with generator -A then for all α ∈ [0, 1 / 2 ), A α is zero-class admissible. This follows from the inequality A α T (t) ≤ M t -α . Consequently, if C is bounded on such a fractional domain space D(A α ) and thus

CT (t)x Y ≤ M [ A α T (t)x + T (t)x ] ,
C is zero-class admissible. It is also a obvious fact that every bounded operator C : X → Y is zero-class admissible. Here (T (t)) t≥0 is merely a strongly continuous semigroup on X. Consider now the linear operator Ψ τ : X → L 2 (0, τ ; Y ) defined by Ψ τ x = CT (•)x. Then admissibility (i.e., M (τ ) < ∞) means that Ψ τ is a bounded operator. If in addition m(τ ) > 0, then Ψ τ is injective and has closed range. Therefore, we may consider the operator Ψ τ :

X → R( Ψ τ ), Ψ τ = Ψ τ . We have (2.1) 1 m(τ ) = sup x∈D(A), x =1 x Ψ τ x = sup x∈D(A), x =0 x Ψ τ x = Ψ -1 τ .
We introduce the following definition.

Definition 2.2. We say that the system (A, C) has the backward-forward conditioning property or shortly that (A, C) is a BFC-system if there exists some 0 < η < τ such that C is admissible and exactly observable in time τ and if

(BFC) Ψ -1 τ Ψ η < 1.
The condition (BFC) is clearly a conditioning property for the output operator with different times η and τ which correspond to a backward and forward evolution of the system. It also follows from (2.1) that (BFC) is equivalent to

(2.2) M (η) < m(τ ) for some η < τ.
Therefore, if C is exactly observable in some time τ and of zero-class, then (2.2) holds trivially by letting η sufficiently small. Hence, the system is BFC. If C is admissible at any τ > 0 and if m(t) → +∞ for t → +∞, then (2.2) holds and again the system is BFC. Zero-class admissible operators are introduced and studied in [START_REF] Xu | Necessary conditions for the exact observability of systems on Hilbert spaces[END_REF]. See also [START_REF] Jacob | Zero-class admissibility of observation operators[END_REF] from which we borrow a concrete example leading to an BFC-system in which C is not zero-class.

       ∂ 2 ∂t 2 z(x, t) = ∂ 2 ∂x 2 z(x, t) for x ∈ (0, 1), t ≥ 0 z(0, t) = z(1, t) = 0 for t ≥ 0 z(x, 0) = z 0 (x) and ∂ ∂t z(x, 0) = z 1 (x) for x ∈ (0, 1) y(t) = ∂ ∂x z(0, t) We rewrite the system as a first order Cauchy problem    ∂ ∂t U = -AU (t), t ≥ 0 U (0) = (z 0 , z 1 ) CU (x, t) = ∂ ∂x f (0)
where

A = 0 -I -∂ 2 ∂x 2 0
and U = (f, g). The latter Cauchy problem is considered on the Hilbert space H = H 1 0 (0, 1) × L 2 (0, 1) endowed with the norm (f, g)

= 1 0 |f ′ | 2 dx + 1 0 |g| 2 dx.
Note that by the Poincaré inequality,

1 0 |f ′ | 2 dx defines a norm on H 1 0 (0, 1)
which is equivalent to the usual one. It is a standard fact that -A generates a strongly continuous semigroup T (t) on H. It is easy to see that A has compact resolvent and the eigenvalues are

λ n = -inπ, n ∈ Z \ {0} with normalised eigenfunctions U n (x) = sin(nπx)
inπ , sin(nπx) which form an orthonormal basis of H. Fix (f, g) ∈ H and denote by

α n = (f, g), U n H (the scalar product in H). Then (f, g) 2 = n∈Z,n =0 |α n | 2 and CT (t)(f, g) = n α n e inπt CU n = -i n α n e inπt .
Using the well known Ingham inequalities (see e.g. [22, p. 162] or [10, Theorem 4.3]) we obtain the following estimates for all τ > 2,

m(τ ) 2 n |α n | 2 ≤ τ 0 n α n e inπt 2 dt ≤ M (τ ) 2 n |α n | 2 with m(τ ) 2 ≥ 2τ π 1 - 4 τ 2 , M (τ ) 2 ≤ 8τ π 1 + 4 τ 2 .
This shows that C is admissible at any time τ > 0 and exactly observable in time τ > 2 with constant m(τ ) → +∞ as τ → +∞. This shows (2.2) and hence the system (A, C) is backward-forward conditioning. In order to see that C is not zero-class, we consider small τ > 0 and f ∈ H 1 0 (0, 1) with Fourier coefficients α n and note that

τ 0 n α n e inπt 2 dt = χ [0,τ ] f 2 2 ,
where

χ [0,τ ] denote the indicator function of [0, τ ].
From this equality it is clear that

1 = sup f 2=1 χ [0,τ ] f 2 2 = sup f 2=1 τ 0 n α n e inπt 2 dt = M (τ ) 2 .
Therefore, the right hand side does not converge to 0 as τ → 0.

Remark 2.4. The above example is a special case of the following situation: let C be admissible in some arbitrary time τ > 0 and exactly observable in some time

η > 0 for a group U (t) t∈R . Observe that x = U (-t)U (t)x ≤ U (-t) U (t)x whence U (t)x ≥ U (-t) -1 x . From nη 0 CU (t)x 2 dt = n-1 j=0 η 0 CU (t)U (jη)x 2 dt ≥ m(η) 2 n-1 j=0 U (-jη) -2 x 2 ,
we then infer m(nη) → +∞ for n → +∞ whenever the sum in the last expression diverges. This is in particular the case for bounded groups U (t) t∈R . By the admissibility of the system (A, C) one then obtains (BFC) by letting n sufficiently large.

We thank Hans Zwart for pointing out this remark to us.

Spectral properties of BFC-systems

We consider the same notation X, Y , A, (T (t)) t≥0 and C : D(A) → Y as in the previous section. Or aim here is to study spectral properties of BFC-systems. We will extend some results which have been proved in [START_REF] Xu | Necessary conditions for the exact observability of systems on Hilbert spaces[END_REF] in the context of zero-class operators. We note also that related ideas and results were obtained previously by Nikolski [START_REF] Nikolski | Rudiments de la thorie du contrle : vision operationnelle[END_REF] in the particular case of bounded observation operators C on X. Let us introduce the classical function ε :

R+ → R + defined by ε(t) := inf x =1 T (t)x .
It is clear that ε(t) is strictly positive for all t > 0 if this holds for a single t 0 > 0. Indeed, from

T (s) T (t)x ≥ T (t + s)x ≥ ε(t) T (s)x ≥ ε(t)ε(s) x one infers that ε(t) T (s) ≥ ε(t + s) ≥ ε(t)ε(s),
for all t, s ≥ 0. For this reason we distinguish the cases that ε(t) is strictly positive for all t > 0 of that it vanishes for all t > 0 and we note this by ε(t) > 0 or ε(t) = 0 respectively. The following lemma is essentially contained in [START_REF] Nikolski | Rudiments de la thorie du contrle : vision operationnelle[END_REF] and [START_REF] Xu | Necessary conditions for the exact observability of systems on Hilbert spaces[END_REF].

Lemma 3.1. If (A, C
) is an admissible and exactly observable BFC-system, then ε(t) > 0.

Proof. By the definition of BFC-system, there exist 0 < η < τ such that

δ := m(τ ) 2 -M (η) 2 > 0.
By the semigroup property,

m(τ ) 2 x 2 ≤ τ 0 CT (t)x 2 Y dt = η 0 CT (t)x 2 Y dt + τ -η 0 CT (t)T (η)x 2 Y dt ≤ M (η) 2 x 2 + M (τ -η) 2 T (η)x 2
which immediately yields

T (η)x 2 ≥ M (τ -η) -2 (m(τ ) 2 -M (η) 2 ) x 2 ≥ M (τ -η) -2 δ x 2 .
Therefore, ε(η) > 0, and hence ε(t) > 0 for all t > 0.

Lemma 3.2. Suppose that (A, C) is an admissible and exactly observable BFCsystem. Then T (t) * is injective for one (and thus all) t > 0 if and only if T (t) extends to a group on X.

Proof. We know by Lemma 3.1 that ε(t) > 0. This implies that T (t) is injective and has closed image for all t ≥ 0. Thus, T (t) is bijective if and only if T (t) * is injective. The latter is clearly independent of t > 0 by the semigroup law. Indeed, if T (t 0 ) * is injective for some t 0 > 0, so are all T (s) * for s < t 0 since T (t 0 ) * = T (t 0 -s) * T (s) * . If s > t 0 then we find n ∈ N, δ ∈ [0, t 0 [ such that T (s) * = (T (t 0 ) * ) n T (δ) * , and the injectivity of T (s) * follows from that of T (t 0 ) * and T (δ) * . We saw that T (t) * is injective for one (and thus all) t > 0 if and only T (t) is bijective which in turn by

S(t) := T (t) if t ≥ 0 T (t) -1 if t < 0 is equivalent to a group extension of T (t) on X.
For a closed operator S on X recall the notions of point spectrum

σ P (S) = λ ∈ C : ker(λI -S) = {0} , the approximate point spectrum σ A (S) = λ ∈ C : inf x∈D(S), x =1
λx -Sx = 0 and the residual spectrum

σ R (S) = λ ∈ C : range(λ -S) is not dense in X .
It is easy to see that σ R (S) = σ(S)\σ A (S). Of course, σ P (S) ⊆ σ A (S).

Proposition 3.3. Let (A, C) be an admissible and exactly observable BFC-system. Then there exist no approximate point spectrum of A with arbitrary large real parts.

In particular, if C is an admissible zero-class operator and A has a sequence of approximate point spectrum with arbitrary large real parts then C is not exactly observable.

Proof. Recall that exp(-tσ A (A)) ⊆ σ A (T (t)) (see [5, p. 276], note that -A is the generator). If we find a sequence λ n ∈ σ A (A) with Re(λ n ) → +∞, then e -tλn ∈ σ A (T (t)). Hence, inf x =1 T (t)x -e -tλn x = 0. By

ε(t) x ≤ T (t)x ≤ T (t)
x -e -tλn x + e -tRe(λn) x we get ε(t) = 0 which is incompatible with exact observability by Lemma 3.1.

Since -A is the generator of a strongly continuous semigroup, σ(A) is contained in a right-half plane. On the other hand, it is well known that the boundary of the spectrum ∂σ(A) is contained in σ A (A). We obtain from the previous proposition that Re(∂σ(A)) is bounded, i.e., ∂σ(A) is contained in a vertical strip. Thus Corollary 3.4. Let (A, C) be an admissible BFC-system. Then

Re(∂σ(A)) := {Re(λ) : λ ∈ ∂σ(A)} is bounded.
The following lemma is known. Lemma 3.5. Let S ∈ B(X) satisfy Sx ≥ γ x for some γ > 0 and all x ∈ X and assume 0 ∈ σ(S). Then there exists δ > 0 such that B(0, δ) ⊆ σ R (S).

