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

AND Duc-Trung Hoang
email: duc-trung.hoang@math.u-bordeaux.fr

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

CONTROLLABILITY AND OBSERVABILITY FOR NON-AUTONOMOUS EVOLUTION EQUATIONS: THE AVERAGED HAUTUS TEST

We consider the observability problem for non-autonomous evolution systems (i.e., the operators governing the system depend on time). We introduce an averaged Hautus condition and prove that for skew-adjoint operators it characterizes exact observability. Next, we extend this to more general class of operators under a growth condition on the associated evolution family. We give an application to the Schrödinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient.

Introduction

Observability is an important concept in system and control theory. It treats the question to which extent an observation, i.e., partial knowledge of the solution of an evolution equation, determines its initial or final state. The theory has been studied for several decades for systems of the form:

(1.1)

   x ′ (t) + Ax(t) = 0 t ∈ [0, T ] x(0) = x 0 y(t) = Cx(t)
in which the two operators A and C are independent of time t and satisfy appropriate conditions such as -A, with domain D(A), generates a strongly continuous semigroup on a Hilbert space H and C is bounded from D(A) into another Hilbert space Y . Observability consists of unique determination or recovery of the initial (or final) time state under the knowledge of the observed solution y(•). Recall that in the case of matrices A and C (finite dimensional setting), all observation concepts coincide and can be characterized in various manners. The Kalman rank condition is certainly the most known version; it states that C is observable if and only if the matrix

[C | CA | CA 2 | . . . | CA n-1 ]
has full rank. An equivalent statement is the Hautus lemma: it characterizes observability by the condition ∀λ ∈ C : rank[λI -A, C] = n that clearly is equivalent to the condition

(1.2) Cx 2 + (λI -A)x 2 ≥ κ x 2 .
In an infinite-dimensional setting with operators A, C, instead of matrices, rank conditions are not appropriate. However, the Hautus test in the form (1.2) can be generalized, and has actually been proposed in [START_REF] Russell | A general necessary condition for exact observability[END_REF] as a criterion for observability. Russell and Weiss conjectured in [START_REF] Russell | A general necessary condition for exact observability[END_REF] that this inequality characterizes exact observability. They proved in [START_REF] Russell | A general necessary condition for exact observability[END_REF] that the conjecture is valid for bounded and invertible operators A. Later, Jacob and Zwart [START_REF] Jacob | Observability of diagonal systems with a finite-dimensional output operator[END_REF] showed equivalence for diagonal semigroup generator on a Riesz basis if the output space Y is finite dimensional. The general conjecture was later proved to be wrong, see [START_REF] Jacob | Counterexamples concerning observation operators for C0-semigroups[END_REF]. Note however, that if C is admissible and A has a bounded H ∞ -calculus on a suitable sector (which is, in turn a consequence of admissibility and exact observation, see Proposition 5.1 in [START_REF] Haak | Exact observability, square functions and spectral theory[END_REF]), then it does not seem to be known whether the Hautus condition implies observability. There exist other formulations of the Hautus condition (or spectral condition) and there are several cases where it implies exact observability. This holds for example if A generates a unitary group. We refer to [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF][START_REF] Zhou | Hautus condition on the exact controllability of conservative systems[END_REF] for early results with bounded observations, and [START_REF] Burq | Zworski Geometric control in the presence of a black box[END_REF][START_REF] Miller | Controllability cost of conservative systems: resolvent condition and transmutation[END_REF] for successive extensions. These have subsequently been generalized (see [START_REF] Jacob | On the Hautus test for exponentially stable C 0 -groups[END_REF]) to groups with certain growth bounds. See also [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF] for more information and references on this subject.

In this paper we consider first order non-autonomous evolution equations of the following form:

(A,C)

   x ′ (t) + A(t)x(t) = 0 t ∈ [0, τ ] x(0) = x 0 y(t)
= C(t)x(t).

The difference with (1.1) is that we allow operators A and C to depend on time t. To be precise, let τ > 0 and assume that for t ∈ [0, τ ], the operator A(t) generates a strongly continuous contraction semigroup (e -sA(t) ) s≥0 on the Hilbert space H. We suppose further that there exists a densely and continuously embedded subspace D ֒→ H such that that for all t ∈ [0, T ], D(A(t)) = D and that t → A(t)v is continuously differentiable in H for every v ∈ D. These assumptions are sufficient to guarantee that the Cauchy problem x ′ (t) = A(t)x(t), x(0) = x 0 admits a solution, see e.g. [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]Sections 5.3 and 5.4]. For each t, C(t) : D → Y is a bounded operator. Then, for initial data x 0 ∈ D, the solution x to (A,C) satisfies x(t) ∈ D for each t ≥ 0 and hence y(t) is well defined. We define observability concepts (and controllability concepts for the adjoint system) as in the autonomous case (1.1).

In the case of time-dependent matrices, a famous result of Silverman and Meadows [START_REF] Silverman | Controllability and observability in time-variable linear systems[END_REF] characterizes exact observability and controllability. Their arguments have been adapted to certain infinite dimensional settings, see for example [START_REF] Alabau-Boussouira | Olive Internal controllability of first order quasi-linear hyperbolic systems with a reduced number of controls[END_REF][START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF][START_REF] Aksikas | Dochain Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems[END_REF]. Our main objective is different. We seek to prove observability from a certain Hautus type condition. In order to do this, we introduce the following averaged Hautus conditions: for all λ ∈ C and all x ∈ D, or

x 2 ≤ m 2 1
x 2 ≤ m 2 1 τ τ 0 C(s)x 2 ds + M 2 1 τ τ 0 (iξ + A(s))x 2 ds
for all ξ ∈ R and x ∈ D. These inequalities do coincide with the usual Hautus conditions if the operators A and C are independent of t. We prove that these averaged Hautus conditions imply exact observability when the operators A(t) are skew-adjoint. This result is refined to the case of invertible evolution families (not necessarily unitary) under certain growth constraints. We apply these results to Schrödinger equations with time dependent potentials and to a damped wave-equation with time-dependent damping. Finally, we mention the papers [START_REF] Yu | Èmanuilov, Controllability of parabolic equations (Russian)[END_REF], [START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF]and the references therein on observability (or controllability) of parabolic equations (with time dependent coefficients). The approach in these papers is based on Carleman estimates and it differs from ours.

Preliminary results

Recall that we suppose A(t) : D → H to have a fixed domain, that t → A(t)v is continuously differentiable in H for every v ∈ D and each semigroup e -sA(t) is a contraction on H. By [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]Sections 5.3 and 5.4] there exists a unique evolution family (U (t, s)) 0≤s≤t≤τ on H generated by A(t) 0≤t≤τ . This evolution family satisfies the following properties.

(1) U (t, s) ≤ M e -ω(t-s) for some

ω ∈ R (2) For all v ∈ D, ∂ + ∂t U (t, s)v| t=s = -A(s)v, ∂ + ∂t U (t, s)v = -A(t)U (t, s)v. (3) For all v ∈ D, ∂ ∂s U (t, s)v = U (t, s)A(s)v. (4) U (t, s)D ⊆ D (5) For all v ∈ D, (s, t) → U (t, s)v is continuous in D for 0 ≤ s ≤ t ≤ T . For every v ∈ D, the evolution equation (CP) d dt η(t) + A(t)η(t) = 0 0 ≤ s ≤ t ≤ τ η(s) = v
has a unique solution. This solution is given by η(t) = U (t, s)v. For f ∈ L 1 (0, τ ; H), the non homogeneous problem (NHCP)

d dt η(t) + A(t)η(t) = f (t) 0 ≤ s ≤ t ≤ τ η(s) = v ∈ H.
has then a mild solution given by (2.1)

η(t) = U (t, s)v + t s U (t,
C(t)U (t, s)x 2 dt ≤ M 2 τ x 2 ∀x ∈ D, s ∈ [0, τ ].
(one can also consider a weaker admissibility notion by requiring the above inequality for s = 0, only). For a single operator C(t 0 ) such that

τ 0 C(t 0 )U (t, s)x 2 dt ≤ M τ x 2 ∀x ∈ D we say that C(t 0 ) is admissible for (A(t)) t∈[0,T ] .
For averaged admissible observations, Ψ s,τ extends to a bounded operator from H to L 2 (s, τ ; Y ) which we denote again by Ψ s,τ .

In this definition the norm inside the integral is taken in Y and the norm of x is taken in H. We always use the same notation • for both, the difference will be clear from the context. Definition 2.2. Suppose that (C(t)) t is an averaged admissible observation for (A(t)) t . We say that the system (A, C) is a) exactly averaged observable in time τ if the map Ψ s,τ is bounded from below in the sense that there exists a constant κ τ > 0 such that for all x ∈ D τ 0

C(t)U (t, 0)x 2 dt ≥ κ τ x 2 .
For a given t 0 ∈ [0, τ ], the system (A, C(t 0 )) is exactly observable at time τ if

τ 0 C(t 0 )U (t, 0)x 2 dt ≥ κ τ x 2 .
b) final-time averaged observable in time τ if there exists a constant κ τ > 0 such that

τ 0 C(t)U (t, 0)x 2 dt ≥ κ τ U (τ, 0)x 2 ∀x ∈ D.
As above we define final observability for the simple operator C(t 0 ) for some t 0 as

τ 0 C(t 0 )U (t, 0)x 2 dt ≥ κ τ U (τ, 0)x 2 .
c) approximately averaged-observable in time τ if ker Ψ s,τ = {0} for all 0 ≤ s < τ . Again we define approximate observability for a single operator C(t 0 ) if (A, C(t 0 )) is approximate observable in average as above.

In order to justify the use of the term "averaged" in the previous notions of observability, we note that it might be possible that (A, C(t 0 )) is not exactly (or final or approximately) observable for some C(t 0 ) or even for all t 0 ∈ J for some subset J of [0, τ ] but (A, C) is exactly (or final or approximately) observable in average. In order to see this, we consider the autonomous case A(t) = A and an observation operator C such that the autonomous system is exactly (or null or approximately) observable at time τ 0 . Define

C(t) = C, t ∈ [0, τ 0 ] 0, t ∈ (τ 0 , τ ]. Then τ 0 C(t)e -tA x 2 dt ≥ τ0 0 C(t)e -tA x 2 dt ≥ κ τ x 2 .
Hence the averaged observability property for (A, C(t)) at time τ holds but the system (A, C(t 0 )) is not observable for t 0 ∈ (τ 0 , τ ] at any time. The same observation is valid for null and approximate average observability.

