Florent Di Meglio

Federico Bribiesca Argomedo

Long Hu

Miroslav Krstic

Stabilization of coupled linear heterodirectional hyperbolic PDE-ODE systems

Keywords:

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Introduction

The interest for coupled Ordinary Differential Equations-Partial Differential Equations (ODE-PDE) systems has first emerged when considering delays in the actuating and sensing paths of ODE. Delays can be seen as first-order hyperbolic PDEs. There are many approaches to deal with input or measurement delays, usually divided into two categories: memoryless controllers, which extend standard control techniques without explicitly accounting for the delay in the control design [START_REF] Michiels | Stability and stabilization of timedelay systems. An eigenvalue based approach[END_REF][START_REF] Yue | Delayed feedback control of uncertain systems with time-varying input delay[END_REF][START_REF] Fridman | H-∞ control of linear state-delay descriptor systems: an LMI approach[END_REF]; and prediction-based controllers aiming at explicitly compensating the delay [START_REF] Smith | A controller to overcome dead time[END_REF][START_REF] Bresch-Pietri | Robust control of variable time delay systems[END_REF][START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF].

The use of Lyapunov and backstepping methods enabled dealing with more involved PDEs in the actuating and sensing paths. In [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion pdes[END_REF], an output feedback control law is derived for an ODE having a heat equation in the actuating and sensing paths. The coupled PDE-ODE system is stabilized using an observer-controller structure relying on a backstepping approach. The same approach has been used to deal with ODEs coupled (rather than cascaded) with parabolic PDEs [START_REF] Tang | State and output feedback boundary control for a coupled pde-ode system[END_REF], uncertain parabolic PDEs [START_REF] Li | Adaptive control of the ode systems with uncertain diffusion-dominated actuator dynamics[END_REF], or ODE-Schrödinger cascades [START_REF] Ren | Stabilization of an odeschrödinger cascade[END_REF]. Lyapunov methods Email addresses: florent.di meglio@mines-paristech.fr (Florent Di Meglio), federico.bribiesca@insa-lyon.fr (Federico Bribiesca Argomedo), longhu@ipm6.fr (Long Hu), krstic@ucsd.edu (Miroslav Krstic).

enable the design of static output feedback controllers for nonlinear ODE-parabolic PDE cascades, as in [START_REF] Wu | Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion pde-governed sensor dynamics[END_REF].

The first application of the backstepping approach to deal with hyperbolic PDE-ODE couplings is [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF] where actuator and sensor delays are explicitly compensated. While this problem had already been tackled by, e.g., the Smith predictor [START_REF] Smith | A controller to overcome dead time[END_REF], the reformulation of the delay as a linear ODE enabled numerous related problem to be tackled, most notably non-constant and uncertain delays [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF][START_REF] Bresch-Pietri | Robust control of variable time delay systems[END_REF]. In [START_REF] Tsubakino | predictor-feedback for multi-input lti systems with distinct delays[END_REF], the problem of stabilizing a multi-input ODE with distinct delays is tackled using a backstepping approach. In [START_REF] Castillo Buenaventura | Dynamic Boundary Stabilization of Hyperbolic Systems[END_REF], an observer is designed for an ODE having a homodirectional 1 hyperbolic PDE in the sensing path, relying on a Lyapunov analysis requiring to solve Linear Matrix Inequalities (LMI). As will appear, the systems of [START_REF] Tsubakino | predictor-feedback for multi-input lti systems with distinct delays[END_REF] are particular cases of the system considered here, although the control design approaches are different.

Here, we solve the problem of stabilizing an ODE with a system of first-order linear hyperbolic PDEs in the sensing 1 i.e. where all the states transport in the same direction Fig. 1. Schematic view of the ODE-PDE coupling structure and actuating paths, i.e. we consider the following system

Ẋ(t) = AX(t) + Bv(t, 0) (1) u t (t, x) = -Λ + u x (t, x) + Σ ++ u(t, x) + Σ +-v(t, x) (2) v t (t, x) = Λ -v x (t, x) + Σ -+ u(t, x) + Σ --v(t, x) (3) u(t, 0) = Q 0 v(t, 0) + CX(t) (4) v(t, 1) = R 1 u(t, 1) + U(t) (5)
where t > 0 and x ∈ [0, 1] are respectively the time and space variables, X ∈ R p is the ODE state, u(t, x) ∈ R n and v(t, x) ∈ R m are the PDE states and U(t) is the control input. The matrices Λ + and Λ -are such that

Λ + = diag (λ 1 , . . . , λ n ), Λ -= diag (µ 1 , . . . , µ m ) (6) with -µ 1 < • • • < -µ m < 0 < λ 1 < • • • < λ n (7)
The system naturally features several feedback loops or couplings that can be sources of instabilities:

• Inside the ODE itself (the A matrix in Equation ( 1))

• Coupling between hyperbolic states inside the spatial domain (the Σ •• matrices in (2),( 3)) • Coupling between hyperbolic states at the boundary (the Q 0 and R 1 matrices in (4),( 5)) • Coupling between the PDE and the ODE (the B and C matrices in (1),( 4)) • A combination of all the above. This structure is schematically depicted on Figure 1. This problem is motivated by applications in the drilling industry, more precisely the suppression of mechanical vibrations. Drilling systems are composed of long flexible strings subject to axial and torsional vibrations that propagate upwards and downwards. At the bottom end, the so-called drill bit crushes rock to create the borehole and is subject to friction and cutting forces. The ODE state X then corresponds to the drill bit axial and torsional positions while the PDE states represent the propagation of torsional and axial waves from and to the drill bit.

When damping of the vibrations along the drillstring is neglected and the axial and torsional vibrations are coupled, the PDE reduces to two delay equations. Several contributions have taken advantage of this simplification and designed stabilizing feedback laws, e.g. relying on neutral system approaches [START_REF] Saldivar | Stick-Slip Oscillations in Oillwell Drillstrings: Distributed Parameter and Neutral Type Retarded Model Approaches[END_REF], flatness approaches [START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF] or predictor-based approaches [START_REF] Bekiaris-Liberis | Compensation of wave actuator dynamics for nonlinear systems[END_REF][START_REF] Bresch-Pietri | Output-feedback adaptive control of a wave pde with boundary anti-damping[END_REF]. However, no existing solution simultaneously allows stabilization

• taking into account damping inside the PDE domain • for a model of both axial and torsional vibrations, yielding 4 coupled PDE states rather than two delay equations [START_REF] Meglio | A distributed parameter systems view of control problems in drilling[END_REF][START_REF] Germay | Multiple mode analysis of the self-excited vibrations of rotary drilling systems[END_REF].

Here, we solve these problems within the general setting of Equations ( 1)- [START_REF] Bresch-Pietri | Output-feedback adaptive control of a wave pde with boundary anti-damping[END_REF]. The system is mapped to an exponentially stable target system using a Volterra transformation. The target system has a cascade structure ensuring its convergence to the zero equilibrium. The design is based on a recent result on heterodirectional systems of hyperbolic PDEs [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF].

The paper is organized as follows. In Section 2 we present the backstepping control design. In Section 3 we present a general well-posedness result for a class of hyperbolic PDEs on a triangular domain. In Section 4 we apply these results to the considered problem and state the main result. We conclude in Section 5 with perspectives for future work.

Control design

The control design is based on a Volterra transformation mapping the state (X, u, v) to a target system (X, α, β) with desirable properties. The target system equations are described in the next section.

Target system

We design the target system as follows

Ẋ(t) =(A + BK)X(t) + Bβ(t, 0) (8) α t (t, x) = -Λ + α x (t, x) + Σ ++ α(t, x) + Σ +-β(t, x) + D(x)X(t) + x 0 C + (x, y)α(t, y)dy + x 0 C -(x, y)β(t, y)dy (9) β t (t, x) =Λ -β x (t, x) + G(x)β(t, 0) + Σβ(t, x) (10) α(t, 0) =Q 0 β(t, 0) + C 0 X(t) (11) β(t, 1) =0 (12)
where C 0 , C + , C -and D have yet to be defined and G(•) and Σ are defined as The target system has a cascade structure schematically depicted on Figure 2. Its stability properties are assessed in the following Lemma.