The main result in [START_REF] Xu | Necessary conditions for the exact observability of systems on Hilbert spaces[END_REF] which states that if C is zero-class admissible and σ R (A) is empty, then T (t) extends to a group. The next propositions extend this result. Proposition 3.6. Let (A, C) be an admissible BFC-system. If Re(σ R (A)) := {Reλ : λ ∈ σ R (A)} is bounded, then (T (t)) t≥0 extends to a group on X.

Proof. We know by Lemma 3.1 that ε(t) > 0. If T (t) was not boundedly invertible for some t > 0, then 0 ∈ σ(T (t)). By Lemma 3.5, there exists δ t > 0 such that B(0, δ t ) ⊆ σ R (T (t)). Since σ R (T (t))\{0} = exp(-tσ R (A)) (see [5, p. 276]) we obtain B(0, δ t )\{0} ⊆ exp(-tσ R (A)), Therefore, there exists a real sequence (λ n ) ∈ σ R (A) such that Reλ n → +∞. This contradicts the assumption. Proposition 3.7. Assume that (A, C) is an admissible BFC-system. If σ(A) does not contain a half-plane then (T (t)) t≥0 extends to a group on X.

Proof. By Corollary 3.4 we see that σ(A) is either contained in vertical strip or contains a half-plane. Now we apply Proposition 3.6 to conclude.

Considering the right shift semigroup T (t) on L 2 (R + ) with the identity observation C = Id provides an example of a BFC-system (even a zero-class admissible one, see [21, Remark 3.1]) for which no group extension is possible. In this example, it is not difficult to check the spectrum of A satisfies σ(A) = σ R (A) = C + (the right half-plane). This shows that the spectral condition in Proposition 3.7 cannot be omitted. Assume that (A, C) is an admissible BFC-system. If T (t) is analytic, differentiable or merely eventually continuous then by [5, p. 113] σ(A) does not contain a half-plane. Thus, we conclude by Proposition 3.7 that (T (t)) t≥0 extends to group on X. Proposition 3.8. Assume that (A, C) is an admissible BFC-system. If T (t) is compact for some t > 0, then X has finite dimension.

Proof. If T (t) is compact for some t > 0 then σ(A) = σ P (A) is discrete. It follows from Proposition 3.3 that σ(A) is bounded. We conclude by Proposition 3.6 that (T (t)) t≥0 extends to a group on X. Thus, I = T (t)T (-t) is compact on X and therefore X has finite dimension.

Sufficient conditions for exact observability

Our aim in this section is to derive conditions on C and A which imply exact observability. Our condition reads as follows

(4.1) CA -(1-β) x Y ≥ δ x for all x ∈ D(A) ∩ R(A).
Here β ∈ (0, 1) and δ > 0 are constants. Since we shall assume that A is injective, it may be convenient to understand (4.1) in the sense

CA -1 x Y ≥ δ A -β x . Of course, R(A) ∩ R(A β ) = R(A).
In the sequel we need some basic properties of the H ∞ functional calculus for sectorial operators. This functional calculus goes back to the work of McIntosh [START_REF] Mcintosh | Operators which have an H∞ functional calculus, Miniconference on operator theory and partial differential equations[END_REF]. More recent publications of the meanwhile rich theory can be found in [START_REF] Haase | The functional calculus for sectorial operators, Operator Theory[END_REF] or [START_REF] Peer | Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus, Functional analytic methods for evolution equations[END_REF] and the references given therein. We briefly sketch the needed results and definitions. Notice that if -A generates a bounded semigroup T (t) t≥0 , then A is sectorial of angle π / 2 by the Hille-Yosida theorem. Moreover, the semigroup is (bounded) analytic if and only if A is sectorial of angle < π / 2 . Let H ∞ (S ω ) denote the holomorphic and bounded functions on S ω and Let H ∞ (S ω ) denote the holomorphic and bounded functions on S ω that are continuous and bounded on S ω . We further consider the ideal

H ∞ 0 (S ω ) (respectively H ∞ 0 (S ω )) of all functions f ∈ H ∞ (S ω ) (re- spectively H ∞ 0 (S ω )) that allow an estimate |f (z)| ≤ M max(|z| ε , |z| -ε ). The class of H ∞ 0 (S ω
) functions admits a natural functional calculus for sectorial operators A of angle ω. Indeed, if f ∈ H ∞ 0 (S θ ) for some θ > ω and if Γ = ∂S θ denotes the orientated path with strictly decreasing imaginary part, the Cauchy integral

f (A) := 1 2πi Γ f (λ) R(λ, A) dλ
converges absolutely in norm and defines therefore a bounded operator f (A). If A has, say, dense range, one obtains then a functional calculus for all functions f ∈ H ∞ 0 (S θ ).

Definition 4.2. We say that A admits a bounded H ∞ (S ω ) (respectively H ∞ (S ω )) functional calculus if f (A) is a bounded operator on X and there exists a constant M such that

(4.2) f (A) ≤ M f ∞ for all f ∈ H ∞ (S θ ) and θ > ω (respectively for all f ∈ H ∞ (S ω )).
Let us mention that by an approximation argument, if (4.2) holds for all f ∈ H ∞ 0 (S θ ) then it holds for all f ∈ H ∞ (S θ ). Therefore, it is enough to check the validity of (4.2) for all f ∈ H ∞ 0 (S θ ) to obtain bounded H ∞ (S ω ) functional calculus. Similarly, if (4.2) holds for for all f ∈ H ∞ 0 (S ω ) we obtain a H ∞ (S ω ) functional calculus. We say that A admits upper square function estimates on S θ if there is a constant M > 0 such that

(4.3) ∀x ∈ X : ∞ 0 ϕ(tA)x 2 dt t ≤ M 2 x 2
for all ϕ ∈ H ∞ 0 (S θ ). In the same way we speak of lower square function estimates on S θ if one has (4.4) ∀x ∈ X :

x 2 ≤ K 2 ∞ 0 ϕ(tA)x 2 dt t
If X = H is a Hilbert space, then upper square function estimates for A and for A * for H ∞ 0 (S µ ) functions for all µ > ω are equivalent to a bounded H ∞ 0 (S µ ) functional calculus for all µ > ω. Moreover, by an approximate identity argument and a duality estimate, lower square function estimates for A follow from upper estimate of its adjoint A * . We will go into some details on the Hilbert space theory of the functional calculus in the last section and refer at this point to [START_REF] Mcintosh | Operators which have an H∞ functional calculus, Miniconference on operator theory and partial differential equations[END_REF][START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF] for more details. Before stating our first result of this section we discuss the following estimate for A

(SQ β ) x 2 ≤ K 2 ∞ 0 (tA) -β (T (2t 2β ) -T (t 2β ))x 2 dt
t that we will need to formulate the theorem. Here, K is a positive constant and β ∈ (0, 1). In case β = 1 / 2 , letting ϕ(z) := z -1 /2 (e -2z -e -z ), this corresponds to a lower square function estimate (4.4) for ϕ ∈ H ∞ 0 (Sπ /2 ). As mentioned above, (4.4) follows from H ∞ -functional calculus when X is a Hilbert space. We will discuss again this in the next section. Assume that -A is the generator of bounded strongly continuous semigroup on X and is injective. If we let ϕ(z) := z -β (e -2z -e -z ), then (SQ β ) can be seen as

x 2 ≤ K 2 ∞ 0 (tA) -β (T (2t 2β ) -T (t 2β ))x 2 dt t = K 2 ∞ 0 t 2β 2 -β ϕ(t 2β A)x 2 dt t (letting s = t 2β ) = K 2 2β ∞ 0 s -( 1 /2-β) ϕ(sA)x 2 ds s ,
i.e. as 'weighted' lower square function estimate for ϕ ∈ H ∞ 0 (Sπ /2 ).

By [START_REF] Haase | The functional calculus for sectorial operators, Operator Theory[END_REF]Theorem 6.4.6], the completion X1 /2-β,2 of X with respect to the seminorm

[x] β = ∞ 0 s -( 1 /2-β) ϕ(sA)x 2 ds s 1 /2
is independent of the choice of ϕ and coincides (with equivalent norms) with the real interpolation space:

(4.5) Ẋ-1 (A), Ẋ1 (A) 3 4 -β 2 ,2 = X1 /2-β,2 .
Here Ẋ-1 (A) is the completion of R(A) with respect to A -1 x and Ẋ1 (A) is the completion of D(A) with respect to Ax . From (4.5), it follows that (SQ β ) is equivalent to the continuous embedding

(4.6) Ẋ-1 (A), Ẋ1 (A) 3 4 -β 2 ,2 ֒→ X.
For the rest of this discussion, we assume for simplicity that A is invertible. In this case, Ẋ1 (A) = D(A) and Ẋ-1 (A) = X -1 . It is a known fact that the semigroup (T (t)) extends to a strongly continuous semigroup (T -1 (t)) on the extrapolation space X -1 , whose (negative) generator A -1 is an extension of A (see [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF] Chapter II, Section 5). In addition A is the part of A -1 on X and hence D(A 2 -1 ) = D(A) with equivalent norms. Indeed,

x ∈ D(A 2 -1 ) ⇔ x ∈ D(A -1 ) = X, A -1 x ∈ D(A -1 ) ⇔ x ∈ X, A -1 x ∈ D(A -1 ) ⇔ x ∈ D(A).
The fact that A -1 : X → X -1 an isometry implies that

Ax X = A 2 -1 x X-1 . Assume now that β < 1 2 . We have (X -1 , D(A)) 3 4 -β 2 ,2 = X -1 , D(A 2 -1 ) 3 4 -β 2 ,2 = D(A -1 ), D(A 2 -1 ) 1 2 -β,2
= (X, D(A)) 1 2 -β,2 ֒→ X. Note that the second equality follows from [19, p. 105]. Hence for β < 1 2 (4.7)

X1 /2-β,2 = (X -1 , D(A)) 3 4 -β
2 ,2 ֒→ X, which means that (SQ β ) always holds for β < 1 2 . Let us finally mention that for β > 1 2 , (SQ β ) never holds for non-negative selfadjoint operators with compact resolvent in infinite dimension separable Hilbert spaces. Indeed, consider such an operator A. The spectrum is discrete σ

(A) = {λ n } with λ n → +∞. Applying (SQ β ) to a normalised eigenvector x = ϕ n (associated with λ n ) yields 1 ≤ K 2 ∞ 0 (tA) -β ϕ n 2 (e -2t 2β -e -t 2β ) 2 dt t = K 2 λ 2β n ∞ 0 t -2β (e -2t 2β λn -e -t 2β λn ) 2 dt t = K 2 λ n 2βλ 2β n ∞ 0 (e -2s -e -s ) 2 ds s 2 = Cλ 1-2β n .
For β > 1 2 , the last term goes to 0 as n tends to +∞. This shows that (SQ β ) cannot hold. Now we come to our main result of this section concerning exact observability.