Along with (A,C) we consider a controlled evolution equation. First, we recall the following: one can construct an extrapolation space H -1 and extrapolated operators A -1 (t) such that the following diagram commutes

H H -1 (t) D H A-1(t) A(t) i i
One way to realize H -1 (t) is to take the completion of H with respect to a resolvent norm (λ -A(t)) -1 x H or via its identification with D(A(t) * ) ′ . For all this we refer to [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]Chapter II.5].

In order to keep the abstract setting simple we will suppose for the rest of this section that D(A(t) * ) =: D * is independent of time as well and equivalent norms with constants independent of t. Note that if for all t ∈ [0, τ ], A(t) = A(0) + R t with a bounded operator on H, then

A(t) * = A(0) * + R * t with domain D * := D(A(0) * ) independent of t.
In the setting of the averaged Hautus test we consider later, we will make the assumption A(t) = A(0) + R t with a family of uniformly bounded operators R t on H. In this case H -1 (t) = H -1 and have equivalent norms with constants independent of t.

Let U be another Hilbert space and let B(t

) : U → H -1 is bounded for each t ∈ [0, τ ]. We consider in H -1 the evolution equation (A,B) x ′ (t) + A(t)x(t) = B(t)u(t) t ∈ [0, τ ] x(s) = 0.
Since the mild solution is of the form (2.1), we have the naturally associated operator

(2.2) Φ s,τ u = τ s U (τ, r)B(r)u(r) dr (τ ≤ τ ) to (A,B).
Definition 2.3 (Averaged admissible controls). Let (B(t)) t∈[0,τ ] be a family of bounded operators in L (U ; H -1 ). We say that (B(t)) t are averaged admissible controls for (A(t)) t∈[0,τ ] if there exists a constant M τ > 0 such that the solution x to (A,B) satisfies x(t) ∈ H and for all s ∈ [0, τ )

τ s U (τ, r)B(r)u(r) dr 2 ≤ M 2 τ u 2 L2(s,τ ;U)
for all u ∈ D(0, τ ; U ) (one can also consider a weaker admissibility notions by requiring the above inequality for s = 0, only).

Let us consider the retrograde final-value problem

(2.3) z ′ (t)-A(t) * z(t) = 0 z(τ ) = z τ .
Observe that for x ∈ D and Proof. The following calculation is standard.

x * ∈ D * , d dt x, U (τ, t) * x * = d dt U (τ, t)x, x * = -U (τ, t)A(t)x, x * = x, -A(t) * U (τ, t) * x * so that z(t) = U (τ, t) * z τ solves the retrograde equation (2.3) on [s, τ ] for all 0 ≤ s < τ .
sup u 2 ≤1 τ s U (τ, r)B(r)u(r) dr = sup u 2 ≤1 sup x * ≤1 τ s U (τ, r)B(r)u(r), x * dr = sup x * ≤1 sup u 2 ≤1 τ s u(r), B(r) * U (τ, r) * x * dr = sup x * ≤1 τ s B(r) * U (τ, r) * x * 2 dr 1 /2
. Definition 2.5. Let (B(t)) t be averaged admissible controls for (A(t)) t∈[0,τ ] . We say that (A,B) is a) Exactly averaged controllable in time τ if for any s ∈ [0, τ ) and x s , x τ ∈ H, there exist u ∈ L 2 (s, τ ; U ) such that the mild solution x satisfies x(s) = x s and x(τ ) = x τ . This definition coincides with the usual one in the autonomous case, that is, given two states x s , x τ ∈ H we find a control u such that the solution takes the value x s at the initial time t = s and the value x τ at time t = τ . b) approximately averaged controllable in time τ if for any 0 ≤ s < τ and any x s , x τ ∈ H and ε > 0, there exist u ∈ L 2 (0, τ ; U ) such that x(s) = x s and x(τ )x τ < ε. c) averaged null controllable in time τ if for every 0 ≤ s < τ and every x s ∈ H, there exist u ∈ L 2 (s, τ ; U ) such that the mild solution x satisfies x(s) = x s and x(τ ) = 0.

Since the mild solution is given by

x(t) = U (t, s)x s + t s U (t, r)B(r)u(r) dr
it is clear that in order to obtain exact averaged controllability it suffices to consider the case where x(s) = 0.

U (τ, s) * z τ 2 ≤ δ 2 Φ * s,τ z τ 2 = δ 2 τ s B(t) * U (τ, t) * z τ 2 dt
for some constant δ > 0. But U (τ, s) * z τ = z(s) where z(•) is the solution of the retrograde equation (2.3).

The averaged Hautus test: skew-adjoint operators

Throughout this section, the family of operators A(t) 0≤t≤τ is as before. Let C(t) 0≤t≤τ be a family of bounded operators from D to a Hilbert space Y . In the autonomous case A(t) = A and C(t) = C for all t, it is well known that for admissible C the exact observability of the system (A, C) implies the so-called Hautus test (or spectral condition)

(3.1) x 2 ≤ m 2 Cx 2 + M 2 (iξ + A)x 2
for some positive constants m and M and all ξ ∈ R and x ∈ D(A). There is also another condition with λ ∈ C in place of iξ, see below. In the general non-autonomous situation we introduce an integrated (or averaged) version of this test. We also study, as in the autonomous case, when the averaged Hautus test is necessary and/or sufficient for averaged observability. We start with the "necessary" part.

Proposition 3.1. Suppose that (C(t)) is averaged admissible for (A(t)). If the system (A,C) is exactly averaged observable at time τ > 0 then there exist positive constants m and M such that:

(AH.1) x 2 ≤ m 2 1
(AH.2) x 2 ≤ m 2 1 τ τ 0 C(s)x 2 ds + M 2 1 τ τ 0 (iξ + A(s))x 2 ds
for all ξ ∈ R and x ∈ D.

Remark 3.2. If C(s) = C for all s then (AH.1) can be written as:

(AH.3) x 2 ≤ e 2τ Re(λ) -1 2τ Re(λ) m 2 Cx 2 + M 2 1 τ τ 0 e Reλ.s (λ+A(s))x ds 2
If, in addition, A(s)=A then both assertions coincide with the classical Hautus (or spectral) conditions. We call the conditions (AH.1) and (AH.2) averaged Hautus tests.

Proof. The proof is similar to the autonomous case. We start from d ds e λs C(t)U (t, s)x = λe λs C(t)U (t, s)x + e λs C(t)U (t, s)A(s)x for x ∈ D. Integrating on [0, τ ] yields

e λt C(t)x -C(t)U (t, 0)x = t 0 C(t)U (t, s)(A(s) + λ)xe λs ds. Hence, τ 0 C(t)U (t, 0)x 2 dt ≤ 2 τ 0 C(t)xe λt 2 dt + 2 τ 0 t 0 C(t)U (t, s)(λ+A(s))xe λs ds 2 dt Since (A,C
) is exactly averaged observable on [0, τ ], the left hand side is bounded below by m 0 x 2 for some constant m 0 > 0. We estimate the second term on the right hand side

I := τ 0 t 0 C(t)U (t, s)(λ+A(s))xe λs ds 2 dt 1 /2 = sup τ 0 t 0 C(t)U (t, s)(λ+A(s))xe λs , g (t)
H ds dt :

g L2(0,τ ;H) ≤ 1 = sup g L 2 ≤1 τ 0 (λ+A(s))x e λs , τ s U (t, s) * C(t) * g(t) dt H ds ≤ sup g L 2 ≤1 τ 0 (λ+A(s))x e λs H τ s U (t, s) * C(t) * g(t) dt H ds .
By Lemma 2.4 and the admissibility assumption of (C(t)), there exists a constant K τ > 0 such that Now we study the converse. In the autonomous case i.e., A(s) = A and C(t) = C, it is well known that condition (AH.2) implies the exact observability if the single operator A is skewadjoint. We extend this result to our more general situation. Theorem 3.3. Suppose that A(t) ∈ L (D; H) be a family of skew-adjoint operators generating an evolution family U (t, s) 0≤s≤t≤τ . Suppose that the differences of the operators A(t) are bounded and satisfy the estimate

I ≤ K τ τ 0 (λ+A (
A(t) -A(s) L (H) ≤ L ∀t, s ∈ [0, τ ]
for some constant L < 1 √ 2M . Assume that C(t) ∈ L (D; Y ) is a family of averaged admissible observation operators and that the second averaged Hautus condition (AH.2) holds with positive constants m and M . Then, for all τ > τ * := 2πM √ 1-2L 2 M 2 there exists κ τ > 0 depending on M, L and τ such that, for all x ∈ D the exact averaged observability estimate

(3.2) 1 τ τ 0 τ 0 C(s)U (t, 0)x 2 dt ds ≥ κτ m 2 x 2
holds. In particular, if C(s) = C is constant, then the system (A,C) is exactly averaged observable for τ > τ * , i.e, for all x ∈ D,

τ 0 CU (t, 0)x 2 dt ≥ κτ m 2 x 2 .
Proof. We proceed in a similar way as in the autonomous case. Let τ > 0, ϕ ∈ H 1 0 (0, τ ) and x ∈ D. For t, s ∈ [0, τ ], let h(t) := ϕ(t)U (t, 0)x and f (t, s) := h ′ (t) + A(s)h(t). Note that h and f (., s) can be extended continuously by zero outside (0, τ ) since ϕ ∈ H 1 0 (0, τ ). We write f (ξ, s) for the partial Fourier transform of f with respect to the first variable, and observe that

f (ξ, s) = R e -itξ f (t, s) dt = R e -itξ h ′ (t) dt + R e -itξ A(s)h(t) dt = iξ h(ξ) + A(s) h(ξ)
where we use the fact that each operator A(s) is closed in order to have A(s)h(ξ) = A(s) h(ξ). We apply (AH.2) with z 0 = h(ξ) to obtain

h(ξ) 2 ≤ m 2 τ τ 0 C(s) h(ξ) 2 ds + M 2 τ τ 0 (iξ + A(s)) h(ξ) 2 ds = m 2 τ τ 0 C(s) h(ξ) 2 ds + M 2 τ τ 0 f (ξ, s) 2 ds.
We integrate over all ξ ∈ R and use Plancherel's theorem together with the fact that C(s) h(ξ) = C(s)h(ξ) to deduce

(3.3) τ 0 h(t) 2 dt ≤ m 2 τ τ 0 τ 0 C(s)h(t) 2 dt ds + M 2 τ τ 0 τ 0 f (t, s) 2 dt ds.
We estimate the last term on the right hand side as follows

