Andrey Polyakov

On Optimal Control Problems for Generalized Homogeneous Systems

The paper deals with optimal control problems for plants modeled by nonlinear ordinary differential equations with homogeneous (in a generalized sense) right hand sides. The classical tools of optimal control theory, namely, Dynamic Programming and Maximum Principle are refined for generalized homogeneous control systems.

Introduction

An object (e.g. an operator or a vector field) is homogeneous in a generalized sense if it is symmetric with respect to a certain family of transformations (called dilations) [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]. On the one hand, a lot of well-known models of mathematical physics are homogeneous in a generalized sense [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF], e.g. heat, wave, Burgers, Navier-Stocks, Saint-Venant and Korteweg-de-Vries equations as well as Fast Diffusion equation. On the other hand, homogeneity is one of the desirable properties for nonlinear control system, since it allows stability to imply robustness of the system [START_REF] Ryan | Universal stabilization of a class of nonlinear systems with homogeneous vector fields[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] and simplifies the time-constrained stabilization providing the finite-time stability [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] to the closed-loop control system with negative homogeneity degree [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF].

Optimal control design is one of classical problems of mathematical control theory [START_REF] Boltyanskii | Towards a theory of optimal processes[END_REF], [START_REF] Bellman | The theory of dynamic programming[END_REF]. Geometry of a system is important for optimal control design [START_REF]Mathemtical Control Theory[END_REF], [START_REF] Agrachev | Control Theory from the Geometric Viewpoint[END_REF]. Being a kind of a symmetry, the homogeneity is expected to simplify analysis and design optimal control systems.

The optimal regulation problem for a class of homogeneous control systems is considered in [?], where some sampled-time control is designed. The proposed feedback comes out of the solution of an infinite horizon optimization problem in discrete time. This paper refines Dynamic Programming and Maximum Principle under the assumption that the control system is modeled by homogeneous ordinary differential equation.

Notation:

• R is the field of real numbers, R + = {x ∈ R : x ≥ 0}; N is the set of natural numbers;

• rank(M ) denotes the rank of the matrix M ∈ R n×m ;

• I n ∈ R n×n is the identity matrix;

• diag{λ i } n i=1 denotes the diagonal n × n matrix with the elements λ i ∈ R on the main diagonal;

• the inequality P > 0 for P ∈ R n×n means that P is positive definite symmetric matrix.

• let W 1,1 ([0, T ), R n ) be the Sobolev space of absolutely continuous functions (0, T ) → R n , where T ≤ +∞, and the set W 1,1 loc ((0, T ), R n ) consists of functions which restriction to any interval (0, T ) with 0 < T < T belongs to W 1,1 ((0, T ), R n );

• let L ∞ ((0, T ), R m ) be the space of uniformly essentially bounded functions, where T ≤ +∞;

• let • be a norm in R n and • A be the matrix norm induced by • , i.e.

A A = sup x∈R n Ax x if A ∈ R n×n ; • L ∞ and • W 1,1 loc • B(ε) = {x ∈ R n :
x < ε} is the ball of the radius ε > 0.

• S = {u ∈ R n : u = 1} is the unit sphere.

Problem Statement and Basic Assumptions

Let us consider a control system modeled by the ordinary differential equation

ẋ(t) = f (x(t), u(t)), t ∈ (0, T ), (1)
x(0) = x 0 \{0}, (2)
where T ≤ +∞ is the time horizon, x(t) ∈ R n is the vector of system states, u(t) ∈ R m is the vector of control inputs, the vector-valued function f : R n \{0} × R m → R n is assumed to be at least continuous. We study the optimal control problem (OPC)

J(x, u, T ) = T 0 L(x(σ), u(σ))dσ → inf x,u (3)
subject to (1), ( 2), the control constraint given by

u(t) ∈ U, t ∈ (0, T ), (4)
and the terminal set described as follows

lim t→T g i (x(t)) ≤ 0, i = 1, 2, ..., l, (5)
where the set U ⊆ R m is compact and the functions g i : R n \{0} → R and the Lagrangian L : R n \{0} × R n → R + are assumed to be continuous. The time horizon T can be fixed or non-fixed, finite T < +∞ or infinite T = +∞.