Theorem 4.3. Let -A be the generator of a bounded semigroup on the Banach space X and assume that A is injective and has dense range. Let C : D(A) → Y be bounded and suppose that there exists β ∈ (0, 1) such that the lower square function estimate (SQ β ) and (4.1) are satisfied. Then there exists a constant m > 0 such that

(4.8) m 2 x 2 ≤ ∞ 0 CT (t)x 2 Y dt for all x ∈ D(A) ∩ R(A).
Proof. Fix x ∈ D(A) ∩ R(A) and apply the lower square function estimate (SQ β ) to obtain

x 2 ≤ K 2 ∞ 0 -β (T (2t 2β ) -T (t 2β ))x 2 dt t ≤ K 2 δ 2 ∞ 0 CA -(1-β) (tA) -β (T (2t 2β ) -T (t 2β ))x 2 Y dt t = K 2 δ 2 ∞ 0 CA -1 (T (2t 2β ) -T (t 2β ))x 2 Y dt t 1+2β .
Using the fact that x ∈ D(A), we can write

CA -1 (T (2t 2β ) -T (t 2β ))x = -CA -1 2t 2β t 2β
AT (s)x ds = -

2t 2β t 2β
CT (s)x ds.

Using this and the previous estimates yields

x 2 ≤ K 2 δ 2 ∞ 0 2t 2β t 2β CT (s)x ds 2 Y dt t 1+2β ≤ K 2 δ 2 ∞ 0 2t 2β t 2β 1 • CT (s)x Y ds 2 dt t 1+2β ≤ K 2 δ 2 ∞ 0 2t 2β t 2β CT (s)x 2 Y ds dt t = K 2 δ 2 2 1 ∞ 0 CT (t 2β u)x 2 Y t 2β-1 dt du = log(2) K 2 2βδ 2 ∞ 0 CT (r)x 2 Y dr.
This shows (4.8) with m = log(2)/2β K δ . Notice that if 0 ∈ ̺(A) then R(A) = X and hence D(A) ∩ R(A) = D(A). In this case (4.8) holds for all x ∈ D(A) and this means that C is exactly observable for A.

Corollary 4.4. Let -A be the generator of a bounded semigroup on the Banach space X and assume that A is boundedly invertible. Let C : D(A) → Y be a bounded operator and assume that there exists β ∈ (0, 1) such that the lower square function estimate (SQ β ) and (4.1) are satisfied. Then C is exactly observable for A.

Corollary 4.5. Let -A be the generator of a bounded semigroup on the Banach space X and assume that A is injective and has dense range. Let C D(A) → Y be infinite-time admissible for A and assume that there exists β ∈ (0, 1) such that the lower square function estimate (SQ β ) and (4.1) are satisfied. Then C is exactly observable for A.

Proof. It remains now to extend the estimate (4.8) from Theorem 4.3 for all x ∈ D(A). This is an easy task. For x ∈ D(A) the sequence

x n := n(n+A) -1 x -n -1 (n -1 +A) -1 x ∈ D(A) ∩ R(A)
converges to x in D(A) (for the graph norm). Therefore, Cx n converges in Y to Cx and

∞ 0 CT (t)x n 2 Y dt converges to ∞ 0 CT (t)x 2
Y dt since C is supposed to be infinite-time admissible. We obtain (4.8) for all x ∈ D(A).

In the next corollary we obtain a criterion for finite time exact observability.

Corollary 4.6. Let -A be the generator of a bounded semigroup on the Banach space X and assume that there exists a constant ω > 0 such that A + ω satisfies (SQ β ). Assume that C : D(A) → Y is bounded and that

C(ω + A) -(1-β) x Y ≥ δ x for x ∈ D(A). Then C is exactly observable in finite time.
Proof. We apply the previous theorem to ω + A and obtain

x 2 ≤ M ∞ 0 Ce -ωt T (t)x 2
Y dt for all x ∈ D(A). We split the right hand side into two parts and write

∞ 0 Ce -ωt T (t)x 2 Y dt = τ 0 Ce -ωt T (t)x 2 Y dt + ∞ τ Ce -ωt T (t)x 2 Y dt ≤ τ 0 CT (t)x 2 Y dt + ∞ 0 Ce -ω(τ +t) T (t + τ )x 2 Y dt = τ 0 CT (t)x 2 Y dt + e -ωτ ∞ 0 Ce -ωt T (t)T (τ )x 2 Y dt.
Since C is infinite-time admissible for e -ωt T (t) and the semigroup T (t) is bounded we have for some constants

M ′ , M ′′ ∞ 0 Ce -ωt T (t)T (τ )x 2 Y dt ≤ M ′ T (τ )x 2 ≤ M ′′ x 2 .
Therefore,

x 2 ≤ M τ 0 CT (t)x 2 Y dt + M M ′′ e -ωτ x 2 .
If we choose τ large enough such that M M ′′ e -ωτ < 1 we obtain the desired inequality.

As explained at the beginning of this section, (SQ β ) holds for all β < [START_REF] Mcintosh | Operators which have an H∞ functional calculus, Miniconference on operator theory and partial differential equations[END_REF]. For the particular functions ψ α (z) := z α /(1 + z) for α ∈ (0, 1) this means that for some positive constants k α

(5.2) k α ∞ 0 ψ α (tA)x 2 dt t ≤ x 2 . Given now a function ψ ∈ H ∞ 0 (Sπ /2 ), we choose ε > 0 small such that |ψ(z)| ≤ M max(|z| 2ε , |z| -2ε ) and write ψ(z) = ψ ε (z) × z -ε ψ(z) + ψ 1-ε (z) × z ε ψ(z). Notice that z ±ε ψ(z) ∈ H ∞ 0 (Sπ /2
). Therefore, upper square function estimate for A with the function ψ follow from (5.2) using the boundedness of the H ∞ (Sπ /2 ) calculus. All what we explain here works also for the adjoint A * . By a duality argument as in [START_REF] Mcintosh | Operators which have an H∞ functional calculus, Miniconference on operator theory and partial differential equations[END_REF] or [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF], we pass from upper square function estimates for A * to lower square function estimates for A. As a particular case, (SQ β ) holds with β = 1 2 . We then apply Theorem 4.3.

Again as in Corollary 4.6, we can obtain observability in finite time by adding a constant w to A. That is Corollary 5.3. Let -A be the generator of a semigroup (T (t)) on the Hilbert space X. Suppose that there exists a constant ω > 0 such that A + ω is accretive and that C : D(A) → Y is bounded. If C(A + ω) -1 /2 x ≥ δ x for all x ∈ D(A) and some δ > 0, then C is exactly observable in finite time. with bounded measurable coefficients a jk ∈ L ∞ (R d ). The operator A is defined by sesquilinear form techniques (see for example [START_REF] Maati | Analysis of heat equations on domains[END_REF]) and note that A is not necessarily self-adjoint. It is a standard fact that -A generates a contraction semigroup on L 2 (R d ). Consider the problem ∂u ∂t = -Au y(t) = Cu(t) = ∇u(t) with X = L 2 (R d ) and Y = (L 2 (R d ) d . By the solution of the Kato's square root problem (see [START_REF] Auscher | Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on R n[END_REF]), it is known that (5.4) ∇u 2 ≈ A 1 /2 u ∀u ∈ W 1,2 (R d ).

On the other hand, the accretivity of A implies an H ∞ functional calculus on L 2 (R d ). Hence it satisfies upper and lower square function estimate (4.3) and (4.4). In particular,

(5.5)

f 2 2 ≈ ∞ 0 A 1 /2 e -tA f 2 2 dt.
This implies that C = ∇ is both admissible and exactly observable for A in infinite time.

Note that the semigroup e -tA is bounded holomorphic on some sector S ω and e -te iw A is a contraction on L 2 (R d ) (see [START_REF] Maati | Analysis of heat equations on domains[END_REF]). Taking the maximal angle w, we obtain a contraction semigroup e -te iw A which is not holomorphic. Since (e iw A)

1 /2 = e iw/2 A 1 /2 we see that (5.4) holds with e iw A in place of A. By Corollary 5.2 we have

(5.6) δ f 2 2 ≤ ∞ 0 ∇e -te iw A f 2 2 dt
for f ∈ D(A) ∩ R(A). We may consider the same problem on a bounded Lipschitz domain instead of R d . In this case, A is invertible. Hence e iw A is also invertible and we obtain (5.6) for all f ∈ D(A). This means that C is exactly observable for e iw A.

Example 2 . 3 .

 23 The following example is taken from Jacob, Partington and Pott [9, Example 3.9]. We shall use Ingham inequalities to prove that our system is BFC. Similar ideas could be used in a more general class of examples. Consider an undamped wave equation on [0, 1] with Dirichlet boundary conditions and Neumann type observation of the form

Definition 4 . 1 .

 41 We denote by S ω the open sector {z ∈ C * : | arg(z)| < ω} and by S ω the closure of S ω in C. We call a closed operator A on X sectorial of angle ω if A is densely defined having its spectrum in S ω such that λR(λ, A) := λ(λ-A) -1 of A is uniformly bounded on the complement of each strictly larger sectors S θ , θ > ω.

Examples 5 . 4 . 1 -∇e it∆ f 2 2

 5412 Consider the Schrödinger equation  ∂Ω y(t) = Cu(t) = ∇u(t) with X = L 2 (Ω), Y = (L 2 (Ω)) d and Ω is any open subset of R d with boundary ∂Ω. In this problem, ∆ denotes the Laplacian with Dirichlet boundary conditions. It is (the negative of ) the associated operator with the symmetric form a(u, v) := Ω ∇u∇vdx, D(a) = W 1dt for all f ∈ D(∆) ∩ R(∆). Note that we can replace here the Dirichlet boundary condition by the Neumann one. The are the same, we just need to replace D(a) = W 1,2 0 (Ω) by D(a) = W 1,2 (Ω). The same method applies for more general boundary conditions. 2-Let A be the uniformly elliptic operator A = -

  This is essentially von Neumann's inequality for contractions on a Hilbert space. Indeed, if A is accretive, T := (A -1)(A + 1) -1 is a contraction von Neumann's inequality states p(T ) ≤ p H ∞ (D) for every polynomial p. From this, the H ∞calculus can be derived by approximation arguments. Two different direct proofs for the boundedness of the H ∞ (Sπ /2 ) calculus are given in[START_REF] Haase | The functional calculus for sectorial operators, Operator Theory[END_REF] Theorem 7.1.7]. One uses a dilation theorem of the semigroup into a unitary C 0 -group due to Sz.-Nagy. The second exploits accretivity of A from a 'numerical range' viewpoint and can be seen as the most simple case of the Crouzeix-Delyon theorems[START_REF] Crouzeix | Some estimates for analytic functions of strip or sectorial operators[END_REF][START_REF] Badea | Convex domains and K-spectral sets[END_REF]. Having the boundedness of the functional calculus on Sπ /2 in hands we certainly have upper square function estimates for functions in H ∞ 0 (S θ ) when θ > π / 2 . This is well known and proved by McIntosh

	means that for every bounded holomorphic function ϕ on Sπ /2 and continuous on
	Sπ /2 , ϕ(A) is well defined and
	ϕ(A) L(X) ≤ M sup	|ϕ(z)|.
	z∈Sπ / 2
	1 2 and all generators of bounded semigroup (see (4.7)). Therefore, applying Theorem 4.3
	with β = 1 2 -ε, we obtain the following corollary.