τ 0 τ 0 f (t, s) 2 dt ds = τ 0 τ 0 h ′ (t) + A(s)h(t) 2 dt ds = τ 0 τ 0 ϕ ′ (t)U (t, 0)x -ϕ(t)A(t)U (t, 0)x + ϕ(t)A(s)U (t, 0)x 2 dt ds ≤ 2τ τ 0 U (t, 0)x 2 |ϕ ′ (t)| 2 dt + 2 τ 0 τ 0 (A(t) -A(s))U (t, 0)x 2 |ϕ(t)| 2 dt ds. (3.4) By skew-adjointness, d dt U (t, s)x 2 = -2Re A(t)U (t, s)x, U (t, s)x = 0
for x ∈ D and so U (t, s) is unitary for 0 ≤ s ≤ t ≤ τ . Therefore (3.3) can be rewritten as

x 2 τ 0 ϕ(t) 2 dt ≤ m 2 τ τ 0 τ 0 C(s)U (t, 0)x 2 ϕ(t) 2 dt ds + 2M 2 x 2 τ 0 ϕ ′ (t) 2 dt + 2L 2 M 2 x 2 τ 0 ϕ(t) 2 dt.
Hence

κ(ϕ) x 2 ≤ m 2 τ τ 0 τ 0 C(s)U (t, 0)x 2 ϕ(t) 2 dt ds
where

κ(ϕ) = (1 -2L 2 M 2 ) τ 0 ϕ(t) 2 dt -2M 2 τ 0 ϕ ′ (t) 2 dt .
We have to chose ϕ such that the constant κ(ϕ) is positive. Taking the first eigenfunction of the Dirichlet Laplacian on (0, τ ), i.e., ϕ(t) := sin tπ τ , we maximize κ(ϕ) and obtain from ϕ ∞ = 1

κτ m 2 x 2 ≤ τ 0 τ 0 C(s)U (t, 0)x 2 dt ds where κ = (1 -2L 2 M 2 ) τ 2 -π 2 M 2 τ
. To ensure κ > 0 we need L 2 < 1 2M 2 and τ > τ * . Remark 3.4.

(1) In (3.4) we have used for simplicity the inequality

(a + b) 2 ≤ 2(a 2 + b 2 ) but we could instead use (a + b) 2 ≤ (1 + r)a 2 + (1 + r -1
)b 2 for any r > 0. In this case, we obtain the theorem (with the same proof ) with the conditions L <

1 M √ 1+r and τ * = πM √ 1+r -1 √ 1-M 2 (1+r)L 2 .
(2) If A(t) = A and hence L = 0 we obtain (from the previous remark) as minimal control time τ * = πM . This is the usual minimal time in the case of unitary groups.

(3) In the last assertion of theorem, if instead of C(s) = C, we assume that

C(s) -C(t) ≤ L 0 |t -s| α
for some positive constants α and L 0 we obtain that for L 0 small enough, the system (A,C) is exactly averaged observable. Indeed, we have from (3.2) (5) We have assumed in the theorem that A(t) are skew-adjoint operators in order to have U (t, s) is a unitary operator on H. The previous proof works under the assumption that

κ x 2 ≤ 2 τ 0 τ 0 (C(t) -C(s))U (t, 0)x 2 ds dt + 2 τ 0 τ 0 C(t)U (t, 0)x 2 ds dt ≤ 2L 0 τ 0 τ 0 |t -s| 2α ds dt x 2 + 2τ τ 0 C(t)U (t, 0)x 2 dt = 2L 0 τ 2α+2 (2α + 1)(α + 1) x 2 + 2τ τ 0 C(t)U (t, 0)x 2 dt. ( 4
K 0 x ≤ U (t, 0)x ≤ K 1 x , x ∈ H
for some positive constants K 0 and K 1 . The statement of the theorem holds with different conditions L and τ * (depending on K 0 and K 1 ).

The averaged Hautus test: a more general class of operators

In this section we extend Theorem 3.3 to a more general class of operators. More precisely, we consider operators A(t) for which the corresponding evolution family U (t, s) is not necessarily an isometry but satisfies an estimate of the form (4.1) ke α(t-s) x ≤ U (t, s)x ≤ Ke β(t-s) x , x ∈ H for some constants k, K, α and β. This question was considered in the autonomous case A(t) = A and C(t) = C by Jacob and Zwart [START_REF] Jacob | On the Hautus test for exponentially stable C 0 -groups[END_REF]. We shall follow similar ideas as in their paper. Note however, even in this autonomous case, the result is very much less precise than in the case of unitary groups. In particular, the minimal time for observability obtained in [START_REF] Jacob | On the Hautus test for exponentially stable C 0 -groups[END_REF] is 1 β-α . This value becomes large as α and β are close and this is not consistent with the result on unitary groups.

The main tool is the following optimal Hardy inequality.

Theorem 4.1 ( Gurka [START_REF] Gurka | Petr Generalized Hardy's inequality[END_REF], Opic-Kufner [START_REF] Opic | Hardy-type inequalities[END_REF] ). Let v, w ≥ 0 be weight functions on [0, τ ]. Then the weighted Hardy inequality

(4.2) ϕ L2(0,τ ;w(x)dx) ≤ C H ϕ ′ L2(0,τ ;v(x)dx)
holds for all ϕ ∈ H 1 0 (0, τ ) if and only if

B := sup y x w(t) dt min x 0 1 v(t) dt, τ y 1 v(t) dt : 0 < x, y < τ is finite. In this case, the optimal constant C H in (4.2) satisfies B √ 2 ≤ C H ≤ 4B.
We make a basic remark on evolution families U (t, s) 0≤s≤t . Given U (t, s) which is exponentially bounded, i.e., U (t, s)x ≤ Ke β(t-s) x . If in addition each U (t, s) is invertible then writing V (t) := U (t, 0) gives

V (t) = U (t, 0) = U (t, s)U (s, 0) = U (t, s)V (s) ⇐==⇒ U (t, s) = V (t)V (s) -1 . Then I = V (t)V (t) -1 gives x ≤ Ke βt V (t) -1 x and so V (t) -1 x ≥ 1 K e -βt x so that (4.3) ke α(t-s) x ≤ U (t, s)x ≤ Ke β(t-s) x .
holds for α = -β and k = 1 K . If A is 'shifted', i.e., replaced by A+ω, this symmetry α = -β will break, and we will therefore use only (4.3) for some constants k, K > 0 and α ≤ β. Theorem 4.2. Let A(t) 0≤t≤τ ∈ L (D; H) be a family of operators generating an evolution family U (t, s) and let 0 < k ≤ K and α < β be such that (4.3) holds. We suppose that the differences A(t) -A(s) are bounded operators with A(t) -A(s) ≤ L for some L such that

L < k √ 2KM e (β-α)τ . Let C ∈ L (D; Y ).
Then the averaged Hautus condition (AH.3) implies exact observability for all τ > τ * * , i.e.,

τ 0 CU (t, 0)x 2 dt ≥ κ m 2 x 2 ∀x ∈ H
for some τ * * > 0 provided that there exist 0 ≤ x ≤ y ≤ τ * * such that

f (x, y) := k 2 4K 2 M 2 (e -2(β-α)x -e -2(β-α)y ) + L 2 (x -y) min(x, τ -y) > 2.
Proof. Observe that exact (averaged) observability is invariant under spectral shifts (replacing A by A+ω), which in turn allows to assume β = 0 and α = -ω for ω = βα > 0. We follow the lines of the proof of Theorem 3.3 until (3.4). Using (4.3) instead of unitarity leads to consider a new function

κ(ϕ) := τ 0 |ϕ(t)| 2 (k 2 e -2ωt -2K 2 M 2 L 2 ) dt -2K 2 M 2 τ 0 |ϕ ′ (t)| 2 dt. Then κ(ϕ) > 0 is equivalent to (4.4) τ 0 |ϕ ′ (t)| 2 dt < τ 0 |ϕ(t)| 2 ( k 2 2K 2 M 2 e -2ωt -L 2 ) dt.
This is an 'inverse Hardy inequality', when compared to (4.2). To establish such an estimate for at least one function ϕ ∈ H 1 0 (0, τ ), we consider on [0, τ ] the weight function

w(t) = k 2 2K 2 M 2 e -2ωt -L 2 . Observe that w is positive if (4.5) 0 ≤ L < k √ 2KM e (β-α)τ .
In order to obtain (4.4) we use the optimality statement in Theorem 4.1 with v(x) = 1: if √ 2 < B < ∞, the optimal constant guaranteeing (4.2) is larger than one. Hence, for any C < 1 there exists a ϕ ∈ H 1 0 ([0, τ ]) for which (4.2) fails. This function will then satisfy (4.4), and provides a strictly positive constant κ(ϕ), yielding exact averaged observability with κ := κ(ϕ), as in the proof of Theorem 3.3 (by rescaling we may suppose ϕ ∞ = 1). Clearly, √ 2 < B is equivalent to our condition on f (x, y) to be larger than 2 for some 0 ≤ x ≤ y.

On the compact set

T = {0 ≤ x ≤ y ≤ τ } ⊂ R 2 we consider the function f (x, y) := y x w(t) dt min x 0 dt, τ y dt = k 2 4K 2 M 2 ω (e -2ωx -e -2ωy ) + L 2 (x -y) min(x, τ -y).
It is continuous and satisfies f | ∂T = 0 so that the maximum is taken inside T . However, due to the many parameters and the mixture of power-type functions with exponentials it may be difficult to calculate explicitly the maximum of f in T . We therefore concentrate on a sufficient condition that ensures f (x, y) > 2 for some x and y. We consider for example the case where f ( 1 4 τ, 3 4 τ ) > 2, i.e., τ e -ωτ 2 1-e -ωτ ωτ

> 1 τ 32K 2 M 2 k 2 + τ 2L 2 K 2 M 2 k 2
.

By numerical calculations * , we see that if ωτ ≤ 1 √ 2 , then the left hand side is larger than τ 2 , so that for

2L 2 K 2 M 2 k 2 < 1 4 , τ 2 = 128K 2 M 2 k 2
gives a concrete observation time. We obtain the following corollary. . In particular, if k=K=1 and L, M are such that 8L 2 M 2 < 1 and 0 ≤ βα ≤ 1 16M , then we have exact observability at time τ > τ * * where τ * * = 8 √ 2M .

In the autonomous case A(t) = A with A is a generator of a group we have L = 0, hence for 0 ≤ βα ≤ k 16KM we obtain exact observability at time τ > τ * * = 8

√ 2KM k
. This might be better than the observation time given in [START_REF] Jacob | On the Hautus test for exponentially stable C 0 -groups[END_REF] which is 1 β-α .