G(x) =                          0 • • • • • • 0 g 2,
Lemma 1 Denote T = {0 ≤ y ≤ x ≤ 1}. Under the following assumptions

(i) A + BK is Hurwitz (ii) C + , C -∈ L ∞ (T ) (iii) G, D ∈ L ∞ ([0, 1])
the zero equilibrium of System (8)-( 12) is exponentially stable in the L 2 sense.

Proof Consider the following Lyapunov functional

V(t) = 1 0 e -δx α(t, x) Λ + -1 α(t, x)dx + 1 0
e δx β(t, x) Λ --1 Rβ(t, x)dx + X(t) PX(t) [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF] where the symmetric definite positive matrix P, the diagonal matrix R = diag(r 1 , ..., r m ) and the design parameter δ > 0 are yet to be determined. Differentiating with respect to time and integrating by parts yields V(t) = -e -δx α(t, x) α(t, x) + e δx β(t, x) Rβ(t, x)

1 0 - 1 0 δe -δx α(t, x) α(t, x)dx - 1 0 δe δx β(t, x) Rβ(, x)dx + 2 1 0 e -δx α(t, x) Λ + -1 Σ ++ α(t, x) + Σ +-β(t, x) + D(x)X(t) + x 0 C + (x, y)α(t, y)dy + x 0 C -(x, y)β(t, y)dy dx + 2 1 0 e δx β(t, x) Λ --1 R G(x)β(t, 0) + Σβ(t, x) dx + X(t) P(A + BK) + (A + BK) P X(t) + X(t) PB + B P β(t, 0) (15)
Using Young's inequality yields the existence of M, γ > 0 such that

2 1 0 e -δx α(t, x) Λ + -1 Σ ++ α(t, x)dx ≤ M 1 0 e -δx α(t, x) Λ + -1 α(t, x)dx (16) 2 1 0 e -δx α(t, x) Λ + -1 Σ +-β(t, x)dx ≤ M 1 0 e -δx α(t, x) R -1 α(t, x)dx + 1 0 e δx β(t, x) Rβ(t, x)dx (17) 2 1 0 x 0 e -δx α(t, x) Λ + -1 C + (x, y)α(t, y)dxdy ≤ M δ 1 0 e -δx α(t, x) Λ + -1 α(t, x)dx (18) 2 1 0 x 0 e -δx α(t, x) Λ + -1 C -(x, y)β(t, y)dxdy ≤ M δ 1 0 e -δx α(t, x) R -1 α(t, x)dx + 1 0 e δx β(t, x) Rβ(t, x)dx (19) 1 0 e -δx α(t, x) Λ + -1 D(x)X(t)dx ≤ M 1 γ |X(t)| + γ 1 0 e -δx α(t, x) Λ + -1 α(t, x)dx (20) X(t) PB + B P β(t, 0) ≤ M 1 γ |X(t)| + γβ(t, 0) β(t, 0) (21)
where γ > 0 is a design parameter to be defined. Further, given the structure of the G matrix given by ( 13), there exists M > 0 such that

1 0 e δx β(t, x) Λ --1 RG(x)β(t, 0)dx ≤ M 1 0 e δx β(t, x) Rβ(t, x)dx + e δ β(t, 0) Cβ(t, 0) (22) 3
where

C = diag (c 1 , • • • , c m ), c i = m j=i+1 r j , 1 ≤ i ≤ m -1 0, i = m (23)
Besides, let S = diag (s 1 , ..., s m ) such that

Q 0 Q 0 -S ≺ 0 (24)
Finally, plugging ( 8),( 11),( 12) into ( 15) and denoting Q =

-P(A + BK) + (A + BK) P > 0 yields V(t) ≤ -β(t, 0) R -MCe δ -γMI m×m -S β(t, 0) -X(t) Q - 2 γ I p×p X(t) - x 0 e -δx α(t, x) (δ -γ)I n×n -M 1 + 1 δ I n×n + R -1 α(t, x)dx -δ -M 2 + 1 δ 1 0 e δx β(t, x) Rβ(t, x)dx (25) thus, picking γ, R such that Q - 2 γ I p×p ≺ 0 (26) ∀i = 1, ..., m -1 r i > Me δ m j=i+1 r j + γM + s i (27)
r m > γM + s m (28)
and δ large enough concludes the proof.

Volterra Transform

To map the original system (1)-( 5) to the target system ( 8)-( 12), we use the following Volterra transformation

α(t, x) ≡u(t, x) (29)
β(t, x) =v(t, x) - x 0 K(x, y)u(t, y)dy - x 0 L(x, y)v(t, y)dy -γ(x)X(t) (30)
where the kernels K, L and γ have yet to be defined. Differentiating (30) w.r.t. space and time yields the following kernel equations

for 1 ≤ i ≤ m, 1 ≤ j ≤ n µ i ∂ x K i j (x, y) -λ j ∂ y K i j (x, y) = -σ -- ii K i j (x, y) + n k=1 σ ++ k j K ik (x, y) + m k=1 σ -+ k j L ik (x, y) (31) for 1 ≤ i ≤ m, 1 ≤ j ≤ m µ i ∂ x L i j (x, y) + µ j ∂ y L i j (x, y) = -σ -- ii L i j (x, y) + m k=1 σ -- k j L ik (x, y) + n k=1 σ +- k j K ik (x, y) (32)
along with the following set of boundary conditions, for i = 1, ..., m

∀ j, K i j (x, x) = - σ -+ i j µ i + λ j (33) ∀ j i, L i j (x, x) = - σ -- i j µ i -µ j ∆ = l i j (34) ∀ j ≥ i µ j L i j (x, 0) = n k=1 λ k K ik (x, 0)q k, j + p k=1 b k j γ ik (x) (35)
where the q k, j in (35) are the elements of Q 0 . Besides, γ satisfies the following ODE

∀ j, µ i γ i j (x) = p k=1             a k j + m l=1 l<i b kl κ l j             γ ik (x) -σ -- ii γ i j (x) - m k=1 k<i µ k L ik (x, 0)κ k j + n k=1 λ k             c k j + m l=1 l<i q kl κ l j             K ik (x, 0) (36)
with initial condition

∀ j, γ i j (0) = κ i j ( 37
)
where the κ i j are the entries of the control matrix gain K. To ensure well-posedness of the system, we add the following arbitrary boundary conditions

∀ j < i, L i j (1, x) = l i j (x) (38)
These are degrees of freedom in the control design. However, their effect on the closed-loop performances are still unclear, thus, to study well-posedness, which we do in the next section, we only impose l i j ∈ L ∞ ([0, 1]). Besides, provided the K and L kernels are well-posed, the coefficients of G, C 0 C + , C -and D are given by

∀ j < i, g i j (x) = µ j L i j (x, 0) - n k=1 λ k K ik (x, 0)q k, j - p k=1 b k j γ ik (x) (39) C + (x, y) = Σ +-K(x, y) + x y C -(x, s)K(s, y)ds (40) C -(x, y) = Σ +-L(x, y) + x y C -(x, s)L(s, y)ds (41) D(x) = Σ +-γ(x) + x 0 D(y)γ(y)dy (42)
C 0 = C + Q 0 K (43)
We prove well-posedness of the kernel equations over the next two sections. First, we study a relatively general class of hyperbolic PDEs on a triangular domain.

3 A general class of kernel equations

Problem setup

We consider the following class of equations on a triangular domain

i (x)∂ x F i (x, y) + ν i (y)∂ y F i (x, y) = Σ i (x, y)F(x, y) (44)
where

F = F 1 • • • F n T . Each unknown F i satisfies bound- ary conditions on a subset Ω i ⊂ ∂T of the following form ∀i = 1, ..., n F i | Ω i = f i + n j=1 Γ i j (•)F j | Ω i (45)
where f i and Γ i j are defined on Ω i . The functions Γ i j , defined on the boundaries of the triangular domain T , are boundary couplings between the different kernels F i . The well-posedness of (44),(45) depends on the sparsity of the matrix Γ = (Γ i j ). More precisely, consider the following definition.

Definition 3.1 Let G be the directed graph whose vertices are the F i and whose edges are defined by the matrix Γ i j ∞ . In other words, there is an edge between nodes i and j iff Γ i j ∞ 0. Thus, a valid path of length p in the graph is a p-uplet a = (a 1 , ..., a p ) such that

p-1 k=1 Γ a k ,a k+1 ∞ 0 (46)
By convention, a path (a 1 ) of length p = 1 is the single node F a 1 .