BERNHARD H. HAAK AND EL MAATI OUHABAZ

The first author was partially supported by the ANR project ANR-09-BLAN-0058-01.

Corollary 4.7. Let -A be the generator of a bounded semigroup (T (t)) t≥0 on a X and assume that A is invertible. Then, if CA -1 /2+ε x ≥ δ x for some ε, δ > 0 and all x ∈ D(A), C is infinite-time exactly observable for A.

Exact observability on Hilbert spaces

Proposition 5.1. X and Y be Hilbert spaces. Then, if (A, C) is exactly observable and admissible in infinite time, T (t) t≥0 is similar to a contraction semigroup.

Proof. Denote by x, y Y the scalar product of Y and define for x, y ∈ D(A)

This is clearly a bilinear (or sesquilinear) form on D(A) × D(A). Admissibility and exact observability imply that x X and x ∼ X are equivalent. By density, we extend this to all x ∈ X and • ∼ X is associated with a scalar product on X. With respect to the new norm,

and so T (t) t≥0 is a contraction semigroup with respect to • ∼ X . Now we turn back to Theorem 4.3. As mentioned at the beginning of the previous section, the lower square function estimate (SQ β ) holds for small β for all generators of bounded strongly continuous semigroups. It is then tempting to use the theorem for small β in order to include a large class of semigroups T (t). On the other hand, if we assume that 0 ∈ ̺(A) and CA -α-ε x ≥ δ x , one also has

That is, the invertibility condition on CA -(1-β) becomes more restrictive when β decreases. To admit more observation operators C one therefore seeks for values of β large enough. Combining both conditions forces to play with different values of β in different situations. In the following corollary we choose β = 1 / 2 . Corollary 5.2. Let -A be the generator of a semigroup of contractions (T (t)) t≥0 on a Hilbert space X. If A has dense range and CA -1 /2 x ≥ δ x for all x ∈ D(A) ∩ R(A), then

Notice that in view of Proposition 5.1, the hypothesis of a semigroup of contractions is necessity to be able to conclude in the case that C is admissible.

Proof. By the Lumer-Phillips theorem, A is an accretive operator, i.e. Re Ax, x ≥ 0 for all x ∈ D(A). Since A has dense range and X is reflexive, A is actually injective (cf. [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF]Theorem. 3.8]). We need to verify that A admits lower square function estimates (4.4) with functions ϕ that are bounded holomorphic on Sπ /2 and continuous on Sπ /2 . We shall explain how this follows from the functional calculus. First notice that A has a bounded H ∞ (Sπ /2 ) functional calculus. This


LATMOS E Marcq

F Montmessin

F Lefèvre

LATMOS. A Reberac

LOOKING FOR SO 2 : A SPECTROSCOPIC STUDY OF MARTIAN UV ALBEDO USING SPICAM

Introduction

The detection of SO2 in the Martian atmosphere would have far-reaching implications in our knowledge of the planet, much akin to the (actual) detection of CH4. First of all, it would be a major point in discussing the level of geologic activity under the surface of Mars, since SO2 photochemical is short enough -less than a year according to Wong et al.(2003) -so that it would need to be constantly replenished from a sub-surface source. Its presence would also help in discriminating the origin of CH4 in the atmosphere; if SO2 were present and correlated to CH4, the source of both gases could be assigned to geological processes, thus ruling out the need for a biological source of CH4.

Unfortunately, all recent measurements have failed to detect any significant amount of SO2. Krasnopolski (2005) set an upper limit at 1 ppbv using TEXES at NASA IRTF, and more recently Nakagawa et al. (2009) found an upper limit at 2 ppbv in the submillimeter range using ASTE. Nevertheless, these upper limits were found by integration upon the whole Martian disk, and thus do not preclude higher mixing ratios locally, provided localized sources exist.

Our goal is to use SPICAM-UV in nadir geometry to yield upper limits (or mixing ratios in case of detection) of SO2 in the atmosphere of Mars, but this time on a localized basis. Even though SPICAM is not a real mapping spectrometer, its proximity to Mars enables a good spatial resolution and the duration of the mission ensures that most of the surface has been observed several times since 2003.

Observations

The UV channel of SPICAM is fully described in Bertaux et al. ( 2006), and we give here only a brief summary of its relevant characteristics: observed range is between 115 and 310 nm. Resolving power si about 1 nm. Typical FOV at pericenter is 1.2 × 2.4 km. Several thousand orbits are available now Pre-processing of data is very similar to Perrier et al. (2006). Correction of DC and straylight are performed first, and then relative radiance factors are derived using a reference obseration of Olympus Mons recorded during orbit #1448. Proceeding to spectral ratios enables many calibiration issues to be solved, but require a very good knowledge of the reference observation used as a denominator.

Modelling

Our spectrum forward model is based on Perrier et al. (2006) model, devised to measure O3 column density. We used the same radiative core (SHDOM), and (P, T ) profiles are extracted from the LMD-GCM as well as surface pressure and O3 relative vertical profile. UV cross-sections are kept the same, except that we now add SO2 using Wu et al. (2000) values. CO2 clouds are not taken into consideration, and dust is assumed to follow a Conrath distribution. Sensitivity measurements yield a detection threshold at around 10 µm-atm for SO2 column density, mainly thanks to its 215 nm band (see Fig. 1). If vertically uniform, this column density translates into a mixing ratio threshold ranging from 10 to 100 ppbv. This is one or two orders of magnitude higher that globally averaged upper limits, so that we do not expect to find SO2 in more than a few percents of SPICAM spectra.

Strategy

Our first strategy to derive SO2 was an extension of Perrier et al. (2006). A Levenberg-Marquardt χ 2 optimization was realized in a 5-dimensional parameter Looking for SO2 using SPICAM-UV space: O3 and SO2 column densities, dust opacity, surface albedos at 200 and 320 nm. However, degeneracies cannot be avoided with such a high number of fitted parameters. A way to test this is to compare the fitted parameters when constraining SO2 to be null and without this constraint: where high discrepancies occur, they cast doubt on the interpretation. The assumption of a surface albedo varying linearly between 200 and 320 nm is especially dubious.

That is why we are currently implementing another approach based on directly fitting radiance factors (instead of radiance factor ratios as we did previously). The spectral distortion induced by O3, dust, CO2 is then corrected using externally derived values, for example using Perrier et al. (2006) values. The corrected radiance factor would then contain information about mostly two parameters: ground albedo in the UV and, sometimes, SO2 spectral signature. This method should be more robust than the first.

Preliminary Results

Provisional results are available only following the first strategy at the time of abstract submission, but neverthe- less encouraging as can be shown on Fig. 2 and3. The spectral slope near 210 nm is especially encouraging in our search for SO2. Nevertheless, the robustness of the fitting must be checked before any detection claim, especially since this fetaure seems present on a higher proportion of data than expected. We checked that this spectral feature is not primarily due to a faulty straylight removal in the pre-processing stage, but did not check its sensitivity versus CO2 pressure -a strong CO2 band below 200 nm could interfere with our fit. And of course, some ground albedo feature at the observed location and/or at our Olympus Mons reference spectrum cannot be ruled out at the present stage.

Figure 1 :

 1 Figure 1: Synthetic radiance factors with 10 µm-atm (black) and without SO2 (red). Absorption near 250 nm is due to O3.

Figure 2 :

 2 Figure 2: Observed relative radiance factor during orbit #61 and its best fit allowing SO2

Figure 3 :

 3 Figure 3: Observed relative radiance factor during orbit #61 and its best fit without SO2

Bibliography•

  Bertaux et al., SPICAM on Mars Express: Observing modes and overview of UV spectrometer data and scientific results, JGR 111 (2006) • Krasnopolski, A sensitive search for SO2 in the martian atmosphere: Implications for seepage and origin of methane, Icarus 178 (2005) • Nakagawa et al., Search of SO2 in the Martian atmosphere by ground-based submillimeter observation, PSS 57 (2009) • Perrier et al., Global distribution of total ozone on Mars from SPICAM/MEX UV measurements, JGR 111 (2006) • Wong et al., Chemical markers of possible hot spots on Mars, JGR 108 (2003) • Wu et al., Measurements of High-, Room-, and Low-Temperature Photoabsorption Cross Sections of SO2 in the 2080-to 2950-Å Region, with Application to Io, Icarus 145 (2000)


Y Dahan

Y Chastel
email: yvan.chastel@ensmp.fr

P Duroux

J Wilsius
email: joel.wilsius@arcelor.com

P Hein

E Massoni

PROCEDURE FOR THE EXPERIMENTAL DETERMINATION OF A FORMING LIMIT CURVE FOR USIBOR 1500 P

Keywords: Formability, hot stamping, forming limit curve, high strength steel

USIBOR 1500 P® (1) is a coated C-Mn steel, micro-alloyed with boron, with excellent processing properties in hot stamping, both in terms of formability, quenchability and surface protection, and leading to superior mechanical properties on the formed part. Arcelor Research is developing a numerical tool to support the feasibility analysis and to optimize the design of hot stamped parts made of USIBOR 1500P®. Among the numerous bricks of such a simulation methodology, formability data of the material are of primary importance as the forming analysis relies on them to make a decision in terms of feasibility. Up to now, there was no experimental procedure available in literature to determine the forming limits of hot stamping material, taking into account the specificities of this process. This paper reports about the research performed in this field.

(1) USIBOR® is a registered trademark belonging to the Arcelor Group and protected throughout the world. Furthermore, Arcelor Group has filed numerous patents covering the principle of direct or indirect hot-stamping process of either aluminized-coated steel sheets and zinc or zinc-alloy coated steel sheets, and of the issued parts.

First, an efficient experimental set-up which allows varying all the desired process conditions has been developed. Several hundred Nakazima hot stamping tests have been carried out for various process parameters (stroke, velocity, friction and heat exchange) and blank parameters (temperature, thickness and shape). The third step consisted in developing and validating an accurate analysis scheme to determine the critical strain values based on the Bragard method. Finally, the critical strain values have been confirmed and comparisons with industrial parts for various process conditions have been performed. In this paper these two last steps are mainly presented.