Applications to the wave and Schrödinger equations with time dependent potentials

In this section we give applications of our results to observability of the Schrödinger and wave equations both with time dependent potentials. We also consider the damped wave equation with time dependent damped term. Before going into these examples we explain the general idea. It is based on a perturbation argument which shows that the Hautus test carries over from the time independent operator to time dependent ones. Once the Hautus test is satisfied by the perturbed operator we appeal to the results of the previous sections and obtain observability of the system.

Let A be the generator of unitary group on H. We assume that C : D(A) → Y is an admissible operator and such that the system (A, C) is exactly observable at time τ 0 . Therefore the Hautus test is satisfied by the operators A and C. Now let R(t) 0≤t≤τ be a family of uniformly bounded operators on H. By classical bounded perturbation argument (see, e.g., [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]Theorem 9.19]). the operators given by A(t) = A+R(t), t ∈ [0, τ ], generate an evolution family U (t, s) on H. Note that for every x ∈ H (5.1) e -β(t-s) x ≤ U (t, s)x ≤ e β(t-s) x with β = sup t∈[0,τ ] R(t) . Indeed, one has for every x ∈ D(A), Re (A + R(t))x, x = Re R(t)x, x ans hence -β x 2 ≤ Re (A + R(t))x, x ≤ β x 2 . We apply this with U (t, s)x at the place of x and obtain

-β U (t, s)x 2 ≤ 1 2 ∂ ∂t U (t, s)x 2 ≤ β U (t, s)x 2 .
We integrate and obtain (5.1). Note that if Re R(t)x, x = 0, then U (t, s) is unitary. Let now x ∈ D(A) and ξ ∈ R. The Hautus test for (A, C) gives

x 2 ≤ m 2 Cx 2 + M 2 (iξ + A)x 2 ≤ m 2 Cx 2 + 2M 2 (iξ + A + R(s))x 2 + 2M 2 R(s) 2 x 2 .
Integrating on [0, τ ] with respect to s gives

x 2 ≤ m 2 Cx 2 + 2M 2 1 τ τ 0 (iξ + A + R(s))x 2 ds + 2M 2 1 τ τ 0 R(s) 2 ds x 2 .
Suppose in addition that there exists τ 1 > 0 and µ < 1 such that for τ ≥ τ 1

(5.2) 2M 2 1 τ τ 0 R(s) 2 ds ≤ µ.
Then we obtain

(5.3) (1 -µ) x 2 ≤ m 2 Cx 2 + 2M 2 1 τ τ 0 (iξ + A + R(s))x 2 ds . * The function g(x) = e -x/2 ( 1-e -x x
) is larger than 1 /2 for x ≤ 0.7143 and 1 √ 2 ≤ 0.70711.

Note that we could also replace iξ by λ ∈ C and obtain the Hautus test (AH.3). Next we assume that C is admissible for the unitary group e tA generated by A. That is there exists a constant K τ > 0 such that (5.4)

τ 0 Ce tA x 2 dt ≤ K τ x 2 , x ∈ D(A).
We prove that C is admissible for (A + R(t)). In order to do so, we start from Duhamel's formula †

(5.5) U (t, s)xe (t-s)A x = t s e (t-r)A R(r)U (r, s)x dr.

We use (5.4) so that

τ 0 CU (t, s)x 2 dt ≤ 2 τ 0 Ce (t-s)A x 2 dt + 2 τ 0 t s Ce (t-r)A R(r)U (r, s)x dr 2 dt ≤ 2K τ x 2 + 2τ τ s τ r Ce (t-r)A R(r)U (r, s)x 2 dt dr ≤ 2K τ x 2 + 2K τ τ s R(r)U (r, s)x 2 dr ≤ K ′ τ x 2 ,
where we use the fact that the operators R(r) are uniformly bounded and U (t, s) is exponentially bounded.

We have admissibility of C and the averaged Hautus test (5.3). Now we conclude either by Theorem 3.3 or Corollary 4.3 that, as soon as R(t) -R(s) are small enough, we have exact observability of the system (A + R(.), C) at time τ > τ * for some τ * > 0. Note that (5.2) holds if R(t) = 0 for t ≥ t 0 for some t 0 > 0.

The Schrödinger equation. Let Ω be a bounded domain of R d with a C 2 -boundary Γ. Let Γ 0 be an open subset of Γ and Y = L 2 (Γ 0 ). It is known that for appropriate condition on Γ 0 , the Schrödinger equation

(5.6)    z ′ (t, x) = i∆z(t, x) (t, x) ∈ [0, τ ] × Ω z(0, .) = z 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω) z(t, x) = 0 (t, x) ∈ [0, τ ] × Γ satisfies the observability inequality (5.7) τ 0 Γ0 | ∂z ∂ν (t, x)| 2 dσ dt ≥ κ τ z 0 2 H 1 0 (Ω)
for every τ > 0, see for example [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]Chapter 7]. Let C be the normal derivative ∂ ∂ν on Γ 0 , Y = L 2 (Γ 0 , dσ) and ∆ D the Laplacian with Dirichlet boundary conditions. The previous inequality means that the system (i∆ D , C) is exactly observable at time τ . Let now R(t)f = iV (t)f where V (t, .) ∈ W 1,∞ (Ω) is a real-valued potential which depends on time. Then under appropriate conditions on V we obtain from the discussion above that the non-autonomous system (i(∆ D + V (t)), C) is exactly observable at time τ > τ * for some τ * > 0. This means that (5.7) is satisfied for the solution of the Schrödinger equation with time dependent potential (5.8)    z ′ (t, x) = i∆z(t, x) + iV (t)z(t, x) (t, x) ∈ [0, τ ] × Ω z(0, .) = z 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω) z(t, x) = 0 (t, x) ∈ [0, τ ] × Γ.

Note however that our method does not give observability at any time τ > 0. If V (t) = V is independent of t then observability for the Schrödinger equation perturbed by the potential V holds at any time τ > 0, see [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]Chapter 7] and the references there.

The wave equation. Let again Ω be a bounded smooth domain of R d . We consider the wave equation (5.9)    z ′′ (t, x) = ∆z(t, x) ∈ [0, τ ] × Ω z(0, .) = z 0 ∈ H 1 0 (Ω), z ′ (0, .) = z 1 ∈ L 2 (Ω) z(t, x) = 0 (t, x) ∈ [0, τ ] × Γ.

Let Γ 0 be a part of the boundary Γ. Observability for the wave equation with the observation operator C = ∂ ∂ν |Γ0 have been intensively studied. Under appropriate geometric conditions on Γ 0 , there exists τ 0 > 0 such that for τ > τ 0 there exists a positive constant κ τ such that (5.10)

κ τ Ω |z 1 | 2 + Ω |∇z 0 | 2 ≤ τ 0 Γ0
| ∂z ∂ν | 2 dσ dt.

We refer to [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF][START_REF] Lions | Controlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF][START_REF] Komornik | Exact controllability and stabilization (the multiplier method)[END_REF] and the references therein. Let A 0 = 0 I -∆ D 0 on H := H 1 0 (Ω) × L 2 (Ω).

It is a standard fact that A 0 generates a unitary group U (t) t∈R on H. Set C(f, g) := ( ∂f ∂ν |Γ0 , 0). Then the energy estimate (5.10) is precisely the observability inequality (5.11) κ τ (z 0 , z 1 Going to the first order system on H, the wave equation (5.12) can be rewritten as Z ′ = A(t)Z with A(t) = 0 I ∆ + V (t) b(t) = A 0 + R(t) where R(t) = 0 0 V (t) b(t)

. As in the case of the Schrödinger equation we can apply the previous discussion to see that the Hautus test for A 0 implies our averaged Hautus test for (A(t)) t . In order to do so we need to verify (5.2). This property holds if

1 τ τ 0 V (t) 2 W 1,∞ (Ω) + b(t) 2 L ∞ (Ω) dt
is small enough. The norms R(t) -R(s) are small if the quantities V (t) -V (s) W 1,∞ (Ω) + b(t)b(s) L ∞ (Ω) are small. In this case, we obtain exact averaged observability for (5.12). That is, we obtain the energy estimate (5.10) for τ large enough for solution z to (5.12). If V and b are independent of t then observability results are known (see [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]). If b(t) = 0 and V depends on t, then a more precise result can be found in [START_REF] Puel | Global Carleman inequalities for the wave equation and applications to controllability and inverse problems[END_REF] for a special class of Γ 0 . The proof in [START_REF] Puel | Global Carleman inequalities for the wave equation and applications to controllability and inverse problems[END_REF] is different from ours and it is based on Carleman estimates.

e

  Reλ.s (λ+A(s))x ds 2

Lemma 2 . 4 .

 24 The family (B(t)) t∈[0,τ ] are admissible controls for (A(t)) t∈[0,τ ] if and only if the family (B(t) * ) t∈[0,τ ] are admissible observations for the retrograde equation (2.3).

e

  Reλ.s (λ+A(s))x ds 2 for all λ ∈ C and all x ∈ D,

  s))xe λs ds = K τ τ 0 (λ+A(s))x e Reλ.s ds. and (AH.1) follows. The second assertion is obtained from the first one by taking λ = iξ and using the Cauchy-Schwarz inequality.

) If we define Cx := 1 τ τ 0 Cds x 2 + M 2 1 τ τ 0 (

 020 (s)x ds then we can apply Proposition 3.1 and Theorem 3.3 to the time independent operator C. We obtain equivalence between iξ -A(s))x 2 ds .

Corollary 4 . 3 . 2 √

 432 Suppose that L < k 2KM and 0 ≤ βα ≤ k 16KM . Then we have exact observability at time τ > τ * * where τ * * = 8 √ 2KM k

) 2 H ≤ τ 0 CU 2 L 2 ( 5 z

 0225 (t)(z 0 , z 1 ) Γ0) dt. Now we consider the damped wave equation without a potential(′′ (t, x) = ∆z(t, x) + b(t, x)z ′ (t, x) + V (t, x)z(t, x) ∈ [0, τ ] × Ω z(0, .) = z 0 ∈ H 1 0 (Ω), z ′ (0, .) = z 1 ∈ L 2 (Ω) z(t, x) = 0 (t, x) ∈ [0, τ ] × Γ.