The following Theorem gives a sufficient condition on the structure of G for the system to be well-posed.

Theorem 3.2 Consider system (44) with boundary conditions (45). Assume (i) that the uncoupled system, obtained by taking Σ(x, y) ≡ 0 in (44) and Γ i j = 0, ∀i, j in (45), is well-posed;

(ii) that there exists α > 1 such that, for all i = 1, ..., n, the following inequality holds

∀(x, y) ∈ T α i (x) -ν i (y) > δ > 0 (47) (iii)
The graph G is acyclic, i.e. is does not contain any cycles.

Then there is a unique solution F ∈ L ∞ (T ).

Remark 1 A necessary and sufficient condition for Assumption (i) to be satisfied is that, for every i = 1, ..., n the characteristics defined by the i , ν i connect each point of T to Ω i .

Remark 2 Assumption (ii) is a simple geometric condition for the well-posedness of the system: the tangent vector ( i (x), µ i (y)) to all the characteristics, at all points (x, y) ∈ T must lie in the half-space such that the scalar product with (α, -1) T is negative. In other words, the characteristics leaving the boundaries where (45) are defined must always "point away" from a certain line y = αx, with α > 1. Examples of such characteristics are pictured on Figure 3. The proof of Theorem 3.2 is quite involved and spans over the next few sections. It relies on the transformation of (44),(45) into integral equations. For this, we define in the next section the characteristic curves.

Transformation into integral equations

Assumption (i) of Theorem 3.2 yields the existence and uniqueness of characteristic curves, defined as follows Definition 3.3 For each i = 1, ..., n and any (x, y) ∈ T there exists (χ 0 i (x, y), ξ 0 i (x, y)) ∈ Ω i and s

F i (x, y) ∈ R + such that                dχ i (s; x, y) ds = i (χ i (s; x, y)) χ i (0; x, y) = χ 0 i (x, y) ∈ Ω i χ i (s F i (x, y); x, y) = x (48)                dξ i (s; x, y) ds = ν i (ξ i (s; x, y)) ξ i (0; x, y) = ξ 0 i (x, y) ∈ Ω i ξ i (s F i (x, y); x, y) = y (49)
The curves (χ i (s), ξ i (s)) are the characteristic curves associated with F i . For any two points (M 1 , M 2 ) ∈ T , we denote

C i (M 1 , M 2 ) the characteristic curve associated with F i start- ing in M 1 = (x 1 , y 1 )
and ending in M 2 = (x 2 , y 2 ), if such a curve exists, i.e. if

χ i (s F (x 1 , y 1 ); x 1 , y 1 ) = x 2 , ξ i (s F (x 1 , y 1 ); x 1 , y 1 ) = y 2 (50)
In the absence of the boundary couplings Γ i j , the proof of well-posedness would consist in integrating (44) along ( 48),(49) and using a method of successive approximations. Here, this yields

F (x, y) = f i (χ 0 i (x, y), ξ 0 i (x, y)) + n j=1 Γ i j (χ 0 i (x, y), ξ 0 i (x, y))F j (χ 0 i (x, y), ξ 0 i (x, y)) + s F i (x,y) 0 Σ i (χ i (s;
x, y), ξ i (s; x, y))F(χ i (s; x, y), ξ i (s; x, y))ds

The second term still contains unknowns, and the method of successive approximations does not straightforwardly apply. Rather, the second term must, again, be integrated along the characteristics of the F j 's for which Γ i j is non-zero. This situation is depicted on Figure 3.2 for an example.

To avoid this situation repeating infinitely (infinitely many "rebounds"), we use the following basic results from Graph Theory.

Basic results from Graph Theory

The following Definitions and Lemmas are classical results, see e.g. [START_REF] Brent | Introduction to graph theory[END_REF]. This allows us to add the following two definitions Definition 3.4 For any node F i , we define its depth d i as the length of the longest valid path to a terminal node. We also define d max as the maximum length of any path d max = max i=1,...,n d i .

Definition 3.5 Let a = (a 1 , ..., a p ) be a (not necessarily valid) path. Then, we recursively define the sequence of points M a k (x, y) ∈ T , k = 0, ..., p such that M a 0 (x, y) = (x, y) (53)

M a k (x, y) = χ 0 a k (M a k-1 ), ξ 0 a k (M a k-1 ) ∈ Ω a k ( 54
)
where χ 0 a k (•) and ξ 0 a k (•) are defined by (48),(49). In other words, M a k is the point on the boundary of T such that the characteristic curve C a k (M a k , M a k-1 ) exists.

where (b, a) denotes the concatenation of the two paths.

We are now ready to prove Theorem 3.2.

Proof of Theorem 3.2

Proof Classically, the proof consists in transformation the PDEs into integral equations and using a method of successive approximations.

Transformation into integral equations

The proof relies on the following transformation of ( 44),(45) into integral equations. For any i = 1, ..., n, any M = (x, y) ∈ T , one has 56) can be rewritten as

F i (x, y) = d i p=1 a=(a 1 ,...ap ) a 1 =i         p-1 k=1 Γ a k ,a k+1 M a k (x, y)         •        f a p M a p (x, y) + C ap (M a p (x,y),M a p-1 (x,y)) Σ a p F        (
F i (x, y) = 1 p=1 a=(a 1 ) a 1 =i f a 1 (M a 1 (x, y)) + C a 1 (M a 1 (x,y),M a 0 (x,y)) Σ a 1 F = f i (M a 1 (x, y)) + C i (M a 1 (x,y),M a 0 (x,y)) Σ i F (58)
Which exactly corresponds to integrating (44) along the characteristics associated to F i since for all j = 1, ..., n, Γ i j = 0 for a terminal node (see (51) with Γ i j = 0). 56) is true for all nodes of depth less or equal to d, for some d ∈ {1, ..., d max }. Consider now F i of depth d + 1. Integrating (44) along the characteristics and plugging in the boundary conditions (45) yields

d → d + 1. Assume now that (
F i (x, y) = f i (M (i) 1 (x, y)) + n j=1 Γ i j (M (i) 1 (x, y))F j (M (i) 1 (x, y)) + C i (M (i) 1 (x,y),M (i) 0 (x,y)) Σ i F (59)
Notice that all the F j for which Γ i j 0 are of depth d j ≤ d.

Applying equation (56) to them yields

F i (x, y) = f i (M (i) 1 (x, y)) + C i (M (i) 1 (x,y),M (i) 0 (x,y)) Σ i F + n j=1 d p=1 a=(a 1 ,...ap ) a 1 = j Γ i j M (i) 1 (x, y)         p-1 k=1 Γ a k ,a k+1 M a k M (i) 1         •        f a p (M a p M (i) 1 ) + C ap M a p M (i) 1 ,M a p-1 M (i) 1 Σ a p F        (60)
which yields

F i (x, y) = f i (M (i) 1 (x, y)) + C i (M (i) 1 (x,y),M (i) 0 (x,y)) Σ i F + d p=1 n j=1 a=(a 1 ,...ap ) a 1 = j         Γ i j (M (i) 1 (x, y)) p-1 k=1 Γ a k ,a k+1 M a k M (i) 1         •        f a p (M a p M (i) 1 ) + C ap M a p M (i) 1 ,M a p-1 M (i) 1 Σ a p F        (61)
i.e.

F i (x, y) = f i (M (i) 1 (x, y)) + C i (M (i) 1 (x,y),M (i) 0 (x,y))
Σ i F + d p=1 n j=1 a=(a 1 ,...ap ,a p+1 ) a 2 = j, a 1 =i        p k=1 Γ a k ,a k+1 M a k (x, y)        •        f a p+1 (M a p+1 (x, y)) + C a p+1 M a p+1 (x,y),M a p (x,y) Σ a p+1 F        (62)
i.e.

F i (x, y) = f i (M (i) 1 (x, y)) + C i (M (i) 1 (x,y),M (i) 0 (x,y)) Σ i F + d p=1 a=(a 1 ,...,a p+1 ) a 1 =i        p k=1 Γ a k ,a k+1 M a k+1 (x, y)        •        f a p+1 (M a p+1 (x, y)) + C a p+1 M a p+1 (x,y),M a p (x,y) Σ a p+1 F        (63)
i.e.