INTRODUCTION

Safety improvement, weight reduction and cost saving are the major objectives of automotive manufacturers in the design of new bodies-in-white. Compared with several others industrial solutions [START_REF] Kolleck | Manufacturing Methods For Safety and Structural Body parts for Lightweight Solution[END_REF], hot stamping of a quenchable steel comes out as a quite novel process which is increasingly used for automotive applications and intensive research is conducted to support this growth [START_REF] Neugebauer | Sheet Metal Forming at Elevated Temperatures[END_REF]. It allows producing thinner parts which display higher mechanical properties. The principle of hot stamping is intimately linked with the opportunities offered by the chemical composition of the quenchable steel, namely a robust process window for martensitic and bainitic transformation.

The hot stamping process consists first in heating a blank until complete austenitization. The hot blank is then moved quickly into a stamping press where it is simultaneously stamped and quenched by the cold tools. To predict part feasibility and perform process layout, Arcelor Research is developing a numerical tool to support the feasibility analysis and the design of hot stamped parts made of USIBOR 1500P®. This task is challenging because of the large number of process parameters, and of the thermo-mechanical and metallurgical interactions which influence formability [START_REF] Hein | A Global Approach of the Finite Element Simulation of Hot stamping[END_REF][START_REF] Hein | Hot Stamping of USIBOR 1500 P: Part and Process Analysis Based on Numerical Simulation[END_REF][START_REF] Wilsius | Status and Future Trends of Hot Stamping of USIBOR 1500 P[END_REF].

Formability is the ability of the material to be strained without necking and fracture. For a coupled thermo-mechanical process, formability is a function of several parameters such as temperature, strain rate, strain path, thickness. An accurate experimental methodology has been defined to determine a failure criterion. This methodology is divided into 4 steps. The first step consists in the development of a robust experimental set-up which allows varying the process parameters. Several hundred hot stamping tests have been carried out following this procedure for various process parameters (stroke, velocity, friction and heat exchange) and blank parameters (initial temperature, thickness and shape). These steps have already been presented by the authors in [START_REF] Dahan | Formability Investigations for the Hot Stamping Process[END_REF]. Then, an analysis scheme has been developed to determine the critical strain values. The last step consists in the validation and comparisons with industrial parts for various process conditions. In this paper we will mainly focus on these two last steps.

NAKAZIMA HOT STAMPING TESTS

The Nakazima and the Marciniak tests are two usual experimental tests which provide information on formability of sheet material. The main difference between these tests is the shape of the punch which is respectively hemispherical or flat. The Nakazima set-up has been selected for a straight-forward reason: it is much simpler to perform than the Marciniak test. The Nakazima set-up is made of a hemispherical punch, a die, a blank-holder and a draw-bead which prevents any sliding motion (see Figure 1). Several sensor data are recorded during the test, such as the punch load and local temperature histories. A grid is etched on the blank and allows determining the strain distribution by a posteriori analysis using pattern recognition systems. Thanks to this set-up, it is possible to vary several parameters of the blank and of the tools (punch temperature, punch velocity, friction). More details on the set-up and on the test protocol are available in [START_REF] Dahan | Formability Investigations for the Hot Stamping Process[END_REF] and similar approaches are reported in [START_REF] Turetta | Investigation of 22MnB5 Formability in Hot Stamping Operations.Investigation of 22MnB5 Formability in Hot Stamping Operations[END_REF]. In [START_REF] Dahan | Formability Investigations for the Hot Stamping Process[END_REF], the authors show that necking occurs in a zone of nearly homogeneous temperature and obtain several strain distributions as functions of punch stroke, temperature, strain mode and thickness. Strain distributions of safe, necked and fractured specimens are presented in Figure 2 for given process conditions (blank thickness, initial blank temperature, punch velocity). These results are then used to determine the critical strain values.

Figure 2. Influence of the punch stroke on the strain distribution

DETERMINATION OF THE CRITICAL STRAIN VALUES

A methodology has to be defined to determine the critical strain values. The history and the available techniques for the determination of Forming Limit Curves (FLC) can be found in [START_REF] Dahan | Formability Investigations for the Hot Stamping Process[END_REF]. Different FLCs can be given for a single material using various criteria. And very different FLCs can be obtained from different experimental methods for a same material specimen. As a result, establishing a reference method to determine FLC still remains an issue in sheet metal forming [START_REF] Col | FLCs: Are we at a Turn?[END_REF][START_REF]forming limit curves -Part 2: Determination of forming limit curves in laboratory[END_REF]. For hot stamping in particular, there is up to now no consistent testing procedure available in literature.

Bibliography

The Hecker and Bragard methods are well-known experimental techniques. The Hecker method is quite easy to implement since tests series with varying punch strokes are sufficient to determine the critical strains. The blank with the highest punch stroke without any visible necking or fracture is selected. The maximum strain given by the analysis of this specimen is set as the critical strain value.

The Bragard method is based on the analysis of a blank with necking or fracture. Once the strain distribution is determined, the major strain evolution along a section normal to the fracture is drawn [START_REF]forming limit curves -Part 2: Determination of forming limit curves in laboratory[END_REF]. At this stage, a zone has to be defined around the fracture where the strain distribution will be not considered. Two variants may be used to define this region.

The first technique is based on an optical observation of the necking or the fracture. All measured cells which overlap the necking or fracture are deleted. For the USIBOR 1500 P sheets, the detection of necked and fractured cells is not always possible because of the damaging of the grid. In addition, this method is highly subjective since it is quite depended on the user's interpretation.

A second method, called "2 nd derivative", uses the extrema of the second derivative of the major strain function [START_REF]forming limit curves -Part 2: Determination of forming limit curves in laboratory[END_REF]. All points between the extrema of the second derivative are deleted from the curve. The next step consists in an interpolation by a polynomial function (6 th degree) of the remaining strain points. The highest value reached by this polynomial function is considered as the critical major strain value and thus one point of the FLC.

Application of the Bragard method to the Nakazima hot stamping test

Using the 2 nd derivative Bragard method, a critical value in plane strain is estimated on a necked sample. The critical value in plane strain is referred as FLC 0 . This value of FLC 0 is used with the well-known Maximum Modified Force Criterion (MMFC) prediction model which assumes the coefficient of strain hardening n is equal to FLC 0 [START_REF] Ben Tahar | Contribution à l'étude et la simulation du procédé d'hydroformage[END_REF]. This criterion has used so far in this study for first shape representations of the FLC and needs further theoretical consideration in the specific context of hot stamping. This calculated FLC and the results of the Nakazima hot stamping tests are superimposed in Figure 3. The 2 nd derivative Bragard FLC is relatively low and several tests reach higher strain values without visible damage. We conclude that the FLC obtained by this method is too conservative. This is probably due to the fact that in hot Nakazima tests, the strain distribution is quite heterogeneous even on specimens well before necking (straining mainly in the moving punch nose area). For this kind of configuration, high order derivatives of the strain distribution seem too conservative to define the area affected by local necking. This is the reason why a novel analysis procedure has been developed; it is derived from the 2 nd derivative Bragard variant and is consistent with the analysis of the Nakazima tests. This method is referred to as the 1 st derivative Bragard method.

The 1 st derivative Bragard method

In the proposed exploitation procedure, the deletion zone is defined between the extrema of the 1 st rather than the 2 nd derivative. For given process conditions, the critical strain is for example found to be 0,40 instead of 0,31 with the 2 nd derivative Bragard variant: see This alternative method leads to higher values of FLC 0 . This result is more reasonable since all safe and damage points fit respectively under and above the FLC in Figure 5.

Thickness and temperature influence

The 1 st derivative Bragard method has been applied to several specimens with different initial thicknesses and temperatures and which displayed cracks in an area deformed in plane strain. In order to write the critical strain values in plane strain FLC 0 as a function of these two parameters (t for thickness and T for temperature), a multiple regression has been carried out using all specimens analysed with the 1 st derivative Bragard method:

[ ] ) ( ), ( 0 C T mm t function FLC °= . (1)
The higher the initial blank temperature (or thickness), the higher the critical strain. The influence of the initial blank thickness on FLC is shown in Figure 6 for a given temperature.

Confirmation on hot stamped parts

This proposed methodology and the measured formability data have been confronted with real size forming tests on the die of a lower B-pillar. Test parts have been stamped with varying process conditions in order to deliberately produce parts with necking or fracture in the zone shown in Figure 6.

Figure 7. B pillar part and critical zone for given process conditions

Using a pattern recognition system, the strain distributions of the three different specimens (safe, necked and fractured) have been plotted in the Forming Limit Diagram. These measurements are compared to the FLC given by the 1 st derivative Bragard method for given values of initial temperature and thickness (Figure 8). For the safe specimen, the strain distribution is located under the FLC. Necking occurrence has been determined manually or optically. Samples with visible necking have strain distributions which reach the FLC. The strain range given by the fractured specimen confirms this trend, since it crosses the FLC. These observations confirm the FLC 0 values expressed as a function of the initial thickness, temperature and strain path. [START_REF] Dahan | Formability Investigations for the Hot Stamping Process[END_REF], the authors show that pure bi-axial stretching can not be obtained with this Nakazima set-up. That's why another laboratory test based on the Marciniak set-up has been designed.

MARCINIAK HOT STAMPING TESTS

Presentation of the Marciniak set-up

The major differences between the Nakazima and the Marciniak tests are in terms of the punch shape. The Nakazima test uses a punch and the Marciniak test a flat punch. Further-more, a specific design was defined for the Marciniak test: one or two additional rings of sheet metal, called carrier blanks, have been used to prevent the contact between the punch and the tested blank. Experimentally, this test is a real challenge since a hot blank has to be introduced from the furnace in-between two cold carrier blanks within a very short transfer time. This is the reason why a first finite element analysis has been carried out to check the strain distribution and to optimize the driving blank (material, diameter of the central hole). Figure 9 presents a numerical model of this experimental set-up (left) and shows the three types of specimens which can be obtained (right). It is confirmed that the strain localization appears in the central blank zone. Necking and fracture can also easily be detected.

Strain distribution and FLC prediction

The global analysis in Figure 10 confirms that positive ε 1 and ε 2 values can be achieved with this test. The closer to the blank centre, the higher the major and minor strains, the closer to an equi-biaxial strain state. In the outer areas of the blank (close to the punch radius), the strain distribution tends towards a plane strain state. The results of the Marciniak hot stamping tests and the FLC given by the MMFC and the 1 st derivative Bragard method are superimposed in Figure 10. No points corresponding to tests with necking and fracture lie below the FLC. Therefore, the shape and the level of the FLC are globally confirmed. Further experiments are in progress to fully exploit this second test set-up.

Strain distribution and prediction of the FLC for Marciniak hot stamping tests

CONCLUSION

The results of this investigation are the following: • Two experimental tests, based on Nakazima and Marciniak tests, have been specifically developed to study the formability of USIBOR 1500 P at high temperature for the hot stamping process. • A robust method has been developed to determine the critical strain and to build a FLC from the Nakazima hot stamping tests. • This method has been applied for various process conditions: temperature, thickness and strain path. The influences of thickness and temperature on FLC 0 have been determined. • A preliminary FLC has been deduced from the experimental FLC 0 values using the MMFC. Results from Marciniak tests are consistent with these data. • These forming limit data have been checked on several laboratory tests and a few industrial parts. So far, all cases have confirmed the accuracy of the derived FLCs for the industrial hot stamping conditions.