  Averaged null controllability for (A,B) in time τ is equivalent to averaged observability of z(s), 0 ≤ s < τ where z is the solution of the retrograde final-value problem (2.3) with the observation operators C(t) = B(t) For simplicity we extend this function by zero for other values of t. Exact averaged controllability for (A,B) at τ is equivalent to range(Φ s,τ ) = H for all s. Since these operators are bounded, the latter property is equivalent to the fact that their adjoints Φ * s,τ is bounded from below on L 2 (s, τ ; H), i.e., there exists κ s,τ such that U (τ, t) * z s 2 dt ≥ κ s,τ z s 2 for all z s ∈ D * . Approximate averaged controllability is equivalent to range(Φ s,τ ) being dense for all s ∈ [0, τ ), or, equivalently, the respective adjoints being injective. Finally, averaged null controllability in time τ is equivalent to range(U (τ, s)) ⊂ range(Φ s,τ ) for all 0 ≤ s < τ . Applying [26, Proposition 12.1.2], averaged null controllability is equivalent to

	τ
	B(t)
	s

Proposition 2.6. Let B(t) ∈ L (U, H -1 ) be a family of averaged admissible controls for (A(t)) t∈[0,τ ] . Then a) Exact averaged controllability for (A,B) in time τ is equivalent to exact averaged observability of the retrograde final-value problem (2.3) with the observation operators C(t) = B(t) * . b) Approximate averaged controllability for (A,B) in time τ is equivalent to approximate averaged observability of the retrograde final-value problem (2.3) with the observation operators C(t) = B(t) * . c) * . Proof. First note that (Φ * s,τ z s )(t) = B(t) * U * (τ, t)z s for t ∈ [s, τ ]. *

† in order to prove this formula one takes the derivative of f (r) := e (t-r)A U (r, s)x for s ≤ r ≤ t and then integrate from s to t.

The first named author is partially supported by the Collaborative Research DFG and ANR project INFIDHEM ANR-16-CE92-0028 .

1


Jean-Christophe Ono-Dit-Biot

Pierre Soulard

Solomon Barkley

Eric R Weeks

Thomas Salez

Elie Raphaël

Kari Dalnoki-Veress

Rearrangement of 2D aggregates of droplets under compression: signatures of the energy landscape from crystal to glass

We study signatures of the energy landscape's evolution through the crystal-to-glass transition by compressing 2D finite aggregates of oil droplets. Droplets of two distinct sizes are used to compose small aggregates in an aqueous environment. Aggregates range from perfectly ordered monodisperse single crystals to disordered bidisperse glasses. The aggregates are compressed between two parallel boundaries, with one acting as a force sensor. The compression force provides a signature of the aggregate composition and gives insight into the energy landscape. In particular, crystals dissipate all the stored energy through single catastrophic fracture events whereas the glassy aggregates break step-by-step. Remarkably, the yielding properties of the 2D aggregates are strongly impacted by even a small amount of disorder.

Glassy materials are drastically different from crystals in their properties and cannot simply be described as crystals with defects [START_REF] Philips | Amorphous Solids: Low-Temperature Properties[END_REF]. The intrinsic disorder associated with molecules that do not neatly pack, or polydisperse colloidal spheres, prevents glasses from crystallizing [START_REF] Pusey | [END_REF]3]. Intense effort has been devoted to understanding glasses and the transition from an ordered crystal to a disordered glass. Microscopic properties such as the packing configuration can be accessed experimentally and provide insight into the crystal-toglass transition [4][5][6][7]. But, these studies did not yield any conclusion regarding the difference in mechanical properties between crystals and glasses. To answer this question several numerical studies have been conducted, with a consistent conclusion: adding even a small amount of disorder to a system with crystalline packing results in properties that are similar to amorphous structures [8][9][10][11][12][13][14][15]. However, conducting an equivalent experimental study is challenging. A beautiful experiment by Keim et al. showed that a small amount of disorder in a colloidal poly-crystal results in a shear modulus similar to the one observed with a binary mixture of colloids [16]. However, an experimental characterization of the transition from a perfectly ordered single crystal to a disordered glass probed using mechanical properties is still lacking, since experimental systems are often polycrystalline and their properties dominated by grain boundaries. Here, we experimentally study the yielding properties of 2D finite-size aggregates of droplets that vary in the extent of disorder from a perfect crystal to a glass.

We use an emulsion since individual particles can easily be imaged to obtain both structural and dynamical information [17][18][19]. Colloids and emulsions are proven model systems for the study of glasses and jamming [17,[20][21][22][23], force chains [24,25], and phase transitions in crystals [26]. Specifically, we use an emulsion of oil in water (model soft spheres with negligible friction, and an attractive potential) confined to a 2D finite-size aggregate. The amount of disorder is tuned by changing the relative fraction of large and small droplets in aggregates with a total of N tot = 20 or 23 droplets. We investigate the transition from a perfectly ordered monodisperse crystal [27] to a disordered bidisperse glass [28][29][30] by systematically adding defects to the crystalline structure. The transition is studied through the force required to globally compress and fracture the 2D aggregates while simultaneously monitoring microscopic reorganization. The adhesion energy between particles exceeds the thermal energy, thus the particles must be treated as athermal, and the analogue molecular system is that of a glass or a crystal well below the solid-melt transition temperature. The small 2D system of athermal particles provides a unique opportunity to i) prepare perfect single crystals, rather than poly-crystal as is typical, ii) add defects to the single crystals one-by-one, and iii) obtain aggregate-scale force response during compression, while iv) simultaneously capturing local structural re-arrangements. With the addition of even a small number of defects, we find: 1) a rapid increase in the number of fracture events upon compression; and 2) that the yield energy is distributed over many small steps in comparison to a single large step for a crystal. These experimental findings provide a signature of the increasingly complex energy landscape as the system transitions from crystal to glass. An analytical model is developed which supports the experimental data.

RESULTS AND DISCUSSION

The experiment is illustrated in Fig. 1(a) and is described in full detail in the Materials and Methods section. In short, 2D aggregates of buoyant oil droplets are compressed between two thin glass pipettes. By monitoring the deflection of the "force sensing pipette" [see (iii) in Fig. 1(a)] during the compression, the force applied on the aggregate is measured [31,32]. The aggregate rearranges under compression by breaking adhesive bonds between droplets. These fracture events can be directly monitored with optical microscopy and related to the force measurement. We use p ini to refer to the initial number of rows of droplets, defined as parallel to the pipettes as shown in Fig. 1(c), while q ini refers to the initial number of droplets per row. Under compression the aggregate rearranges to have p rows and q columns, while N tot remains fixed. Using two "droplet pipettes" with different tip radii facilitates the preparation of well controlled bidisperse aggregates [33]. To increase the disorder in an aggregate, large droplets are replaced by small droplets (or vice versa). in the force-distance curve of Fig. 2(a) and to a corresponding inter-basin barrier in the energy landscape. Clearly, for a p → (p -1) transition, a crystal made of small droplets will fracture at a smaller spacing between the pipettes (trace 1), compared to a crystal of larger droplets (trace 7). All the bonds are broken in a catastrophic and coordinated manner, in agreement with other studies of crystals under compression [34,35]. For 2D crystals we find that the fracture patterns consist of equilateral triangles with (p -1) droplets on a triangle's side as shown in Fig. 1(c). These equilateral triangles arise because they minimize the number of broken bonds between droplets as p → (p -1). After fracture, the triangles slide past each other and reassemble into a new crystal with (p -1) rows of droplets. By design, the force sensor does not register a friction force during sliding, nor are we sensitive to viscous drag during compression, because slow compression (0.3 µm/s) ensures that viscous drag forces are negligible.

With the introduction of defects in the structure compression forces are no longer homogeneously distributed within the aggregate (see Fig. 2(c-e)). Thus, rather than a single catastrophic fracture, additional fracture events occur and ex-tra peaks appear in the force data, as seen in the traces 2 to 6 of Fig. 2(a). When a single defect is introduced (traces 2 and 6), extra peaks are observed but peaks corresponding to the fracture of the crystalline portion of the aggregate can still be identified (large peaks at the same values of δ). Defects are systematically introduced up to trace 4, which corresponds to the most disordered system that we use to model a glass (equal fraction of large and small droplets). The forcedistance curves are strongly impacted by increasing disorder: i) the number of peaks increases; ii) the overall magnitude of the force peaks decreases; and iii) the peaks corresponding to the underlying crystalline structure can no longer be differentiated from the others. In order to identify peaks as corresponding to a specific transition from p to (p -1) one can invoke the fact that a restructuring event in a bidisperse aggregate must occur within the compression range set by the onset of fracture associated with a crystal of big droplets and the completion of fracture in an aggregate of small droplets. Thus, we invoke the following criterion: a peak corresponds to the p → (p -1) transition if the peak is found in the compression range δ ∈ [δ p min , δ p max ]; where δ p max is defined by the onset of the force peak upon compression of aggregates only made of large droplets and δ p min is defined by the completion of the fracture event in aggregates made of only small droplets [i.e. corresponding to the compression value at the maximum force as detailed in Fig. 2(a)].

From Fig. 2 it is evident that fracture properties are strongly dependent on the aggregate composition. In the simplest case, that of compressing a crystal cluster, the droplets deform and the stored elastic energy increases with compression. Eventually the stored elastic energy exceeds the depletion-induced adhesive energy, and a coordinated fracture occurs as discussed above (shown in Figure 1(c)), such that a minimal number of bonds are broken. We now turn to the more complex bidisperse aggregates. As defects are introduced, the most striking feature is the rapid increase in the number of force peaks [Fig. 2(a)]. To further quantify this observation, we perform experiments for two different aggregate geometries: i) p ini = 4 with q ini = 5, and ii) p ini = 3 with the three rows initially made of 8 -7 -8 droplets. The composition of the cluster is given by the number fraction of small droplets in the aggregate, φ = N small /N tot , which varies from zero to one. Both φ = 0 and φ = 1 correspond to crystals while φ = 0.5 corresponds to the maximum amount of disorder -a model glass. The defects are purposely distributed throughout the whole structure to avoid clumps of defects. In Fig. 3(a), we plot the total (i.e. until we reach p = 1) number N pini→1 of detectable force peaks as a function of the defect fraction.

Model

We propose a minimal model that rationalizes the experimental observations. A given (p, q) aggregate is approximated by an ensemble of q independent columns, of p rows of droplets each. By allowing this simplification, one can treat each column as a random packing of droplets belonging to two different species which correspond to the two different radii: small droplets with R = r and large droplets with R = R. Since droplets are arranged in a nearly hexagonal lattice, each column consists of alternating layers of a single droplet or two droplets side-by-side (see SI for more details). The probabilities associated with finding a small or large droplet at a specific site are given by the number fractions φ and 1 -φ. The total resulting height, H, of a given column depends on the specific composition in that column, and takes values ranging from H r to H R , for columns made of small (φ = 1) and large (φ = 0) droplets. We define the dimensionless height h = 2H/(H R + H r ). One can compute (see SI) the associated probability distribution, P (h), plotted in Fig. 3(c) for various φ (black bars). Compression of an aggregate then proceeds as follows: First, the tallest columns are compressed and broken, which creates a force peak whose magnitude reflects the abundance of these highest columns in the aggregate. Then, the pushing pipette starts compressing the second highest columns and the process repeats.