F i (x, y) = f i (M (i) 1 (x, y)) + C i (M (i) 1 (x,y),M (i) 0 (x,y)) Σ i F + d+1 p=2 a=(a 1 ,...,ap ) a 1 =i         p-1 k=1 Γ a k ,a k+1 M a k+1 (x, y)         •        f a p (M a p (x, y)) + C ap M a p (x,y),M a p-1 (x,y) Σ a p F        (64)
i.e., using (57)

F i (x, y) = d i p=1 a=(a 1 ,...,ap) a 1 =i         p-1 k=1 Γ a k ,a k+1 M a k+1 (x, y)         •        f a p (M a p (x, y)) + C ap M a p (x,y),M a p-1 (x,y) Σ a p F        (65)
which concludes the proof by induction since d i = d + 1.

Method of successive approximations

The end of the proof follows the classical successive approximations method, applied to (56). More precisely, we define the following operators

Φ = Φ 1 • • • Φ n Φ i [F](x, y) = d i p=1 a=(a 1 ,...ap ) a 1 =i         p-1 k=1 Γ a k ,a k+1 M a k (x, y)         C ap (M a p (x,y),M a p-1 (x,y)) Σ a p F (66)
As well as the following vector

φ = φ 1 • • • φ n φ i (x, y) = d i p=1 a=(a 1 ,...ap ) a 1 =i         p-1 k=1 Γ a k ,a k+1 M a k (x, y)         f a p M a p (x, y) (67)
Define now the following sequence for q ∈ N F 0 (x, y) =0 (68)

F q+1 (x, y) =φ(x, y) + Φ[F q ](x, y) (69)
= φ 1 (x, y) • • • φ n (x, y) + Φ 1 [F q ](x, y) • • • Φ n [F q ](x, y) (70)
Finally, define the following sequence for q ≥ 1

∆F q = F q -F q-1 (71)
Provided the limit exists, then

F = lim q →+∞ F q = +∞ q=1 ∆F q (72)
is a solution to (56). To prove that the series is convergent, we rely on the following lemmas Lemma 3 Assume inequality (47) holds. Then for all i = 1, ..., n, (x, y) ∈ T , the following function

ψ i (x,y) : s ∈ [0, s F i (x, y)] → αχ i (s; x, y) -ξ i (s; x, y) (73)
is strictly increasing. In particular, the following inequality holds

ψ i (x,y) (s F i (x, y)) = αx -y > αχ 0 i (x, y) -ξ 0 i (x, y) = ψ i (x,y) (0) (74)
Thus, ψ i (x,y) defines a diffeomorphism of [0, s F i (x, y)] onto its image αχ 0 i (x, y) -ξ 0 i (x, y), αxy .

Proof The proof is trivial since for i = 1, ..., n and (x, y) ∈ T , one has

dψ i (x,y) ds (s) = α i (χ i (s; x, y)) -ν i (ξ i (s; x, y)) (75)
and recalling (47).

Corollary 3.7 For any path a = (a 1 , ..., a p ) of length p ≥ 0 and any k = 0, ..., p, denote M a k (x, y) = (x k , y k ). Then one has

αx k -y k ≤ αx -y (76)
Proof We prove the result by induction. For k = 0, given the definition of the M a k (Equation ( 53)), one has

αx 0 -y 0 = αx -y (77)
Assume now that (76) is satisfied for some k = 1, ..., p -1, then, by definition of the M a k (•, •), one has

αx k+1 -y k+1 = αχ 0 a k (x k , y k ) -ξ 0 a 1 (x k , y k ) (78)
Using Lemma 3 for i = a k+1 , this yields

αx k+1 -y k+1 < αx k -y k (79)
Then, using the induction assumption, this concludes the proof.

Lemma 4 For any i = 1, ..., n and any (x, y) ∈ T , one has

s F i (x,y) 0 αχ i (s; x, y) -ξ i (s; x, y) q ds ≤ 1 δ αx -y q+1 q + 1 ( 80
)
where α and δ are defined by (47).

Proof Consider the following change of variables

τ = ψ i (x,y) (s) (81)
where ψ i (x,y) (•) is defined by (73). It yields s F i (x,y) 0 αχ i (s; x, y) -ξ i (s; x, y) q ds = αx-y αχ 0 i (x,y)-ξ 0 i (x,y) τ p dτ α i ( χi (τ; x, y)) -ν i ( ξi (τ; x, y))

(82)

where we have abusively denoted χi (τ; x, y) = χ i ((ψ i (x,y) ) -1 (τ); x, y) (resp. ξi (τ; x, y) = ξ i ((ψ i (x,y) ) -1 (τ); x, y)). Using (47) this yields

s F i (x,y) 0 αχ i (s; x, y) -ξ i (s; x, y) q ds < 1 δ αx -y q+1 -αχ 0 i (x, y) -ξ 0 i (x, y) q+1 q + 1 (83)
Since (χ 0 i (x, y), ξ 0 i (x, y)) ∈ T and α > 1, one has αχ 0 i (x, y)ξ 0 i (x, y) > 0 which yields the result.

Lemma 5 For any path a = (a 1 , ..., a p ) of length p ≥ 0 and any k = 1, ..., p, one has

C a k (M a k (x,y),M a k-1 (x,y)) αχ a k (•) -ξ a k (•) q ≤ 1 δ αx -y q+1 q + 1 (84)
Proof Denoting M a k (x, y) = (x k , y k ), one has from Lemma 4

C a k (M a k (x,y),M a k-1 (x,y)) αχ a k (•) -ξ a k (•) q ≤ 1 δ αx k-1 -y k-1 q+1 q + 1 (85)
Applying Corollary 3.7 yields the results.

Lemma 6 Define φ = max i=1,...,n φ i (•, •) L ∞ (T ) , Γ = max i, j∈{1,2,...,n} Γ i j ∞ (86) Σ = max (x,y)∈T , i=1,...,n | | |Σ i (x, y)| | | (87) M = d max δ         d max k=0 (n -k)         Γd max -1 Σ (88)
Assume that for some q ≥ 1, one has, for all (x, y) ∈ T ∀i = 1, ..., n ∆F q i (x, y) ≤ φ M q (αxy) q q! (89)

Then, one has

∀i = 1, ..., n ∆F q+1 i (x, y) ≤ φ M q+1 (αx -y) q+1 (q + 1)! (90)
Assume that (89) holds for some fixed q ≥ 1. Then, one has, for all i = 1, ..., n

∆F q+1 i (x, y) (91) = |Φ i [∆F q ](x, y)| (92) ≤ d i p=1 a=(a 1 ,...ap ) a 1 =i
Γp-1

C ap (M a p (x,y),M a p-1 (x,y))

Σ a p ∆F q (93)

Using (89) yields

≤ d i p=1 a=(a 1 ,...ap ) a 1 =i Γp-1 Σ C ap (M a p (x,y),M a p-1 (x,y)) φ M q αχ a p (•) -ξ a p (•) q q! (94)
Using Lemma 5 yields

≤ d i p=1 a=(a 1 ,...ap ) a 1 =i Γp-1 Σ φ M q (αx -y) q+1 (q + 1)! (95)
Noticing that there cannot be more than p k=1

(nk) paths of length p from a given node i, this yields

≤ d max i          d max i k=1 (n -k)          Γd max i -1 Σ φ M q (αx -y) q+1 (q + 1)! (96)
which, in turn, yields the result given the definition of M (Equation (88)). Finally, Lemma 6 ensures that the series (71) is uniformly convergent, thus the kernel equations (44) with boundary conditions (45) are well-posed (see, e.g. [START_REF] Di Meglio | Stabilization of a system of n+1 coupled first-order hyperbolic linear pdes with a single boundary input[END_REF] for a detailed proof). In the next section, we apply Theorem 3.2 to prove well-posedness of (31)-(38).