Figure 3 .

 3 Figure 3. 2 nd derivative Bragard FLC and results from the Nakazima hot stamping tests

  Figure

4

 4

  .

YFigure 4 .

 4 Figure 4. 1 st derivative Bragard (left) and 2 nd derivative Bragard (right) methods

Figure 5 .

 5 Figure 5. 1 st derivative Bragard FLC and results from the Nakazima hot stamping tests

6 Thickness ratio = 1 Figure 6 .

 616 Figure 6. Influence of the initial blank thickness on FLC

Figure 8 .

 8 Figure 8. Safe, necked and fractured test parts

Figure 9 .

 9 Figure 9. Marciniak set-up (left) and experimental safe, necked and fractured blanks (right)

ACKNOWLEDGEMENT

The authors would like to thank S. Jacomet, G. Fiorucci and B. Triger from Centre for Material Forming (CEMEF) for their significant contribution to the experimental works and A. Col, C. Dessain, J.-P. Durbise, J.L. Geoffroy, C. Kerisit and B. Tavernier from Arcelor Research for their important inputs in the discussion on the analysis scheme.


Yoann Dahan
email: yoann.dahan@ensmp.fr

Yvan Chastel

Patrick Duroux

Philipp Hein

Elisabeth Massoni

Joël Wilsius

Formability investigations for the hot stamping process

Keywords: Formability, Hot Stamping, Experiment, Numerical Simulation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The objective of automotive manufacturers is to combine safety improvements and weight reductions of their vehicles in order to comply with CO 2 emission regulations. A non exhaustive list of the industrial solutions is given in [START_REF] Kolleck | Manufacturing Methods For Safety and Structural Body parts for Lightweight solution[END_REF]. As a response to these constraints, the hot stamping process of quenchable steels allows to obtain thinner parts with higher mechanical properties. It is a relatively novel process and it is increasingly used for automotive applications. This process consists in first heating up a blank in a furnace to reach a stable austenitic state. The hot blank is then moved into a stamping press where it is simultaneously stamped and quenched by the cold tools. The cooling rate of this process step allows obtaining the desired martensitic microstructure leading to superior mechanical properties on the formed part. It is the reason why the main target components are in the crash relevant parts of car structures: bumpers, side impact reinforcements, Aand B-pillars (see Figure 1). For this process, Arcelor developed USIBOR 1500P ®1 , a quenchable C-Mn steel micro-alloyed with boron. The distinctive value of USIBOR ® lies in the coating which avoids oxidation during the heating phase and saves a very expensive shot-peening step. With this material, a tensile strength of about 1500 MPa is reached on the quenched formed part. More details on the material, the process and some market trends are available in [START_REF] Hein | Hot Stamping of USIBOR 1500P ® : Part and Process Analysis Based on Numerical Simulation[END_REF].

Figure 1. Crash relevant components hot stamped in USIBOR 1500 P ® (darkened parts) [START_REF] Hein | A Global Approach of the Finite Element Simulation of Hot stamping[END_REF] Arcelor is developing a numerical tool to support the feasibility analysis and the design of hot stamped parts made of USIBOR 1500P ® . This task is challenging due to the high number of process parameters, the thermo-mechanical and metallurgical interactions and their influence on formability (more details are presented in [START_REF] Hein | A Global Approach of the Finite Element Simulation of Hot stamping[END_REF]). This requires both experimental tests and advanced numerical simulation tools. To provide formability data and to support the development of a fracture criterion, experimental hot stamping tests are carried out at Cemef (Centre for Material Forming). In this article we will mainly focus on the experimental results and the comparisons with the numerical simulations. Three types of non-conformities can be observed on hot stamped parts. They prevent from reaching the targeted mechanical properties defined in the specifications. First, the blank may only be partially quenched and thus not fully martensitic. The second type of non-conformity concerns the wrinkling and thickening zone. These defects are usually decreased by reducing the clearance between the die and the blank-holder or optimizing the initial blank shape. In this article, we will focus on the necking and/or fracture of the blank during hot stamping.

Process description

To establish the formability criterion, hot stamping tests are performed with modified Nakazima-type tools and various product/process conditions. The axisymmetrical set-up is made of a hemispherical punch, a die, a blank-holder and a draw-bead which prevents any sliding motion (see Figure 2). Several parameters are recorded in-situ, such as the punch load and the local temperature history. A grid is etched on the blank and allows determining the strain distribution using pattern recognition systems. Numerical simulation has been initially used for the physical understanding of the process. Numerical models of these hot stamping tests have been developed with different finite element softwares (Forge2 ® , Forge3 ® and Abaqus). Forge2 ® and Forge3 ® are being constantly improved at Cemef respectively for 2D and 3D analysis. Besides Arcelor developped a finite element environment dedicated to hot stamping in order to perform coupled thermo-mechanical simulations and make feasibility studies of hot stamped components [START_REF] Hein | A Global Approach of the Finite Element Simulation of Hot stamping[END_REF].

In order to obtain correct results, a large number of model parameters have to be either defined with additional experimental tests (e.g. heat exchange coefficient, flow behavior) or adjusted by inverse identification (e.g. friction, heat radiation). Some of them are defined in [START_REF] Garcia-Aranda | Etude thermo-mécanique et modélisation numérique de l'emboutissage à chaud de l'USIBOR 1500P ®[END_REF]. In this section, we will present the numerical results for a punch velocity of 30 mm/s, a 1,5 mm thick blank and an initial blank temperature of 780°C (when the forming starts). In the post-processing, we can see the temperature and strain rate distribution at different steps of the process (see Figure 3). A high cooling rate is measured at the centre of the blank, which is in contact with the punch. Nevertheless, the temperature of the free blank zone seems to be roughly constant during the process. The distribution of the Von Mises equivalent strain rate indicates an obvious strain localization in a zone called punch radius exit. Indeed, during the hot stamping process, the strains are usually localized in the hotter regions (free blank zone in our case), and especially at the exit of the punch radius. Numerical simulation enables us to give results all along the process at a given location. Selecting a node near the exit of the punch radius, we plotted the nodal evolutions of the temperature and the equivalent strain rate in Figure 4. From 0 to 0,6 s, the punch is not in contact with the chosen node of the blank. So, obviously, it is at a constant temperature. The strain rate is increasing and reaches a maximum value at 0,6 s. At this moment, the node comes in contact with the punch and it is quickly cooled. As soon as the temperature drops down, the strain rate goes down as well. In fact, during the first time of the process, the blank centre is deformed and cooled down simultaneously. When this central zone is cool enough, the surrounding region, which is obviously hotter, is then deformed by the punch. In fact, due to the punch motion, a strain and a cooling wave is propagated away from the blank centre towards the exit of the punch radius. Using an axisymmetric configuration enables us to check the model parameters defined previously. Figure 5 illustrates accurate comparisons between the numerical results and the experimental measurements for the punch load and the temperature at the centre of the blank. The punch force is directly dependent on the viscoplastic flow behavior of the material, the initial blank temperature and the blank thickness. The temperature distribution is mainly defined by the initial temperature gradient between the punch and the blank, and by the heat exchange coefficients. In [START_REF] Garcia-Aranda | Etude thermo-mécanique et modélisation numérique de l'emboutissage à chaud de l'USIBOR 1500P ®[END_REF], Garcia-Aranda et al. determined this coefficient as a function of the contact pressure and the clearance between the tool and the blank. This analysis confirms the accuracy of the parameters used to develop the numerical model. The numerical model being checked and validated, we now focus on the strain distributions which are the key point for the formability analysis.

Predicted and experimental strain distributions

Three types of stamped blanks can be obtained: conformed, with strain localization or fractured blank (see Figure 6). For a safe punch stroke, the blank shows no critical damage. For a higher punch stroke, a strain localization occurs in the exit of the punch radius which causes a local thinning called necking and can be determined by a technique developed by Hecker [START_REF] Hecker | Formability of Aluminium Alloy Sheets[END_REF]. Once the maximum punch stroke is reached, the necking leads to a fracture which spreads over the exit zone of the punch radius. Fractography analysis has been carried out in order to analyze the fracture mode. In Figure 7, we observe a cross-section of the fractured zone. During hot stamping, strain localization occurs in the hotter region which leads to the thinning of the blank. The thinning appears in both a concentrated and continuous manner close to the fracture region. In addition, it seems to be symmetrical with respect to the blank neutral line. The front view of the fracture has been obtained by SEM. At the centre, the damage mode of the steel is observed in terms of intragranular microvoids characterizing a ductile fracture. Thanks to the quality of the coating and to an accurate method to preserve the grid, we managed to assess several strain distributions using pattern recognition systems. Uniform and random grids have been etched or painted on to the blank to be respectively analyzed by Asame and Aramis. This technique enables us to measure the strain distributions. The thinning values are then calculated considering volume conservation. Figure 8 compares the results given by Forge3 ® and Aramis for similar process configurations. The numerical simulations confirm the location and the magnitude of the thinning. Indeed, Aramis and Forge3 ® estimate that the maximum major strain is reached at the exit of the punch radius and that this extreme value was roughly 0,4. We can also notice that the blank centre is only slightly deformed due to the high cooling rate generated by the early contact with the punch. The pattern recognition systems such as Asame and Aramis enable us to plot the strain distribution in the (ε 1 ,ε 2 ) space. This diagram is widely used for sheet metal forming when strain gradients in the thickness direction can be neglected. Figure 9 represents the numerical and the experimental strain states of the blank at the end of the hot stamping process. First of all, we can observe an excellent correlation between the numerical and the experimental results. In our simple configuration, strains are located between plane strain and biaxial expansion. Forge2 ® and Aramis results indicate that the maximum major strain is reached in a mixed mode, close to plane strain. The blank centre is obviously in an expansion state. We see that the major strain is 0,05 at the blank centre which is roughly the ratio of the blank thickness to the punch radius. According to [START_REF] Hecker | Formability of Aluminium Alloy Sheets[END_REF], this value could be a simple bending effect due to the accumulated effects of high friction and heat exchange. The central zone of the blank quenched by the punch reaches a constant minor strain (≈ 0,04) all the way to the punch radius exit. In this region, the further from the blank centre, the higher the major strain. Beyond the punch radius exit zone, the strain quickly decreases.