The simple model predicts that the average number of force peaks observed during the p → p -1 transition of an aggregate can be identified with the average number N p (φ, q) of different column heights present in the aggregate composed of q columns. For a monodisperse aggregate, there is only one possible column height, and thus N p (0, q) = N p (1, q) = 1 resulting in one force peak for the p → p-1 transition. In contrast, as the defect fraction increases, the number of possible different heights and thus the number of force peaks increase. The number of different heights can be calculated numerically according to the scheme described above (see SI). In addition, a simple argument provides an analytical estimate for the average number of force peaks in a sample with a given φ. The increase in the average number of different column heights in comparison to a crystal is expected to be proportional to the standard deviation, σ(φ, p) ∝ φ(1 -φ), of the height distribution centered at µ(φ, p) shown in Fig. 3(c). This results from the random packings of the columns described above and gives:

N p (φ, q) -1 = [N p (φ = 0.5, q) -1]2 φ(1 -φ) . (1)
Finally, in order to determine all the force peaks encountered on average as the aggregate is compressed, we sum Eq. ( 1) over all the transitions starting from a cluster with p ini rows to one row, in order to construct N pini→1 qini (φ) = pini p=2 N p (φ, q), where q = N tot /p. Defining the average number of peaks compared to a crystal, ∆N (φ,

p ini , q ini ) = N pini→1 qini (φ) - N pini→1 qini (0), we obtain (see SI): ∆N ∆N max (φ) = 2 (1 -φ)φ , (2)
where ∆N max = ∆N (φ = 0.5, p ini , q ini ) corresponds to the average maximum excess number of peaks, observed when compressing the most disordered aggregate. The experimental value of ∆N max is obtained by fitting Eq. S20 to each set of data presented in Fig. 3(a). Figure 3(b) shows that this simple model captures well the rapid increase of the number of force peaks as defects are added. The derivative of Eq. S20 at φ = 0 is infinite; thus, a small change in the fraction of defects in an aggregate results in a drastic change in the yield properties as observed in experiments. We note that the minor discrepancy between the data and the model reflects experimental error as well as three main departures of the real aggregate from the proposed idealization: i) neighbouring columns are not independent, ii) the real aggregate has a finite number of columns, and iii) some peaks may not be detected.

Probing the Energy Landscape through the Crystal-to-Glass Transition

The compression experiments can also be used to characterize the yield energy of the aggregate as a function of the defect fraction, which reflects the evolution of the underlying energy landscape through the crystal-to-glass transition.

Specifically, the work W tot exerted (and then fully dissipated in the fluid) in order to generate a p → (p -1) rearrangement is obtained by integrating the force-distance curve (Fig. 2(a)), for the corresponding transition. As explained previously, a p → (p -1) transition corresponds to δ ∈ [δ p min , δ p max ], so the integration is performed over this interval. Moreover, we only consider the rising (along the compression orientation i.e. upon decreasing δ) elastic part F s of the force peaks, as the subsequent decay corresponds to the viscous relaxation of the force-sensing pipette. For this analysis, we focus on the collection of force traces presented in Fig. 2(a) and in particular the transition p = 4 → p = 3. Within our resolution, the total work W tot = δ p max δ p min dδ F s (δ ) = 2.2 ± 0.7 fJ is found to be nearly constant for all the different experiments and is not correlated to the composition of the aggregate when the initial geometry (p ini , q ini ) is kept constant (see SI). The remarkable result that the work is nearly independent of the composition of the aggregate, is an indication that the number of bonds broken must be nearly constant.

While the total work may be nearly constant, there is an important distinction between the disordered and crystalline systems in how that work is distributed during a p → (p -1) transition of the aggregate. To access that information, we consider the partial work

W (δ) = δ p max δ dδ F s (δ ), with δ ∈ [δ p min , δ p max ].
For the crystals, the bonds are broken simultaneously as the system is driven out of a deep minimum in the landscape. For instance, the crystal made of large droplets breaks near δ = δ p max [Fig. 2(a)], with the normalized partial work going abruptly from zero to one upon compression (i.e. decreasing δ) near that point. This fracture event is detailed in the top panel of Fig. 4 where we plot the normalized work as a function of the inter-pipette distance, δ. The crystal made of small droplets exhibits a similar sudden transition, except that the fracture event happens at δ = δ p min . In contrast, when defects are introduced, several intermediate steps are observed. For an aggregate with a single defect, a major step (corresponding to the crystalline fraction) is still observed but rapidly fades away as more defects are added. The curves for 6 defects (φ = 0.3) and for the model glass (φ = 0.5) both show many discrete jumps in the work -thus, the failure of disordered systems is more progressive and has a much lower yield threshold than for crystals. Finally, one can compare the experiment to the theoretical model developed above. In the model, the average normalized partial work is given by the fraction of columns that have a height H larger than δ. Invoking the probability distribution P of column heights, one gets on average:

W (δ) W tot = δ p max Λ δ Λ dh P (h, φ, p) , (3)
where Λ = µ(p, φ = 0.5) = (H r + H R )/2. This expression is plotted in the bottom panel of Fig. 4 for various compositions, and is consistent with the experimental data. The theory predicts more steps than the experiment, this is because the experiment probes one configuration, while the the- ory is an average over all the configurations. We have thus shown that model 2D crystals and glasses are markedly different under compression: crystals deform elastically until a catastrophic global fracture event occurs, whereas glasses rearrange locally with many intermediate fracture events that each have lower individual yield thresholds. This deviation from the well-established response of a crystal to an external stress has also been observed in a recent analytical study [36], and it was shown numerically that materials go from brittle to ductile when transitioning from crystal to glass [12] -a fact that is tested here directly with the idealised microscopic experiments.

CONCLUSIONS

In summary, by systematically adding disorder in finitesize 2D colloidal crystals, we have studied the crystal-to-glass transition. Upon addition of defects the mechanical properties of the aggregates rapidly transition from crystalline to glassy. The number of force peaks, corresponding to fracture events, increases steeply with the defect fraction, before saturating to the glass value. Additionally, the yield energy as a function of disorder has been investigated. We find that for a 2D crystal, a high energy barrier must be overcome, while glasses fracture progressively through failure in many small steps. In the system studied the adhesion energy between particles exceeds the thermal energy, thus the aggregates correspond to a glass or a crystal well below the solid-melt transition temperature. The fracture events observed reflect the substructure introduced by disorder in the underlying energy landscape. This is consistent with the brittle failure of crystals as opposed to the plasticity of glasses. A minimal analytical model captures the essential experimental features. From the combination of experiments and theory, we quantify the crystal-to-glass transition using macroscopic yield observables that are consistent with a simple microscopic picture.

MATERIALS AND METHODS

Experimental setup

A chamber (55 × 30 mm) is made of two glass slides separated by a gap of 2.5 mm, which is 10 3 times greater than the size of droplets. The chamber is filled with an aqueous solution of sodium dodecyl sulfate (SDS) at 3% and NaCl at 1.5%. This concentration of SDS leads to the formation of micelles acting as a depletant resulting in a short-ranged attraction between the droplets [37]. The chamber is placed atop an inverted optical microscope for imaging while the aggregates are compressed.

Micropipettes

Three small micropipettes are inserted into the chamber: the "droplet pipette", "pushing pipette", and "force-sensing pipette". Pipettes were pulled from glass capillaries (World Precision Instruments, USA) with a pipette puller (Narishige, Japan) to a diameter of about 10 µm over several centimeters in length. The "droplet pipette" produces monodisperse droplets, with size directly proportional to the tip radius of the pipette, using the snap-off instability [33]. The droplets are buoyant and form a 2D aggregate under the top glass slide [Fig. 1(b)]. The "pushing pipette" is short and stiff and is used to compress the aggregate. The pushing pipette is affixed to a translation stage and its speed set to 0.3 µm/s for all experiments. The "force-sensing pipette" is a long compliant pipette, and its deflection is used to measure forces applied to the aggregate [31,32]. To be sensitive to forces as small as ≈ 100 pN, the force-sensing pipette needs to be long (≈ 3 cm) and thin (≈ 10 µm). This long straight pipette is locally and temporarily heated to soften the glass such that it can be shaped to fit within the small chamber [see pipette (iii) in Fig. 1 (a)].

Preparation of the aggregates

Aggregates of oil droplets are assembled droplet-by-droplet and thus can be prepared into any arbitrary shape.

Cross-correlation analysis

The distance between the pushing pipette and the forcesensing pipette, δ, is measured using cross-correlation analysis between images with a precision of ∼ 0.1 µm [31,32]. Additionally, correlation analysis provides the deflection of the force-sensing pipette, which is converted to a force using the calibrated spring constant k p = 1.3 ± 0.1 nN/µm of the pipette [31,32]. The typical uncertainty on the force is: δF/F ≈ 2%. Random walk statistics can be applied to this model. To simplify, we consider p being even. The results for p being odd would be similar. Thus, for the even case, building such a column is equivalent to two random walks of p/2 steps: one with the circles and one with the rectangles. Using the random walk statistics formalism [START_REF] Sokolov | First Steps in Random Walks: From Tools to Applications[END_REF], we can express the probability P circ (H 1 , φ, p, r, R) of finding a height H 1 by stacking p/2 circles of two different sizes for a given φ, and P rect (H 2 , φ, p, r, R), the probability of finding a height H 2 by stacking p/2 rectangles of three different heights:

     P circ (H 1 , φ, p, r, R) = 1 2π π -π (φe 2iθr + (1 -φ)e 2iθR ) p 2 e -iθH1 dθ P rect (H 2 , φ, p, r, R) = 1 2π π -π (φ 2 e 2iθr + (1 -φ) 2 e 2iθ R + 2φ(1 -φ)e iθ( R+r) (φe iθ r +(1-φ)e iθ R ) 2 ) p 2 e -iθH2 dθ (S4)    P circ (H 1 , φ, p, r, R) = p 2 k=0 p 2 k φ k (1 -φ) p 2 -k δ(2kr + 2( p 2 -k)R -H 1 ) P rect (H 2 , φ, p, r, R) = p l=0 p l φ l (1 -φ) p-l δ(lr + (p -l) R -H 2 ). (S5)
It turns out that the random walk of p/2 steps with three different step sizes is equivalent to p steps of two different sizes (see Eq. S4). The Dirac δ function in Eq. S5 is a geometrical constraint on the total height. Only the combinations of droplets that leads to the right total heights H 1 and H 2 are considered. The distribution of probability of the total height H is the convolution product of P circ (H 1 , p, φ, r, R) and P rect (H 2 , p, φ, r, R):

P (H, φ, p, r, R, r, R) = ∞ 0 P circ (H -H 2 , φ, p, r, R)P rect (H 2 , φ, p, r, R) dH 2 .
(S6)

Using Eq. S5 we find:

P (H, φ, p, r, R, r, R) = p 2 k=0 p l=0 p 2 k p l φ k+l (1 -φ) 3p 2 -k-l δ(lr + (p -l) R + 2kr + 2( p 2 -k)R -H). (S7)
To simplify the notation we consider r, R, r and R fixed so the probability distribution is P (H, φ, p). Eq. S7 is used to calculate numerically the discrete distribution presented in the main text (Fig. 3(c)-(e) histograms). The height H can take discrete values H i , with probability P i = P (H i , φ, p), ranging from H r for a column made of small droplets (φ = 0) to H R for a column made of large droplets (φ = 1). The total number of different heights H i only depends on p and is noted m p . Finally, the height H is renormalized as follows h = 2H/(H r + H R ). With this renormalization, a column made of 50% large droplets and 50% small droplets (φ = 0.5) has a dimensionless height h = 1.