Below we assume the considered OCP is homogeneous in a generalized sense. Homogeneity is a property of an object (e.g. function or vector field) to be symmetric (in a certain sense) with respect to a group of transformations (called dilations). The generalized homogeneity [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF], [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF] deals with linear transformations (linear dilations) given below.

Definition 1 ([4]) A map d : R → R n×n is called dilation in R n if it satisfies • Group property: d(0) = I n and d(t+s) = d(t)d(s), t,s ∈ R; • Continuity property: d is a continuous map, i.e. ∀t > 0, ∀ε > 0, ∃δ > 0 : |s -t| < δ ⇒ d(s) -d(t) A ≤ ε;
• Limit property: lim Obviously, the dilation d is a continuous group of invertible linear maps

d(s) such that d(-s) = [d(s)] -1 . Definition 2 A vector field g : Ω → R n (a function h : R n → R) is said to be d-homogeneous of degree ν ∈ R if d(s)Ω ⊆ Ω for s ∈ R and g(d(s)z) = e νs d(s)g(z), ∀z ∈ Ω, ∀s ∈ R. (6)
(resp. h(d(s)z) = e νs h(z), ∀z ∈ Ω, ∀s ∈ R. )

More details about standard, weighted and geometric homogeneity of nonlinear systems can be found in [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], and references therein, where a lot of examples of homogeneous control systems are also studied.

We study the considered optimal control problem under assumption that the vectorfield f and Lagrangian L are homogeneous (for the control u treated as a parameter).

Assumption 1 Let d be a dilation in R n . The vector-field f , the Lagrangian L and the functions g i are assumed to be d-homogeneous with respect to the first argument, i.e.

∃ν f ∈ R : f (d(s)x, u) = e ν f s d(s)f (x, u), (7) ∃ν L ∈ R : L(d(s)x, u) = e ν L s L(x, u) (8) ∃ν i ∈ R : g i (d(s)x) = e νis g i (x), (9)
for s ∈ R, x ∈ R n \{0} and u ∈ R m , i = 1, 2, ..., l.
The optimal control problem (1) -( 5) that satisfies the given assumption is called the homogeneous optimal control problem (HOCP) in the Lagrange form. Below (see Section IV) we also make Assumption 2 that implies the uniqueness of solution to the system (1) -(2) in forward time for u ∈ L ∞ ((0, T ), R m ) satisfying (4).

The main goal of the paper is to refine the conventional optimal control design tools (namely, dynamic programming and Pontraygin Maximum Principle) for the case of homogeneous evolution equations, homogeneous terminal sets and homogeneous cost function. x . Limit property implies

• d(s) A → 0 as s → -∞; • d(s) = I n if s = 0; • d(s) A → +∞ as s → +∞; • G d > 0 (ker G d = {0}).
The most popular dilations are the uniform (or standard) dilation (L. Euler) : d(s) = e s , s ∈ R and the weighted dilation [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]): d(s) = diag{e ris }, s ∈ R and r i > 0, i = 1, ..., n. They obviously satisfy Definition 1 with G d = I n and G d = diag{r i }, resp.

Geometric dilation [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF] is more general since it allows the map d(s) : R n → R n to be nonlinear.

Definition 3 ([18]

) The dilation d is monotone if it is a strong contraction for s < 0, i.e. d(s) A < 1 as s < 0.

Monotonicity of dilation depends on the norm • selected in R n . For example, the dilation d(s) = e s cos(s) sin(s) -sin(s) cos(s) with

G d = 1 1 -1 1 is monotone if R 2
is equipped with the weighted norm x P = √

x P x, P =

1 1/ √ 2 1/ √ 2 1
> 0 and it is nonmonotone if, for example, P = 1 3/4 3/4 1 > 0. In the latter case, the curve {d(s)u : s ∈ R} may cross the unit sphere x P = 1 in two different points.