Formability analysis

The experimental determination of formability limits has to take into account several process characteristics in terms of temperature (anisothermal process), phase transformation (unstable austenitic structure), stress triaxiality (thickness effect), process speed (viscoplastic behaviour and heat transfer) and strain path (blank shape and friction). These parameters can be changed by varying the process time, the initial blank temperature (defined as the blank temperature at the start of the stamping), the thickness and the blank shape in order to analyze their respective influence on the formability. Only the influence of the blank shape and the of the initial temperature of the blank will be detailed in this article. Several hot stamping tests were performed for different blank widths. Each stamped blank is then analyzed and classified according to quality criteria. All failed blanks showed necking or fracture in the punch radius region (see Figure 10). The punch stroke is then plotted as a function of the blank width for safe parts, strain localization (necking) or fractured blanks. The formability is strongly dependent on the blank width: the larger the blank width, the lower the critical punch stroke. This observation is mainly due to the modification of the strain paths of the critical region from plane strain towards uniaxial tensile strain. The pattern recognition systems have been used to deduce the influence of the width on the strain distribution. The strain states for several widths of necked blanks are shown in Figure 11. Several remarks can be made from this graph. The lower the blank width reduction ratio, the closer to the uniaxial tensile mode is the strain state and the higher is the maximum major strain. As seen before, the critical and the expansion zones are still respectively the exit of the punch radius and the blank centre for the higher width. The strain value at the centre seems to be constant for all blank widths. Of course, the blank centre zone is not in an expansion state anymore for the narrowest blanks. Asai et al. studied the influence of the heating temperature on the deep drawability of a hardenable C-Mn-B steel sheet [START_REF] Asai | Hot Stamping Drawability of Steel[END_REF]. In our case, varying the initial blank temperature from 600°C to 800°C allows us to analyze the influence of temperature on the formability. Figure 12 confirms the influence of temperature: the higher the initial temperature, the higher is the maximum punch stroke. According to these experimental results and with the given process conditions, a stroke of 27 mm is feasible only if the initial blank temperature is higher than 720°C. Below this temperature, necking and fracture in the critical zone of the blank are strongly likely.

Conclusion

Hot stamping tests have been carried out with a modified Nakazima set-up. Safe parts, strain localization (necking) and fractured blanks were obtained. The fractography analysis of the fractured blanks shows intragranular microvoids characterising a ductile fracture. Thanks to the quality of the blank coating and to an accurate method to preserve the grid, we managed to assess several strain distributions using pattern recognition systems. Numerical simulations have been performed and show an accurate correlation with the experimental results in terms of punch load, temperature at the center of the blank and strain distribution. The numerical model confirms the location and the magnitude of the critical zone. Numerical simulations enable us to analyze the evolution of temperature and strain rate during stamping as a wave which propagates from the blank centre towards the die radius, following closely the exit of the punch radius. Formability analysis results show the influence of the width and the initial temperature of the blank. For the first analysis, the narrower the blank, the higher is the allowable punch stroke. Modifying the blank shape changes the strain path. Indeed, the strain distribution graph shows that the narrower the blank, the closer to the uniaxiale tensile mode is the strain state and the higher is the maximum major strain. A similar study has been performed for the influence of the initial blank temperature. It appears that the higher the initial temperature, the better is the formability. All these tests will be used to develop a necking criterion which takes into account several process parameters such as the blank temperature, the blank thickness and the strain rate.

Figure 2 .

 2 Figure 2. Axi-symmetrical set-up for the modified Nakazima hot stamping tests

Figure 3 .

 3 Figure 3. Results of numerical simulation for 10, 20 and 30 mm punch strokes: Evolution of the temperature [°C] (left) and Von Mises equivalent strain rate [/s] (right)

Figure 4 .

 4 Figure 4. Nodal evolution of the temperature and of the equivalent strain rate during the process

Figure 5 .

 5 Figure 5. Experimental measurements and numerical calculations during the hot stamping process punch force (left) and temperature at the centre of the blank (right)

Figure 6 .

 6 Figure 6. Conformed (left), necked (middle) and fractured (right) hot stamped blanks for various punch strokes.

Figure 7 .

 7 Figure 7. Fractography analysis: binocular cross-section (left) and SEM front view (right) of the fracture zone

Figure 8 .

 8 Figure 8. Major surface strain distributions obtained by Forge3 ® (left) and Aramis (right) Process conditions: punch stroke: 30 mm, punch velocity: 30 mm/s, blank thickness: 1,5 mm, blank initial temperature: 780°C

Figure 10 .

 10 Figure 10. Hot stamped blanks with different width reduction ratios (left) and influence of the blank width on formability (right)

Figure 11 .

 11 Figure 11. Influence of the blank width on punch stroke and formability

Figure 12 .

 12 Figure 12. Influence of the initial blank temperature on the punch stroke and the formability of the blank

  Strain distribution given by Aramis assessments and Forge2 ® results Process conditions: punch stroke: 20 mm, punch velocity: 30 mm/s, blank thickness: 1,5 mm, blank initial temperature: 780°C

		0.2						Plane strain				Punch radius exit
		0.18	Uniaxiale tensile											Expansion
				strain
		0.16
		0.14
	Major strain	0.08 0.12
		0.06
													Blank centre
		0.04		Aramis assesments
		0.02		Numerical simulation
											External blank part
		0
		-0.12	-0.1	-0.08	-0.06	-0.04	-0.02	0	0.02	0.04	0.06	0.08	0.1	0.12	0.14
								Minor strain
				Figure 9.

USIBOR ® is a registered trademark belonging to the Arcelor Group and protected throughout the world. Furthermore, Arcelor Group has filed numerous patents covering the principle of direct or indirect hot-stamping process of either aluminized-coated steel sheets and zinc or zinc-alloy coated steel sheets, and of the issued parts.

Acknowledgements

The authors would like to thank S. Jacomet, G. Fiorucci and B. Triger from Centre for Material Forming (CEMEF) and C. Dessain, J.-P. Durbise and B. Tavernier from Arcelor Research Automotive Applications.


Metod Saniga
email: msaniga@astro.sk

Michel Planat
email: michel.planat@femto-st.fr

A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies

Keywords: 03, 65, Aa -03, 65, Fd -02, 10, Ox Qubit-Qudit -Generalized Pauli Group -Generalized Quadrangle of Order Two -Nested Structure

Following the spirit of a recent work of one of the authors (J. Phys. A: Math. Theor. 44 (2011) 045301), the essential structure of the generalized Pauli group of a qubit-qudit, where d = 2 k and an integer k ≥ 2, is recast in the language of a finite geometry. A point of such geometry is represented by the maximum set of mutually commuting elements of the group and two distinct points are regarded as collinear if the corresponding sets have exactly 2 k -1 elements in common. The geometry comprises 2 k -1 copies of the generalized quadrangle of order two ("the doily") that form 2 k-1 -1 pencils arranged into a remarkable nested configuration. This nested structure reflects the fact that maximum sets of mutually commuting elements are of two different kinds (ordinary and exceptional) and exhibits an intriguing alternating pattern: the subgeometry of the exceptional points of the (k + 2)-case is found to be isomorphic to the full geometry of the k-case. It should be stressed, however, that these generic properties of the qubit-qudit geometry were inferred from purely computer-handled cases of k = 2, 3, 4 and 5 only and, therefore, their rigorous, computer-free proof for k ≥ 6 still remains a mathematical challenge.

Introduction

Although finite geometries/point-line incidence structures behind the generalized Pauli groups of single qudits have essentially all been discovered and thoroughly explored [START_REF] Havlicek | Projective Ring Line of a Specific Qudit[END_REF]- [START_REF] Thas | The Geometry of Generalized Pauli Operators of N-Qudit Hilbert Space, and an Application to MUBs[END_REF], only little is still known about those underlying the groups of multi-qudits and/or tensor products of qudits of different ranks. A number of particular cases have been analyzed by computer [START_REF] Planat | Qudits of Composite Dimension, Mutually Unbiased Bases and Projective Ring Geometry[END_REF]- [START_REF] Planat | Graphs When the Hilbert Space Dimension Contains a Square: Why the Dedekind Psi Function?[END_REF], and certain "easy-to-handle" configurations have also been studied and completely described in a computerfree fashion [START_REF] Thas | The Geometry of Generalized Pauli Operators of N-Qudit Hilbert Space, and an Application to MUBs[END_REF], [START_REF] Planat | On the Pauli Graphs of N-qudits[END_REF], [START_REF] Saniga | Multiple Qubits as Symplectic Polar Spaces of Order Two[END_REF]- [START_REF] Havlicek | Moebius Pairs of Simplices and Commuting Pauli Operators[END_REF]. Based on the recent work of one of us [START_REF] Planat | Graphs When the Hilbert Space Dimension Contains a Square: Why the Dedekind Psi Function?[END_REF], we shall address below a particular case of the tensor product of a qubit and a single qudit, where d = 2 k and an integer k ≥ 2. Analyzing the first four cases in the sequence, viz. k = 2, 3, 4 and 5, by computer was already sufficient for us to infer the generic pattern of geometries behind the corresponding Pauli groups for any k ≥ 2. This sequence of finite geometries is characterized by a remarkable nesting of fundamental building blocks that are nothing but generalized quadrangles of order two, GQ(2, 2)s. This seems to be the most crucial finding given the facts that this generalized quadrangle 1) is the geometry behind the generalized Pauli group of two-qubits [START_REF] Planat | On the Pauli Graphs of N-qudits[END_REF][START_REF] Saniga | Projective Ring Line Encompassing Two-Qubits[END_REF] and 2) also plays an essential role in finite geometrical aspects of the still mysterious black-hole/qubit correspondence [START_REF] Lévay | Black Hole Entropy and Finite Geometry[END_REF].

2 Qudits, Generalized Pauli Groups and the Doily

X (d) =        0 0 . . . 0 1 1 0 . . . 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . 0 0 . . . 1 0        and Z (d) =        1 0 0 . . . 0 0 ω 0 . . . 0 0 0 ω 2 . . . 0 . . . . . . . . . . . . . . . 0 0 0 . . . ω d-1        . (1)
The (generalized) Pauli group generated by X (d) and Z (d) will be denoted as G (d) . For all s ∈ Z d we have

X (d) Z (d) |s = ω s (d) |s + 1 and Z (d) X (d) |s = ω s+1 (d) |s + 1 .
This gives the basic relation

ω (d) X (d) Z (d) = Z (d) X (d)
which implies that each element of G (d) can be written in the unique normal form

ω a (d) X b (d) Z c (d) for some integers a, b, c ∈ Z d . (2)
The uniqueness of this normal form implies that G (d) is a group of order d 3 . From (2) it is also readily seen that

(ω a (d) X b (d) Z c (d) )(ω a ′ (d) X b ′ (d) Z c ′ (d) ) = ω b ′ c+a+a ′ (d) X b+b ′ (d) Z c+c ′ (d) . which implies that commutator of two operators W and W ′ , [W, W ′ ] := W W ′ W -1 W ′ -1 , acquires in our case the form [ω a (d) X b (d) Z c (d) , ω a ′ (d) X b ′ (d) Z c ′ (d) ] = ω cb ′ -c ′ b (d) I (d) . (3)
Let us recall that two operators commute if, and only if, their commutator (taken in any order) is equal to I (d) (the identity matrix); hence, G (d) is a non-commutative group. We also mention in passing that there are two important normal subgroups of G (d) : its centre Z(G (d) ) and its commutator subgroup G ′ (d) , the two being identical

G ′ (d) = Z(G (d) ) = {ω a (d) I (d) : a ∈ Z d }.
It is important to observe that a and a ′ do not occur on the right-hand side of eq. ( 3). So, in order to study the commutation relations between the elements of G (d) one can disregard the complex phase factors ω a (d) and work solely with

d 2 matrix products X b (d) Z c (d) .
The final notion we shall need in the sequel is that of a finite generalized quadrangle of order (s, t), usually denoted GQ(s, t). This is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [START_REF] Payne | Finite Generalized Quadrangles[END_REF]: (i) each point is incident with 1+t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). If s = t, S is said to have order s. A generalized quadrangle with both s > 1 and t > 1 is called thick. The smallest thick generalized quadrangle is obviously the (unique) GQ(2, 2), often dubbed the "doily." This quadrangle is endowed with 15 points/lines, with each line containing three points and, dually, each point being on three lines. Moreover, it is a self-dual object, i. e., isomorphic to its dual. In the older literature this geometry is also known as the Cremona-Richmond configuration (see, e. g., [START_REF] Coxeter | Self-Dual Configurations and Regular Graphs[END_REF]) and essential features of its structure are depicted in Figure 1.