Number of peaks

The column model gives access to the probability P i of finding the height H i in an aggregate for any fraction of defects φ. The number of peaks in the force measurement is calculated from the height distribution. Let us denote the average number of force peaks during the compression of an aggregate with p rows to an aggregate with (p -1) rows by N p (φ, q). Observing a single peak in the force measurement means that all the columns share the same height. Measuring two peaks means that there are two and only two different heights. Thus N p (φ, q) corresponds to the average number of different heights composing an aggregate of p rows and q columns.

For a given fraction of defects φ, an aggregate of size p × q is built by choosing randomly q columns from a pool of columns. Correlation between two adjacent columns are neglected. Experimentally, a small fraction of the peaks in the force measurement is due to the correlation between columns but most of the peaks are indeed due to compression of independent columns. From the discrete probability distribution, Eq. S7, there is a finite number m p of possible heights H i with a non-zero probability. To predict the number of peaks we calculate the probability A n (φ, p, q) of finding strictly n different columns heights in an aggregate of size p × q at a given fraction of defects φ.

Building an aggregate is equivalent to drawing q columns which can take m p different heights H i with probability P i . {ijk...} n defines n different numbers between 1 and m p . Let P{ijk...}n denote the probability that the aggregate is composed only of the n heights {H i , H j , H k , ...} and each height appears at least once. As the order in which the heights are drawn is not important, one gets: Hence, the probability that an aggregate of q columns is composed only of the heights {H i , H j , H k , ...} is given by (P {ijk...}n ) q . However, this probability is not equal to P{ijk...}n since it does not take into account that each height must appear at least once.

A n (φ, p, q) = Θ(q -n) n! {ijk...
The difference between (P {ijk...}n ) q and P{ijk...}n is the probability that one or more of the heights {H i , H Noticing that the sum over the κ and the other sums can be switched and using Eq. S8, one finds:

A n (φ, p, q) = Θ(q -n) n! {ijk...
B κ = κ! n κ n-1 β=κ (m p -β) A κ , (S13)
leading to:

A n (φ, p, q) = Θ(q -n) n! {ijk...}n⊂ 1,mp (P {ijk...}n ) q - 1 n! n-1 κ=1 n κ κ! n-1 β=κ (m p -β) A κ . (S14)
The average number of peaks N p (φ, q) is given by:

N p (φ, q) = mp n=1
nA n (φ, p, q). (S15)

Equation S14 and S15 can be evaluated numerically, see Fig. 6. The importance of this distribution A n (φ, p, q) can be easily understood for both extreme values of φ. If φ = 0, there is only one possible height for the column meaning that A n (φ = 0, p, q) = δ 1n , where δ ij is the Kronecker symbol. On the other hand, if φ = 0.5, it is unlikely to find only one height so A 1 0. It is more likely to find all the different heights in the aggregate leading to A mp 1. This is illustrated in Fig. 6(a) which shows the probabilities, A n , for p = 2 (with n takes values from 1 to m 2 = 6) as a function of φ. Note that in Figs. 6 (a)-(b) we restrict the range to φ ∈ [0, 0.5] as the function is symmetric about φ = 0.5. For φ = 0, only A 1 = 0. As φ increases, finding two different heights becomes more likely and A 2 becomes dominent. For φ = 0.5, it is very likely to find the maximum number of columns, m 2 = 6, in the aggregate and A 6 1.

Finite size effect

The number of force peaks is a function of the size of the cluster: in the model, the number of force peaks depends explicitly on the number of columns q since A n depends on q (in Fig. 6, q = 50). In an infinitely large cluster (q → ∞) all the heights will appear and so :

N p (φ > 0, q → ∞) = m p ,
with m p the number of possible heights one can get with the random packing described previously at a given p. For a finite q, one has N p (φ, q) < m p because all the possible heights will not appear simultaneously in the same cluster. As a simple consequence, one has: N p (φ, q 1 ) < N p (φ, q 2 ), for q 1 < q 2 . Moreover, even if the total number of different possible heights m p is large, we cannot find more different heights than the number of columns, q. This is the reason for the Heaviside function in the definition of A n . For the experimental aggregates, q varies from 3 to 15. In particular for p = 3 or p = 4, the number of columns is usually ∼ 5 and the value of q gives an upper limit for the number of force peaks. Fig. 7 shows the impact of the number of columns on the number of peaks for p = 2. A simpler approach to estimate the number of force peaks

The approach discussed above predicts accurately the average number of force peaks observed during the compression of an aggregate. However the number of force peaks estimated for a transition p → p -1 is strongly dependent of the size of the cluster. In this section, we take a simpler approach that leads to an analytical expression for the number of force peaks observed during the compression of a cluster. In addition, with this approach we are able to define a quantity that allows us to renormalize our results with respect to the size of the cluster. This analytical expression characterizes the transition of a cluster from being crystal-like to glass-like.

We define the excess number of force peaks as the number of force peaks for a given φ compared to the number of force peaks observed in the crystal case N p (φ = 0, q) = 1 for the same transition p → p -1: N p (φ, q) -1. This quantity can be normalized by its value for a glassy case where φ = 0.5: N p (φ = 0.5, q) -1. The normalized quantity quantifies how crystalline or glassy a cluster is, and takes values ranging from 0 for a crystal to 1 for a glass.

Instead of numerically calculating the average number of different column heights in a p × q cluster, we propose the following statistical argument: the average number of different heights in a p × q cluster is well approximated by the number of different highly probable heights in the probability distribution of heights. To define if a height H i is highly probable, one has to invoke a threshold for the probability, P (H i ), which is strongly dependent on the total number of columns, q, in the cluster. The larger q is, the smaller the threshold must be, and should go to zero in the limit of infinitely large clusters (q → ∞). Since the columns are built as a 1D random walk, the distribution follows a binomial law and the number of probable heights can be characterized using the standard deviation σ of the height distribution. Using σ to define the threshold, the excess number of peaks is then directly proportional to the width of the height distribution. As we are interested in the ratio between the excess number of peaks at a given fraction of defects, φ, and its maximum value, observed at φ = 0.5, the choice of the threshold is not critical, for the range of q values explored in the experiments. This leads to: N p (φ, q) -1 N p (φ = 0.5, q) -1 σ(φ, p) σ(φ = 0.5, p) . (S16)

The standard deviation, σ, as well as the average value, µ, of the continuous height distribution can be calculated analytically in this simplified approach. We find µ(φ, p) = p(α + 1)[(1 -φ)R + rφ] and σ 2 (φ, p) = p(2 + α 2 )(R -r) 2 φ(1 -φ). The analytical expression for σ as a function of φ is tested against the numerically calculated values from the discrete model, described in the previous sections, for different values of φ. Figure 8 shows perfect agreement between both approaches for p = 2. Using the expression found for the standard deviation, σ, Eq. S16 can be rewritten as:

N p (φ, q) -1 N p (φ = 0.5, q) -1 2 φ(1 -φ).

(S17)

Note that the result is now independent of the size of the cluster. Indeed, this ratio simply compares the excess number of peaks to its maximum value, but it does not predict the exact number of peaks observed in the force curves. This simplified approach can be tested against the discrete model by comparing the number of peaks predicted by each model with p = 2. The left hand side of Eq. S17 is calculated numerically for the discrete model and compared to 2 φ(1 -φ) as shown in Fig. 9. Both models are in good agreement for the number of peaks as long as q < 50. The analytical prediction overestimates the number of force peaks in the range of q values experimentally explored. Comparison between the average number of peaks predicted by the discrete calculation (for different values of q) and the continuous approximation, for p = 2. The continuous model does not take into account the number of droplets per row q in the aggregate.

FIG. 1 .FIG. 2 .

 12 FIG. 1. (a)Schematic top view of the experimental chamber. The typical dimensions of the wall (dark grey) are 55×30×2.5 mm. The "droplet pipette", "pushing pipette", and "force-sensing pipette" are labelled as (i), (ii) and (iii) respectively. (b) Schematic side view (not to scale). The buoyant droplets form a quasi 2D aggregate bounded by the top glass plate. The pushing pipette (black circle on the left) and the force-sensing pipette (red circle on the right) are placed near the average equatorial plane of the droplets so forces are applied horizontally. (c) Optical microscopy image of a typical crystal (scale bar is 50 µm). Red dashed lines show observed fracture lines for a crystal when compressed.

30 FIG. 3 .

 303 FIG. 3. (a) Measured total number of force peaks as a function of defect fraction, for a compression from pini to p = 1. ( ) pini = 4 with qini = 5 -two data sets (different colours); ( ) pini = 3 with the three rows initially made of 8 -7 -8 droplets respectively -two data sets (different colours). (b) Evolution of the normalized excess number of peaks compared to a crystal, with the black dashed line corresponding to Eq. S20. (c) Theoretical probability distribution of the dimensionless column height h, in an aggregate with q → ∞ and p = 4, for four defect fractions φ = {0; 0.1; 0.3; 0.5} (see SI).Gaussian curves (blue solid lines) with same standard deviation, σ, and average, µ, as the discrete distribution are overlayed as a guide to the eyes. Typical radii of large and small droplets are ≈ 22 µm and ≈ 18 µm (see SI).