4 Well-posedness of (31)-( 38) and control law

In this section, we apply the results of Section 3 to prove the well-posedness of the kernel equations. The following theorem assesses the well-posedness of the kernel equations. Proof We prove the result by induction on i = 1, ..., n. i = 1. For i = 1, the equations rewrite as follows for 1 ≤ j ≤ n

µ 1 ∂ x K 1 j (x, y) -λ j ∂ ξ K 1 j (x, y) = -σ -- 11 K 1 j (x, y) + n k=1 σ ++ k j K 1k (x, y) + m p=1 σ -+ p j L 1p (x, y) (97) for 1 ≤ j ≤ m µ 1 ∂ x L 1 j (x, y) + µ j ∂ ξ L 1 j (x, y) = -σ -- 11 L 1 j (x, y) + m p=1 σ -- p j L 1p (x, y) + n k=1 σ +- k j K 1k (x, y) (98) ∀ j, K 1 j (x, x) = - σ -+ 1 j µ 1 + λ j ∆ = k 1 j (99) ∀ j 1, L 1 j (x, x) = - σ -- 1 j µ 1 -µ j ∆ = l 1 j (100) ∀ j ≥ 1 µ j L 1 j (x, 0) = n k=1 λ k K 1k (x, 0)q k, j + p k=1 b k j γ 1k (x) (101) ∀ j, µ 1 γ 1 j (x) = p k=1 a k j γ 1k (x) -σ -- 11 γ 1 j (x) (102)
+ n k=1 λ k c k j K 1k (x, 0) (103) ∀ j, γ 1 j (0) = κ 1 j (104)
One can readily check that (97)-(98) are of the form (44). Besides, the ODE (103) can also be put under the form (44) by "embedding" it into T . More precisely, denoting

1 {y=0} (x, y) = 1 if y = 0 0 otherwise (105)
one can define γ j such that ∀(x, y) ∈ T γ j (x, y) = 1 {y=0} (x, y)γ 1 j (x) (106) or, equivalently, the γ j satisfy the following PDEs of the form (44)

µ 1 ∂ x γ j (x, y) = 1 {y=0} (x, y)        n k=1
a k j γk (x, y) -σ -- 11 γ j (x, y)

+ n k=1 K 1k (x, y)λ k c k j        (107)
with boundary conditions γ j (x, x) = 1 {y=0} (x, x)κ 1 j (108)

Besides, boundary conditions (99)-( 101),(108) are of the form (45), with the boundary coupling coefficients Γ i j being zero for every kernel except the L i j on the y = 0 boundary. Therefore, the graph defined by Γ i j is acyclic, and Theorem 3.2 applies to (97)-(104) which is well-posed, i.e. has a unique solution with K 1 j , L 1 j ∈ L ∞ (T ) and γ 1 j ∈ L ∞ ([0, 1]).

{1, ..., i -1} → i. Let i ∈ {2, ..., m} be fixed and assume that for k = 1, ..., i -1 there exist K k j , L k j ∈ L ∞ (T ) and γ k j ∈ L ∞ ([0, 1]), for all j. Then, Equations (31)-(38) are of the form (44),(45) with coefficients in L ∞ since they are linear in the K i j , L i j and γ i j variables with coefficients that depend on the K k j , L k j and γ k j for k < j. Thus, Theorem 3.2 applies again and the equations are well-posed.

This yields the main result of the paper, stated in the following theorem. where K, L and γ are defined by (31)-(38). Then, the zero equilibrium is exponentially stable in the L 2 sense.

Proof Theorem 4.1 ensures the existence of K, L ∈ L ∞ (T ), γ ∈ L ∞ ([0, 1]) such that (29),(30) holds and (α, β) satisfies ( 8), [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF]. Lemma 1 and the invertibility of the Volterra transformation yields the result.

Conclusion and perspectives

We have presented a control design for ODEs with a system of hyperbolic PDEs in the actuating path. The design results in a full-state feedback law needing measurements of the distributed actuator states along the spatial domain. This is not realistic in practice and future contributions will focus on the design of an observer solely relying on (collocated) boundary measurements.

Besides, the result opens the door to control design for other systems involving cascaded hyperbolic PDEs. In particular, networks of systems of hyperbolic balance laws are instrumental in modeling, e.g. oil production systems, networks of open channels [START_REF] Bastin | On lyapunov stability of linearised saint-venant equations for a sloping channel[END_REF] or power transmission lines [START_REF] Günther | A pdae model for interconnected linear rlc networks[END_REF].

Fig. 3 .

 3 Fig. 3. Examples of characteristic lines that satisfy Assumption (ii) of Theorem 3.2.

Lemma 2

 2 If G is acyclic, then the following holds (a) There exist terminal nodes, i.e. there exists a set T N ⊂ {1, ..., n} such that ∀i ∈ T N, ∀k ∈ {1, ..., n} Γ ik = 0 (52) All the valid paths are of finite length. In other words, any valid path can be completed with a valid path that leads to a terminal node and has a uniformely bounded length.

Property 3 . 6

 36 For any two paths a = (a 1 , ..., a p ) and b = (b 1 , ..., b q ) one has M a p M b q x, y = M (b,a) p+q (x, y)

Theorem 4 . 1

 41 System (31)-(38) has a unique solution K, L ∈ L ∞ (T ).

Theorem 4 . 2 1 0K( 1 1 0L( 1

 421111 Consider System (1)-(5) with the following control law U(t) = -R 1 u(t, 1) + , y)u(t, y)dy + , y)v(t, y)dy + γ(1)X(t) (109)

  paths of length p starting from the node F i . However, a large number of the terms of this sum is zero due to the product of Γ a k ,a k+1 inside this sum.

			56)
	Remark 3 The sum	denotes the sum over all (pos-
	a=(a 1 ,...ap ) a 1 =i
	sibly invalid) We now prove Equation (56) by recursion on the depth d.
	0
	Γ a k ,a k+1 ≡ 1	(57)
	k=1
	Equation (

d = 1. Consider a node F i such that d i = 1, i.e. F i is a terminal node.

Assuming by convention that, the empty product is equal to 1, i.e.


Machteld Meulleman
email: machteld.meulleman@univ-reims.fr

Katia Paykin
email: katia.paykin@gmail.com

WEATHER VERBS SIFTED THROUGH A MOTION SIEVE

Keywords: weather verbs, atmospheric use, motion verbs, metaphorical use, semantic conflation, FIGURE-incorporation, V-framed & S-framed languages, MANNER-incorporation

In cognitive frameworks à la Talmy, the weather verb 'to rain' has been analyzed repeatedly as a MOTION verb incorporating the FIGURE 'rain' in the verb root. In this contribution we claim that weather verbs can indeed be analyzed as motion verbs, albeit with a very heterogeneous behavior with respect to the presence of the conceptual components FIGURE and PATH. However, they consistently express MANNER, both in atmospheric and metaphorical contexts. Despite their conceptual structure as motion verbs, cross-linguistic differences in the availability of weather verbs in Russian and several Romance and West-Germanic languages cannot be accounted for through Talmy's classification of verb-framed and satellite-framed languages. Nevertheless this classification is partly verified for weather verbs used metaphorically to express directional motion.

Introduction

The majority of work consecrated to the study of weather verbs analyzes their argument structure (cf. among many others [START_REF] Darden | What rains // Linguistic Inquiry[END_REF][START_REF] Ruwet | Des expressions météorologiques[END_REF] These verbs, whose behavior is unanimously considered as particular because of the conceptual difficulty of distinguishing participants and processes inside the complex atmospheric phenomena they encode 1 , have also received some attention on the margin of theoretical studies tackling more general phenomena, such as, for instance, conceptual or cognitive structure of motion verbs (cf. Jackendoff 1983, [START_REF] Talmy | Toward a cognitive semantics[END_REF]. It does seem, however, that these studies focus exclusively on the verb denoting rain, thus promoting it to the status of the most prototypical weather verb, possibly characteristic of the whole class (cf. [START_REF] Bleotu | Why Does IT Always Rain on Me? On Weather Verbs[END_REF]).

In the present paper, we will examine the semantic structure of the weather verb class as a whole inside the conceptual structure framework à la [START_REF] Talmy | Toward a cognitive semantics[END_REF] and concentrate specifically on three questions: a) are weather verbs motion verbs? b) can the verb 'to rain' represent the entire class of verbs denoting atmospheric phenomena? and c) does [START_REF] Talmy | Toward a cognitive semantics[END_REF] distinction between verb-framed and satellite-framed languages apply to the weather domain and account for cross-linguistic differences in weather verbs?