Figure 1: A picture of the generalized quadrangle of order two, the doily. The circles stand for its points, while its lines are given by triples of points lying on the same segments [START_REF] Saniga | Multiple Qubits as Symplectic Polar Spaces of Order Two[END_REF] and/or arcs of circles [START_REF] Planat | Qudits of Composite Dimension, Mutually Unbiased Bases and Projective Ring Geometry[END_REF]. (The points of different colors represent two distinguished subgeometries, which are, however, irrelevant for our further reasoning.)

3 The Qubit-Qu2 k it Pauli Group and Its Geometry As already mentioned, we shall focus on the generalized Pauli group associated with the Hilbert space of the form C 2 ⊗ C 2 k , where an integer k ≥ 2.

We shall first handle the k = 2 case. Disregarding the phase factors, from eqs. ( 1) and ( 2) one sees that the elements of the group associated with C 2 (qubit) are

{I (2) , X (2) , Z (2) , X (2) Z (2) },
and those for C 4 read {I (4) , X (4) , X 2 (4) , X 3 (4) , Z (4) , X (4) Z (4) , X 2 (4) Z (4) , X 3 (4) Z (4) , Z 2 (4) , X (4) Z 2 (4) , X 2 (4) Z 2 (4) , X 3 (4) Z 2 (4) , Z 3 (4) , X (4) Z 3 (4) , X 2 (4) Z 3 (4) , X 3 (4) Z 3 (4) }.

12 ≡ {22, 36, 50, 68, 82, 96, 118, 143, 157, 171, 185, 203, 217, 239, 253}.

One observes that each twelve-element set features three common elements and the two sets in a pair share a single element, this being the element "4", "32" and "36", respectively. Paralleling the preceding case, we again form the point-line incidence geometry where points are the 87 maximum sets, but where two points are now collinear if the corresponding sets have exactly seven (=2 3 -1) elements in common. This geometry, depicted in Figure 3, consists of seven doilies. These form three pencils, each comprising the doily generated by the 15 exceptional sets and two doilies coming from pair A if the missing line {a, b, c} is added, from pair B if the missing line {d, e, f } is added and from pair C if the missing line {g, h, i} is added. The three lines in questions are pairwise skew; for the reader well-acquainted with the structure of the doily it may be interesting to learn that in the dual of the distinguished doily these lines answer to a tricentric triad of points.

For the next case in the hierarchy, k = 4 (qubit-quhexadecit), our computer calculations showed that 1023 elements of the corresponding Pauli group form 183 maximum sets of 31 elements each, out of which 39 are exceptional. We shall not go into much detail here but mention only the outcome of our analysis, illustrated in line three of Figure 4. The geometry is defined as in the previous two cases, save for the fact that collinearity is now synonymous with sharing fifteen (= 2 4 -1) elements. The three "central" doilies in a pencil answer to the 39 elements of the exceptional set, with the carrier line accounting for the finer, 36+3 split; hence, this subgeometry is isomorphic to the full k = 2 case (see Figure 4). The 12 doilies at sides, underlying 144 ordinary points, form three quadruples, each being a "satellite" to one central doily. A quadruple further splits into two pairs, and the two doilies in each pair form a pencil with the corresponding "central" doily. So we have altogether seven pencils of doilies.

In the k = 5 case, we find 4095 elements of the associated Pauli group forming 375 maximum sets of 63 elements each. The 87 of them are exceptional, and their geometry is fully isomorphic to the k = 3 one (see Figure 4). The 24 satellite doilies account for the remaining 288 ordinary points. They form 12 pairs and with 'six 'central" doilies define 12 pencils; taking into account three "central" pencils, this case is thus seen to be altogether endowed with 15 pencils.

These four cases obviously suffice to infer the geometry for an arbitrary k ≥ 2. This geometry, which features 2 k-2 36 ordinary points, will comprise 2 k -1 copies of the generalized quadrangle of order two that form 2 k-1 -1 pencils arranged into a remarkable nested configuration. The core of this configuration consists of exceptional points and, starting with k = 4, is isomorphic to the full k -2 geometry. From Figure 4 it can readily be discerned how to construct generically the k + 2 geometry around the k one. One simply takes a "peripheral" doily, i. e. any doily that does not consist of exceptional points only, and associate with it two pairs of doilies in such manner that they form two pencils. Obviously, we shall get two distinct sequence patterns according as the initial geometry is that of the k = 2 or k = 3 case.

Conclusion

Following and extending the strategy adopted in [START_REF] Planat | Graphs When the Hilbert Space Dimension Contains a Square: Why the Dedekind Psi Function?[END_REF], a finite geometrical treatment of the generalized Pauli groups associated with the Hilbert spaces of type C 2 ⊗ C 2 k , where an integer k ≥ 2, has been performed. A point-line incidence structure was defined as follows: its points are maximum sets of pairwise commuting group elements and two distinct points are collinear if the corresponding sets share 2 k -1 elements. The points were found to be of two distinct kinds, referred to as ordinary and exceptional; a point of the latter kind being represented by such maximum set that shares with any other maximum set at least 2 k-1 -1 elements. A computer-based analysis of the first four cases in the sequence implies that, in general, this geometry features 2 k -1 copies of the doily that form 2 k-1 -1 pencils arranged into a remarkable nested configuration reminding a fractal-like behaviour with growing k. The core of this geometry, being generated by exceptional points, is isomorphic to the full geometry of the k -2 case. Finding the doily to be the fundamental building block of these geometries is pleasing of its own, all the more that -as already stressed in introduction -this generalized quadrangle is the geometry behind the generalized Pauli group of two-qubits and it plays also an essential role in finite geometrical aspects of the still mysterious black-hole/qubit correspondence. We hope that all the above-discussed properties will soon be given a rigorous, computer-free proof.

1 s=0α 1 s=0|α s | 2 = 1 .

 111 Given an integer d > 1 and Z d := {0, 1, . . . , d -1}, where addition and multiplication of elements from Z d is understood modulo d, we consider the d-dimensional complex Hilbert space C d and denote by { |s : s ∈ Z d } a computational basis of C d . A qudit in C d is a vector |ψ = ds |s , where α s ∈ C and d-Taking ω (d) to be a fixed primitive d-th root of unity (e. g., ω (d) = exp(2πi/d)), we define unitary X (d) ("shift") and Z (d) ("clock") operators on C d via X (d) |s = |s + 1 and Z (d) |s = ω s (d) |s for all s ∈ Z d . In the computational basis

Figure 3 :

 3 Figure 3: A schematic sketch of the geometry behind the qubit-quoctic generalized Pauli group. The seven doilies lie in three pencils on the distinguished doily (middle); the carrier lines of these pencils are shown in boldface.

Figure 4 :

 4 Figure 4: A compact illustration of the properties of the above-described finite geometries for (top to bottom) k = 2, 3, 4 and 5. Each circle represents a doily and each triangle stands for a pencil of doilies; the filled circle means that the corresponding doily contains solely exceptional points.

Acknowledgements

This work was partially supported by the VEGA grant agency projects 2/0092/09 and 2/0098/10.

From the last two equations it follows that the generalized Pauli group of C 2 ⊗ C will feature the following elements (again disregarding the phase factors) {I (2) ⊗ I (4) , I (2) ⊗ X (4) , I (2) ⊗ X 2 (4) , . . . , I (2) ⊗ X 2 (4) Z 3 (4) , I (2) ⊗ X 3 (4) Z 3 (4) , X (2) ⊗ I (4) , X (2) ⊗ X (4) , X (2) ⊗ X 2 (4) , . . . , X (2) ⊗ X 2 (4) Z 3 (4) , X (2) ⊗ X

commutes with every other element, we consider only the remaining 63 elements/operators and, for convenience, number them from 1 to 63 in the order as they appear in the last equation.

To find the geometry behind this Pauli group, we first look for maximum sets of mutually commuting elements. We find that each such set features seven elements and their total number is 39. It is easy to verify that this split into three above-given families is unique as the twelve sets in each family have a single element in common ("2", "8" and "10", respectively).

Next, we define the point-line incidence geometry whose points are these 39 maximum sets of mutually commuting elements and where two points are collinear if the corresponding sets have exactly three (2 2 -1) elements in common. The three exceptional sets then form a distinguished line in this geometry and each family generates a copy of the doily from which one line is deleted; this line being nothing but the distinguished line! Hence, this point-line incidence structure comprises three doilies on a common line, a pencil of doilies -as illustrated in Figure 2. In a completely analogous way, we shall analyze the k = 3 case. Here, the "relevant" elements of the Pauli group related to the Hilbert space C 8 are (after dropping the subscript "8" for better readability of the formulas) {I, X, X 2 , X 3 , X 4 , X 5 , X 6 , X 7 , Z, ZX, ZX 2 , ZX 3 , ZX 4 , ZX 5 , ZX 6 , ZX 7

Z 4 X 3 , Z 4 X 4 , Z 4 X 5 , Z 4 X 6 , Z 4 X 7 , Z 5 , Z 5 X, Z 5 X 2 , Z 5 X 3 , Z 5 X 4 , Z 5 X 5 , Z 5 X 6 , Z 5 X 7 , Z 6 , Z 6 X, Z 6 X 2 , Z 6 X 3 , Z 6 X 4 , Z 6 X 5 , Z 6 X 6 , Z 6 X 7 , Z 7 , Z 7 X, Z 7 X 2 , Z 7 X 3 , Z 7 X 4 , Z 7 X 5 , Z 7 X 6 , Z 7 X 7 }, which yield the following 255 (disregarding the identity) distinct matrices of the group associated