1 FIG. 4 .

 14 FIG. 4. Normalized partial work (see definition in text) as a function of distance, for the p = 4 → p = 3 transition, for aggregates of different relative compositions (number of small droplets / number of large droplets) as indicated. (top) Experimental results corresponding to the force curves shown in Fig. 2(a). (bottom) Corresponding theoretical results, according to Eq. (3).

5 .

 5 Schematics of the columns with p = 6 considered in the theoretical model. The left part shows how rectangles and circles are assembled to build a column. The right part shows the different choices for circles and rectangles along with their probability to appear.

6 FIG. 6 .

 66 FIG. 6. (a) Probability distribution An as a function of φ for an aggregate made of p = 2 rows and q = 50 droplets per row. (b) Prediction of the average number of peaks in the force measurement as a function of φ based on the distribution An. Note that we only plot the function for φ ∈ [0, 0.5] by symmetry about φ = 0.5.

6 FIG. 7 .

 67 FIG.7. Impact of the size of the aggregate on the number of peaks in the force measurement for p = 2 and different values of q.

1 FIG. 8 .

 18 FIG. 8. Comparison between the analytical expression for the standard deviation of the height distribution and the numerically calculated values for different φ for p = 2.

  FIG. 9. Comparison between the average number of peaks predicted by the discrete calculation (for different values of q) and the continuous approximation, for p = 2. The continuous model does not take into account the number of droplets per row q in the aggregate.

  is the Heaviside function and {ijk...}n⊂ 1,mp denotes the sum over all the n-tuples {ijk...} n in 1, m p . We define P {ijk...}n as the probability to draw one of the n heights {H i , H j , H k , ...}, and obtain:

		P{ijk...}n ;	(S8)
	}n⊂ 1,mp
	where Θ P {ijk...}n =	P κ .	(S9)
	κ∈{ijk...}

  j , H k , ...} does not appear. To calculate P{ijk...}n we subtract from (P {ijk...}n ) q the probabilities that the aggregate is only composed of n -κ different types of columns of heights {H a , H b , H c , ...} with {abc...} ⊂ {ijk...} n summed over all the possible (n -κ)-tuples in {ijk...} n and summed over all the κ from 1 to n -1: P{ijk...}n = (P {ijk...}n ) q -

	n-1
	P{abc...}n-κ .	(S10)
	κ=1 {abc...}n-κ⊂{ijk...}n

  Let us focus on the second term of the right hand side, called B κ , in Eq. S11. P{abc...}n-κ depends on n -κ indices and it is summed over n indices. So if we fix the n -κ indices that P{abc...}n-κ depends on, it will appears n κ times. Thus:

	}n⊂ 1,mp	(P {ijk...}n ) q -	1 n!	n-1 κ=1 {ijk...}n⊂ 1,mp {abc...}n-κ⊂{ijk...}n	P{abc...}n-κ	. (S11)
							Bκ
		B κ =	{ijk...}n⊂ 1,mp	n κ	P{ijk...}n-κ .	(S12)
	By splitting the sum into two parts, one finds:

TABLE II .

 II Total work needed to transition from p = 4 to p = 3 for different compositions.

	Composition φ Wtot (fJ)
	20/0	0	1.2
	19/1	0.05	2.8
	14/6	0.3	2.8
	10/10	0.5	2.1
	6/14	0.3	3.1
	1/19	0.05	1.5
	0/20	0	2.0

ACKNOWLEDGEMENTS

The authors thank Maxence Arutkin, Matilda Backholm, and James Forrest for valuable discussions, as well as Yilong Han for sharing a preliminary draft on a similar topic. Financial support from NSERC (Canada), the Joliot chair from ESPCI Paris, and the Global Station for Soft Matter, a project of Global Institution for Collaborative Research and Education at Hokkaido University is gratefully acknowledged. The work of ERW was supported by the National Science Foundation (CBET-1804186).

SUPPLEMENTARY INFORMATION

Height of a column and probability

In this section, the theoretical model used to predict the number of peaks in the force measurement as a function of φ is derived. In this calculation, each transition from p rows of droplets to (p -1) rows is studied individually. In the following, p and q are constant values.

The theoretical model developed for this study is based on geometrical arguments. An assembly of droplets is compressed if its lateral unstrained extent is larger than the spacing between the pipettes. The aggregate is modeled as q independent columns of height H i stacked next to each other, the index i, going from 1 to q, is labelling the columns. The total height of a column depends the composition of droplets. For a crystal, all the columns are the same so they break at the same time, which results in a single peak in the force measurement. When defects are introduced, large droplets are substituted by small ones (or vice versa). Columns constituting the aggregate now have different heights and break for different values of δ resulting in several peaks in the force measurement.

A column is made of alternating layers of two droplets, which are modelled as a rectangle, and single droplets, modelled as circles, as shown in Fig. 5. The number fraction φ of small droplets in an assembly of N small small droplets and N tot -N small large droplets is defined as: φ = N small /N tot . Depending on the composition of the two droplets, the rectangles can take three heights {2 R, 2r, ( R+r)} with probabilities {(1 -φ) 2 , φ 2 , 2(1 -φ)φ} respectively. The circles can only have two diameters resulting in two distinct heights {2R, 2r} with probabilities {(1 -φ), φ} respectively. Finally, we take the relation between (R, r) and ( R, r) to be a geometrical factor α. It is the sum of the heights of the rectangles and the heights of the circles that determine the overall height of a column as shown in Fig. 5.

In the main text, we compare the experimental results and the total number of peaks when compressing a cluster initially made of p ini rows and q ini columns to a single row (p = 1), at a given percentage in defects φ: N pini→1 qini (φ). Equation 2 in the main text is obtained by summing Eq. S17 over the different transitions:

(S18)

During these transitions, the total number of droplets N tot is conserved and thus p × q = p ini × q ini . Noticing that pini p=2 1 is the number of peaks observed when compressing a crystal initially made of p ini rows, this quantity is independent of q ini and will be noted N pini→1 (φ = 0). Equation S18 can then be written as:

leading to:

Finally, we define ∆N (φ, p ini , q ini ) = N pini→1 qini (φ) -N pini→1 (0) as the average excess number of peaks observed when compressing an aggregate with a defect fraction φ in comparison to a crystal of same geometry p ini × q ini . The maximum excess number of peaks ∆N max = N pini→1 qini (φ = 0.5) -N pini→1 (0) corresponds to the excess number of peaks observed when compressing the most disordered aggregates (model for a glass, φ = 0.5). The ratio of these two quantities ∆N/∆N max does not depend on the size of the cluster p ini × q ini but only on the fraction of defects φ. We can thus write Eq. S20 in a simpler form and obtain Eq. 2 of the main text:

The maximum number of peaks, N pini→1 max (q) = N pini→1 (φ = 0.5, q), depends on the system size as it has been shown in the previous section. The experimental value is obtained by fitting Eq. S21 to each set of data presented in Fig. 3(a).

Size of the droplets in the crystal-to-glass transition study

The data shown in Figs. 4(a)-(b), in the main text, come from four different sets of experiments. The droplets used during a given set of experiments are the same, while the defect fraction is varied. New droplets were produced for each new set. Table I summarizes the sizes of the droplets used in these experiments.

Work analysis

In the main text, we study how the work is distributed along a compression as a function of the composition of the aggregate. This analysis relies on the assumption that the total work for a given transition does not depend on φ. We found that within the uncertainty of the experiments, the total work for the transition p → (p -1) is constant and is not correlated to the fraction of defects φ. Table II summarizes the total work, W tot , exerted to go from four to three rows for the different aggregates.


Marc Charlet -Sia
email: marc.charlet@sia.fr

Session 7B: Verification & Validation

Keywords:

Starting with a the state of the art of the validation and verification strategies and the associated tools, Atos Origin has proposed a set of recommendations addressing mainly the specification and verification phases of the development process. For the software requirement phase, the recommendations refer to the splitting of the protocols in subcomponents in order to isolate the automates and therefore to reduce the complexity of the software to test ; the formalization of the finite state machines (mathematical determinism of automates). The verification is based on the progressive testing and integration of the software components. The study is supplemented with the definition of ATPA, a framework to develop a generic testing tool dedicated to a specific protocol. ATPA allows to exchange protocol data with the implementation under test across various communication channels. It offers many generic features to perform degradation of the data, driven by a control language, and therefore permits the addition, deletion, modification and delaying of protocol data in both ways. At this time, the work is successfully exploited by Airbus France for the A380 program which includes an increased number of avionics protocols.

Validating avionics protocols -Methodology and tools for A380 program 1

A methodology for testing avionics protocols for A380 Program

Validating avionics protocols -Methodology and tools for A380 program 2

Context

The CONTEXT Airbus, A380, Capitalisation on A320 and A340 programs (ATSU equipment).

The OBJECTIVE

To improve the quality of the testing protocols for certifying to DO-178B level C.

The classic "functional coverage" approach is not well suited

The APPROACH External and Internal expertise, state of the art studies, Definition of a methodology relying on Atos's experience and Airbus's objectives.

Validating avionics protocols -Methodology and tools for A380 program 5

Context

The problem

The response to a stimulus for a protocol depends on :

The context of execution The date of the stimulus A simple minded structural coverage is not sufficient

The objectives

Isolation and simplification of the automate

In order to reduce the complexity

The methodology

Determinism : f(current state, in event) -> out event with some is (current state, in event) then out event is only

Validating avionics protocols -Methodology and tools for A380 program 7

The method : development phase

  the protocol with automate black boxIn order to minimize the number and complexity (The complexity of testing layer isn't combined.)Testing toolsIn order to concentrate efforts on the description of the tests instead of the execution of the testsTestingValidating avionics protocols -Methodology and tools for A380 program 13The RESULTSIMPROVEMENTof definition of the requirements preciseness Coverage IMPROVEMENT of the coverage of the test cases REDUCTION of the number of tests REDUCTION of the time spent to design the tests Results

-22 -23 January 2004

ABSTRACT :

In the context of the development of avionics embedded computers, it is required to implement avionics communication protocols inside the on board network and also between board and ground systems through the air/ground network. The experience of Airbus France in the field of the development of protocols with the need of avionics certification has demonstrated that the validation and verification of the implementations is a complex task. From this report, the methodology and tools department of Airbus France/EYY contracted with Atos Origin a study in order to define a methodology for the verification of protocols and to propose a testing tool suite to facilitate the validation process. The objectives of the study are to increase the coverage of the validation while reducing the effort and improving the traceability of the protocol automatons. The study has been conducted in relationship with Mr. Richard Castanet who is the head of the laboratory of research in information technology from the University of Bordeaux (LaBRI).

Tracability of requirements

« Tool box »

The testing stack :

• Suited for all testing scenarios

• Warrants the integrity of the testing protocol