Our study is structured as follows. We will start with a short summary of the theoretical background, briefly presenting previous analyses elaborated by Jackendoff (1983) and [START_REF] Talmy | Toward a cognitive semantics[END_REF]. Then, we will examine to what extent weather verbs can be conceptualized as motion verbs and whether they form a homogeneous semantic class in this respect. Finally, we will verify whether the cross-linguistic differences in the availability and use of weather verbs in some Germanic, Romance and Slavic languages can be accounted for by Talmy's distinction between verb-framed and satellite-framed languages.

Theoretical background

In 1983, Jackendoff proposes to consider semantic structure as a conceptual structure and decomposes various lexical verbs into constitutive conceptual elements 2 . Under his analysis the verb rain has the conceptual structure under [START_REF] Barrajón López | On Meteorological Denominal Verbs in Spanish. Syntactic-Semantic Properties and Argument Relationships[END_REF]. [START_REF] Barrajón López | On Meteorological Denominal Verbs in Spanish. Syntactic-Semantic Properties and Argument Relationships[END_REF] [ Event GO ([ Thing RAIN], [ Path DOWNWARD])] (Jackendoff 1983 : 185)

Therefore, according to Jackendoff, the verb to rain denotes a motion event that incorporates the THING 'rain' and the PATH 'downward'. Nothing is said though of other meteorological verbs, which are rather scarce in English as they are limited to precipitation verbs, such as to snow or to hail, and the verbs to thunder and to lightning 3 . While the proposed structure can easily be adapted to all precipitation verbs through the necessary modification of the THING, the case of the verbs to thunder or to lightning is more complicated as there is no clear THING nor clear PATH and the denoted event cannot be reasonably conceived of as a "GO event".

Indeed, one can wonder if the conceptual structure for the verb to rain was intended as a model transposable to other atmospheric phenomena, in which case a structure in (2) can be proposed for the verbs to thunder and to lightning through analogy.

( of the former can be compared to the THING of the latter but nothing is said of the PATH. While

Jackendoff's analysis deals with the conceptual structure of a specific verb, Talmy is interested in lexicalization patterns of motion events used in various languages 4 . The verb to 3 It is questionable whether the verb to freeze should be considered as a weather verb. Indeed, its -ing form in it is freezing has a problematic syntactic status and the clearly verbal form is mostly attested in personal constructions such as the lake froze and the cold weather froze the water pipes. Considering that the verb to rain and, by extension, all weather verbs, have been explicitly treated as motion verbs, the question arises whether the conceptual structures proposed for the description of rain can account for the entire class of weather verbs across languages.

Motion inside weather domain

When dealing with weather verbs, we should bear in mind that on a semantic level they denote phenomena independent from human activity or control, and if they do express motion, it is necessarily autonomous motion, as in their atmospheric use no true agent can be expressed (nor conceived).

Motion verbs only?

The idea that weather verbs express motion events is definitely plausible for the most prototypical meteorological phenomenon, i.e. rain. Indeed, in the case of rain, but also of other precipitation phenomena, such as snow, hail and drizzle, there is no doubt that the phenomenon implies the movement of an entity, i.e. water under any form of condensation, along a top-down PATH from the sky towards the earth.

( Indeed, languages that do not possess a whole array of weather verbs often express precipitation through the combination of a substance noun and a motion verb, such as 'go' or 'fall'. Russian, for example, has no specific weather verb for hailing but encodes this phenomenon through either of these verbs.

Slavic languages, on the contrary, tend to lexicalize the MANNER inside the verb root and encode the PATH respectively through directional complements, particles or prefixes.

(4) RU Idët / Padaet grad goes / falls hail 'It is hailing'

Moreover, even in languages that have specific precipitation verbs, it is often possible to express these phenomena through a motion verb and the atmospheric substance involved.

( Another argument in favor of the motion event analysis of precipitation verbs comes from the fact that, as already noted by [START_REF] Ruwet | Les verbes météorologiques et l'hypothèse inaccusative[END_REF], precipitation verb constructions can contain an explicit goal. In French, it can appear as a dative co-occurring with a locative PP or an adverb, as in [START_REF] Croft | Revising Talmy's Typological Classification of Complex Event Constructions[END_REF], while in English and Dutch, it can be encoded through a particle, expressing either a location or a human being, as illustrated in [START_REF] Darden | What rains // Linguistic Inquiry[END_REF].

( Italian provides yet another argument since its precipitation verbs display an ambiguous auxiliary selection, between 'be' and 'have' (cf. (8a)). As noted by Benincà & Cinque (1992: 158), the presence of an explicit goal entails the exclusive use of the auxiliary 'be', as in (8b), typically reserved to unaccusative motion verbs (such as venire 'come' which exclusively takes the verb 'be'). The construction with 'be' is, therefore, interpreted as the displacement of a substance, while the use of 'have' implies the interpretation of a weather phenomenon as an activity (just as camminare 'walk' which exclusively takes the auxiliary 'have'). Both cases thus fit the motion verb definition, the only real difference being in the directionality of the movement. If precipitation events do seem to behave as motion events, the question arises whether this also applies to other atmospheric phenomena. Some phenomena, like lightning for example, can indeed be encoded through a weather verb or a combination of a noun with the verb 'fall', as illustrated in [START_REF] Fagard | The Expression of Motion Events: A Quantitative Study of Six Typologically Varied Languages[END_REF]. However, for wind, the use of this verb cross-linguistically indicates its disappearance or decrease in strength, as illustrated in [START_REF] Glynn | Iconicity and the grammar-lexis interface[END_REF]. The main difference between lightning and thunder, on the one hand, and wind, on the other hand, has to do with the conceptualization of PATH. If bolts of lightning (and to some extent thunder) can be conceived of as motion along a top-down PATH, wind has a more complex PATH as no clear starting point nor end point can be identified and the orientation of movement is much more horizontal than vertical. This conceptual difference can also be observed in the compatibility of these verbs with directional particles, datives and PPs. As illustrated in (11a-b), a human being can function as the destination of a bolt of lightning, often inside a passive construction, while this is hardly possible with wind, though the latter can have its source and trajectory identified, as illustrated in (11c).

(11) a. NL Alpinist neergebliksemd in de Alpen alpinist down-lightning-V in the Alps 'Mountain climber struck by a bolt of lightning in the Alps' b. FR L'alpiniste a été foudroyé the alpinist has been lightning-V 'The alpinist has been struck by a bolt of lightning' c. NL Er woei een gure wind uit het Westen / door de straten / *op mij there wind-V a cutting wind from the West / through the streets / on me 'A cutting wind blew from the West / through the streets' Yet other phenomena, such as storms or thunderstorms, are completely incompatible with directional elements, as illustrated in [START_REF] Langacker | Foundations of cognitive grammar II: Descriptive application[END_REF]. This might be due to the fact that these weather predicates encode complex atmospheric phenomena involving several FIGURES at a time, such as rain and wind in the case of storms, and thunder, lightning and rain in the case of thunderstorms. Therefore it seems impossible to specify one single given PATH. [START_REF] Langacker | Foundations of cognitive grammar II: Descriptive application[END_REF] a. NL *Het stormt / onweert binnen it storms / thunderstorm-V inside b. NL *Alpinist neergeonweerd in de Alpen alpinist down-thunderstorm-V in the Alps

However, there are also some weather verbs denoting stationary atmospheric phenomena involving We can thus conclude that weather predicates can denote various types of motion events, vertically or horizontally oriented, one-, multi-or non-directional. Some possess a rather clear PATH, like in the case of precipitation verbs, and to some extent of lightning, thunder and wind. Complex weather phenomena, such as storms and thunderstorms, however, denote dynamic events in which no single PATH can be identified. We also find some weather predicates denoting events in which the conceptual MOTION component corresponds to the PRESENCE or APPEARANCE of a substance. Thus, the class of weather verbs is rather heterogeneous with respect to their conceptual structure 7 .

Figure incorporation

According to Jackendoff (1983) and [START_REF] Talmy | Toward a cognitive semantics[END_REF], the verb 'rain' includes a constituent THING or FIGURE referring to the substance integrated into the meaning of the verbal root. As we have already shown supra, in the case of weather verbs denoting precipitation, it is indeed possible to argue that the verb root incorporates an NP denoting the substance falling down from the sky, thus presenting conflated lexicalization, equivalent to the combination of an explicit motion verb, such as 'go' or 'fall' and a meteorological noun.

A possible indication that precipitation predicates incorporate the FIGURE 8 The mere fact that a weather verb can be used with nouns other than cognates, denoting atmospheric or metaphorical entities, concrete or even abstract ones, implies that at least in these particular cases the verb should incorporate something else than a FIGURE. What then is encoded in the verbal root?

What about MANNER?

We claim that the metaphorical use of precipitation verbs, as has been illustrated in [START_REF] Ruwet | Les verbes météorologiques et l'hypothèse inaccusative[END_REF], can best be accounted for if they express the way an object or abstract entity falls down. Indeed, this falling down might be continuous and rapid, as in the case of rain, slow, whirling around, as in the case of snow, or heavy with impact, as in the case of hail. Interestingly, in some languages, even non-precipitation verbs can be used to denote a specific way of falling down.

For example, as illustrated in [START_REF] Talmy | Toward a cognitive semantics[END_REF], in Dutch, the verbs 'thunder' and 'lightning' can be used to denote the falling of an object respectively with a lot of noise or very quickly. Notice that the PATH gets encoded explicitly in a satellite. The weather predicates themselves are used here as MOTION verbs incorporating MANNER rather than FIGURE.

(21) a. NL Het boek is naar beneden gedonderd the book is to down thundered 'The book thundered down' b. NL Hij is van de trap gebliksemd he is of the stairs lightning-V 'He tumbled down the stairs' However, as illustrated in [START_REF] Ruwet | Des expressions météorologiques[END_REF], the metaphorical use of weather verbs is not restricted to the description of entities falling down. Indeed, in English and Dutch, two Germanic languages, the verbs 'thunder' and 'lightning' can also be used to denote the noisy or quick way vehicles move along a horizontal line, as can be seen in ( 22). In addition, both languages use the verb for 'storm' to describe an entity entering or leaving a place in an uncontrolled way, as in (23). Once again, in these constructions the PATH is made explicit through a satellite and the weather verb can be said to encode MANNER. If it can be argued that weather predicates are MOTION verbs incorporating MANNER, their cross-linguistic distribution should, at least to some extent, be in line with Talmy's binary typology between verb-framed and satellite-framed languages.

The impossible language classification

One of the most puzzling characteristics of weather verbs from a cross-linguistic point of view is a big discrepancy in their number from one language to another. Some languages, like Dutch, allow encoding through verbs for most atmospheric phenomena, including black ice, hoarfrost, fog and lightning, as in (28a), while others, like English and Russian, possess very few properly atmospheric verbal forms. Standard French displays very few weather verbs, while archaic, popular and regional varieties make a surprisingly rich use of them (cf. (28b)).

( If weather verbs can be considered motion verbs, the question arises whether Talmy's distinction between V-framed and S-framed languages can explain these cross-linguistic differences in availability of weather verbs. Indeed, some parallels can be drawn. Firstly, some metaphorical uses of weather verbs in Dutch encode MANNER in the verbal root, such as waaien in (30a), which corresponds in French to the encoding of MANNER in a satellite PP en coup de vent, as in (30b).

( However, it is questionable whether the French verbs really indicate different FIGURES rather than differences with respect to MANNER. Indeed, the semantic distinctions between these verbs are operated through intensity and periodicity 10 .

Next to these two rather debatable arguments in line with Talmy's classification, there are also some discrepancies. From a semantic point of view, weather verbs behave in a very similar way across languages. In none of the languages under study are they limited to verbs denoting directional motion events nor to those denoting weather events involving clearly distinguishable FIGURES 11 . In addition, they can all be used for expressing Other metaphorical uses though do not fit into Talmy's classification as they can be found in all languages. For example precipitation verbs encode MANNER in the verb root crosslinguistically, as has been illustrated in [START_REF] Ruwet | Les verbes météorologiques et l'hypothèse inaccusative[END_REF].

In conclusion, the use and availability of weather verbs across languages cannot be satisfactorily explained by Talmy's distinction between V-framed and S-framed languages.

However, when these verbs are used metaphorically as directional motion verbs, the binary language classification holds to a certain extent.

Conclusions

Just like the most prototypical verb 'rain', weather verbs can indeed be analyzed as denoting motion events. These events, however, are rather heterogeneous as they can be vertically or horizontally oriented, one-or multi-directional, or even without any direction at all, dynamic or non-dynamic. Moreover, they can involve one or several substances or no substance at all.

Weather verbs are thus far from being uniform in terms of their conceptual structure with respect to PATH and FIGURE. Talmy's hypothesis about the incorporation of FIGURE inside weather verbs does not apply, as many weather events cannot make their FIGURE explicit through a cognate object, and those who do can encode all types of other FIGURES than the one supposedly incorporated in their verbal root.

What characterizes the majority of weather verbs is their systematic incorporation of MANNER, be it in metaphorical or strictly atmospheric use. Indeed, MANNER seems to be the core component even for weather events that do not contain a clear PATH or FIGURE and for those that are used metaphorically and, on the contrary, possess a clear source or goal.

Finally, considering that weather verbs are MOTION verbs incorporating MANNER, Talmy's binary typology between verb-framed and satellite-framed languages should have applied to describe their distribution across languages but in fact it does not 12 . All languages under study use weather verbs that incorporate MANNER directly in the verb root, although it is true that Sframed languages, such as Dutch and English, have some additional metaphorical uses of weather verbs as directional motion verbs. Still, we cannot really argue that the classification gets verified as weather verbs are most frequent in Dutch and less frequent in Russian, two languages belonging to the S-framed languages, while English, French and Spanish occupy an intermediate position. Thus, weather verbs in their atmospheric use differ from regular MOTION verbs and cross-linguistically display a peculiar puzzling behavior.

  , Paykin 2010, Bleotu 2012, Meulleman & Stockman 2013), often putting emphasis on their impersonal subject and sometimes taking into account possible NP expansions, like in It is drizzling a cold little rain.

(

  Het bliksemt it lightning-V 'It is lightning' b. ES Está relampagueando / Están cayendo rayos are lightning-V / are falling flashes 'It is lightning' c. RU Molnii padali v zemlju lightnings fell in soil 'Lightnings were falling onto the ground' (10) a. EN The wind fell b. NL Als de wind valt, … when the wind falls, … 'When the wind stops blowing, …' c. RU Veter padaet wind falls 'The wind is calming down'

(

  22) a. EN The train thundered past b. NL De moto bliksemde voorbij the motorcycle lightning-V past 'The motorcycle flashed by' it snows huge drops of rain (27) a. EN *It is hailing rose petals b. FR *Il neige des coups it snows blows c. NL *Er waait een traditionele sfeer in die dienst there wind-V a traditional atmosphere in that department

  FIGURES, such as dew, black ice, hoarfrost or fog. Although these phenomena can be encoded through verbs, they rather express the PRESENCE or APPEARANCE of a substance.

	(14)	a. EN It is freezing / #Ice is appearing
		b. FR Il gèle / #Il apparaît de la glace
		it freezes / it appears partitive ice
		c. ES Está helando / #Hay hielo
		is freezing / have-there ice
		d. NL Het vriest / #Er is vorst aan de grond
		it freezes / there is frost on the ground
		e. RU Morozit / Stoit moroz
		frost-V / stands frost

[START_REF] Manente | On the locative component of French weather-ça[END_REF]

a. NL Het dauwt / Het ijzelt / Het rijmt / Het mist it dew-V / it black ice-V / it hoarfrost-V / it fog-V 'There is dew / black ice / hoarfrost / fog' b. ES Rocía / Escarcha dew-V / hoarfrost-V 'There is dew / hoarfrost' Finally, we find weather verbs that denote phenomena for which no MOTION event nor FIGURE can be easily identified. This applies especially to temperature predicates. Indeed, the verb 'freeze', cross-linguistically attested, seems particularly difficult, although not impossible, to paraphrase through a light verb (be it a motion or a posture verb) and a meteorological noun (cf. Ruwet 1990: 54 for French).

  is that this FIGURE has to appear explicitly when it is modified, thus functioning as a cognate object9 . Er waait een jonge sfeer in die dienst there wind-V a young atmosphere in that department 'There is a young atmosphere in that department' b. NL Er donderde een kreet van verschikking door de zaal there thundered a shriek of terror through the hall 'A shriek of terror rent through the hall'

	(20)	a. NL	there fog-V a dense fog
		d. NL ??Er dauwt een lichte dauw
			there dew-V a light dew
		e. NL ??Er onweert een zwaar onweer
			there thunderstorm-V a heavy thunderstorm
	Interestingly, precipitation verbs can also appear cross-linguistically with NP expansions
	denoting atmospheric substances other than those that seem to be incorporated into the verbal
	root. As illustrated in (17), the verbs 'drizzle' and 'rain' may take nouns like 'snow', 'sleet',
	etc. The same construction is marginally possible with wind, but not with other weather
	phenomena, as shown in (18). The mere availability of this construction raises the question
	whether it can be claimed that precipitation verbs indeed incorporate the FIGURE.
	(17)	a. EN It was drizzling cold sleet
		b. FR Il pleuvait de la neige fondue
			it rained partitive snow melted
			'It was raining melted snow'
	(15)	c. NL Het regende hagel a. EN It was raining cold rain it rained hail b. FR Il pleuvait une petite pluie froide 'It was raining hail' it rained a tiny rain cold e. ES Llovían bolas de granizo (Calzado Roldán 2000: 87) 'It was raining little cold rain' rained balls of hail c. ES Estaba lloviendo una lluvia torrencial was raining a rain torrential 'It rained hailstones'
	(18)	'It was raining torrential rain' a. NL Er waait een hevige storm op zee
		d. RU Doždilo melkim doždëm there wind-V a heavy storm on sea
			rained fine rain 'A heavy storm is raging at sea'
		'It was raining little rain' b. NL *Er dondert een hevige regen
			there thunders a heavy rain
	(16)	a. NL Er waait een harde wind
	(19)	there wind-V a strong wind a. EN It is raining bullets / It is snowing bubbles
		'There blows a strong wind' b. NL Het regent kritiek op de hervormingen
		b. FR Il vente un vent froid de fin d'octobre it rains critique on the reforms
		it wind-V a wind cold of end of October 'It is raining critiques on the reforms' 'A cold late October wind is blowing' c. ES Llovía bombas sobre la ciudad / Le llovieron los aplausos c. NL ??Er mist een dichte mist rained bombs on the city / him dat rained the applauses
			'It was raining bombs on the city / He was greatly applauded'
		d. FR Il pleut des mauvaises nouvelles / Il grêle des coups / Il neige des pétales
			it rains some bad news / it hails some hits / it snows some petals
			'It is raining bad news / It is hailing blows / It is snowing petals'
		e. RU Doždilo puljami
			rained bullets
			'It was raining bullets'

However, not all weather verbs allow NP expansions. These expansions are possible with wind, but are ungrammatical (or at least very awkward) with many other phenomena, even if they clearly involve substances, as in the case of fog or dew. The same holds for thunderstorms involving several "substances". Could these verbs incorporate something else than precipitation verbs? Moreover, when genuine precipitation verbs are used in a metaphorical way, they describe the falling of non-meteorological substances, which may denote concrete objects, such as bullets, petals and bubbles, and abstract entities, such as critiques, applauses and news, as can be seen from

[START_REF] Ruwet | Les verbes météorologiques et l'hypothèse inaccusative[END_REF]

. Some other weather phenomena may be used in this construction as well (cf. examples in

[START_REF] Ruwet | Des expressions météorologiques[END_REF]

).

  However, if Dutch weather verbs clearly encode MANNER in the verbal root in metaphorical contexts, it is questionable whether they do it in atmospheric contexts, as the specific MANNER of raining, for example, can be expressed by different prefixes. In these constructions, therefore, MANNER rather gets encoded in satellites and not in the verbal root as expected for Dutch motion verbs.Secondly, within Talmy's classification, if FIGURE, just like PATH, belongs to the core components of the motion event, we can expect it to be encoded in the verbal root more often in verb-framed languages than in satellite-framed ones. Perfectly in line with this hypothesis is the fact that Standard French displays a great number of verbal forms for raining events, which have a clear FIGURE, while it has only very few forms for atmospheric phenomena without a distinct FIGURE component. On the contrary, Dutch, which possesses a large array of weather verbs, has only a limited number of specific rain-verbs, all of which contain prefixes rather than different verb roots.

	30)	a. NL Hij is komen binnenwaaien
		he is come in-wind-V
		b. FR Il est passé en coup de vent
		he is passed in gust of wind
		'He came as a gust of wind'
	(31)	a. NL Het motregent
		it dust-rains
		'It is drizzling'
		b. NL Het stortregent / plasregent
		it dump-rains / puddle-rains
		'It is pouring rain'
	(32)	a. FR pleuviner, pleuvioter, pleuvasser, etc.
		b. NL motregenen, stortregenen, plasregenen etc.

  MANNER, both in atmospheric and metaphorical contexts. Moreover, the fact that precipitation verbs incorporating FIGURE are used in all languages under study, except Russian, cannot be explained by this bipartite language classification as Russian belongs to the same group as Dutch and English and to a different one from French and Spanish.However, Talmy's language classification can apply to the metaphorical use of weather verbs when they are used in personal constructions to denote motion of a real participant.

Indeed, in line with Talmy's classification, Germanic languages encode MANNER of directional motion in the verbal root insisting on the PATH in the satellite, as in (33a-b), while Romance languages rather lack this use, as shown in (33c). Interestingly, English uses metaphorically certain weather verbs not really attested in atmospheric contexts, such as to storm. (33) a. EN He stormed into the room b. NL Hij stormde de kamer binnen c. FR #Il tempêta dans la pièce

According to[START_REF] Ruwet | On Weather Verbs // Papers from the regional Meetings[END_REF]

[202][203], weather events are perceived in a global way, which is incompatible with the analytical character of syntactic encoding. Langacker (1991: 366) calls them "all-encompassing phenomena" lacking the "clear-cut division into setting, process and discrete participants". Recently, Eriksen et al. (2010: 567-568) claimed that "meteorological events differ in terms of whether one may conceptualize the given event as incorporating participant-like entities, but the general feature is that in none of these cases can any such entity be perceived as a participant which is referentially independent from the event itself."[START_REF] Beavers | The Typology of Motion Expression Revisited[END_REF] According toRuwet (1990: 56), a similar analysis has already been presented in his 1972 book.

We will use the ISO-639 language codes.

Whenever our examples illustrate the same phenomena in various languages, we do not provide explicit glosses.

The heterogeneity of the linguistic domain of weather has already been suggested by[START_REF] Ruwet | On Weather Verbs // Papers from the regional Meetings[END_REF][START_REF] Ruwet | Des expressions météorologiques[END_REF]) and extensively shown by[START_REF] Paykin | Il pleut des idées reçues[END_REF] and[START_REF] Meulleman | La inacusatividad en los verbos meteorológicos en español: un análisis comparativo de 'llover' y 'amanecer[END_REF] with respect to other characteristics.

From a diachronic point of view, meteorological verbs are rather denominal verbs resulting from a recategorization process of meteorological nouns. (For more details, cf. Paykin 2003 for Indo-European languages, and in particular French and English, and Barrajón López 2011 for Spanish).

For more information about the cognate object construction in the weather domain, cf. Calzado Roldán(2000) and[START_REF] Paykin | Il pleut des idées reçues[END_REF].

The exact definitions of these verbs contain adverbial modification of the basic phenomenon rain. For example, TLFi and Grand Robert dictionaries define them as follows: pleuvinerto rain softly, in little drops, pleuvioterto rain lightly, pleuvasserto rain lightly, at irregular intervals (our translation).

In fact, according to[START_REF] Eriksen | The Linguistics of Weather. Cross-Linguistic Patterns of Meteorological Expressions // Studies in Language[END_REF], precipitation verbs cross-linguistically get most rarely encoded through a specific weather predicate precisely because, contrary to other events, the substance of precipitation can be conceived as a kind of participant.

This is in line with recent cross-linguistic work showing various limits of the binary typology (cf. among others, Beavers et al. 2010, Croft et al. 2010 and Fagard et al. in press).

In addition, weather predicates denoting thunder and lightning can also be used metaphorically to express the emission of noise and light. The verb 'thunder' is crosslinguistically used for speaking in a noisy and violent way, as illustrated in (24). French even uses the verb 'storm' in similar contexts. In some languages, like Spanish and Dutch for instance, the verb 'lightning' preserves the flashing or blinking component in its metaphorical use, as in (25). In all these contexts, weather verbs express MANNER of a very peculiar kind of MOTION event with a clear human source.