Theorem 1 ([18])

The next four conditions are equivalent 1) the dilation d is monotone;

2) d(s) A > 1 for s > 0;

3) the continuous function d(•)x : R → R + is strictly increasing for any x ∈ S; 4) for any x ∈ R n \{0} there exists a unique pair

(s 0 , x 0 ) ∈ R × S such that x = d(s 0 )x 0 .
Theorem 1 guarantees the functions d(•) A : R → R + and d(•) A : R → R + are also continuous and strictly increasing.

Definition 4 ([18]) The dilation d is called strictly monotone if ∃β > 0 : d(s) A ≤ e βs for s ≤ 0.
The dilation d considered in the above example is strictly monotone if R 2 equipped with the Euclidean norm.

Theorem 2 ([18]

) Let d be a dilation in R n then

• the matrix -G d is Hurwitz, i.e. all eigenvalues λ i of G d are placed in the right complex half-plane;

• there exists

P ∈ R n×n P G d + G d P > 0, P = P > 0; (10)
• the dilation d is strictly monotone with respect to the weighted norm • = •, • induced by the inner product u, v = u P v with P satisfying (10):

e αs ≤ d(s) A ≤ d(s) A ≤ e βs if s ≤ 0, e βs ≤ d(s) A ≤ d(s) A ≤ e αs if s ≥ 0, (11)
where

α = 0.5λ max P 1 2 G d P -1 2 + P -1 2 G d P 1 2 , β = 0.5λ min P 1 2 G d P -1 2 + P -1 2 G d P 1 2 . ( 12
)
Therefore, any dilation d is strictly monotone if R n is equipped with the weighted norm x =

√

x P x provided that the matrix P > 0 satisfies (10).

Homogeneous norm

Dilation d introduces a topology in R n (spheres and balls [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]) by means of the so-called "homogeneous norm".

Definition 5 A continuous function • d : R n → R + is said to be d-homogeneous norm if x d → 0 as x → 0 and d(s)x d = e s x d > 0 for x ∈ R n \{0} and s ∈ R.
For monotone dilations the canonical homogeneous norm can be defined as follows:

x d = e sx with s x : d(-s x )x = 1. (13)
In [START_REF] Polyakov | On finite-time stabilization of evolution equations: A homogeneous approach[END_REF] such a homogeneous norm was called canonical since it is induced by the canonical norm • in R n and x d = x = 1 on the unit sphere S. In this case, obviously,

d(ln x d ) A ≤ x ≤ d(ln x d ) A .
Note also that if d(s) = e s then • d = • . By default, below we deal only with the canonical homogeneous norm. Let B d (r) be homogeneous ball of the radius r > 0 , i.e.

B d (r) = {x ∈ R n : x d < r}. Proposition 1 ([18]) If d is a strictly monotone dilation then • x 1 β d -x 2 β d ≤ x 1 -x 2 for x 1 , x 2 ∈ R n \B d (1),
• the homogeneous norm • d is Lipschitz continuous outside the origin;

• if the norm • is smooth outside the origin then the homogeneous norm • d is also smooth outside the origin,

d d(-s)x ds < 0 if s ∈ R, x ∈ R n \{0} and ∂ x d ∂x = x d ∂ z ∂z | z=d(-s)x d(-s) ∂ z ∂z | z=d(-s)x G d d(-s)x s=ln x d (14)

Generalized Homogeneous Functions and Vectors Fields

Homogeneous functions and vector fields (see, Definition 2) have a lot of properties useful for control design and state estimation of both linear and nonlinear plants as well as for analysis of convergence rates [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF], [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF]. It is worth stressing that essentially non-linear vector-fields and functions may be d-homogeneous.

Example 1 Let the dilation d be defined as follows

d(s) = e s 1 0 0 0 cos(s) sin(s) 0 -sin(s) cos(s)
.

It is strictly monotone with respect to the Euclidean norm x = √

x T x and G d =

1 0 0 0 1 1 0 -1 1
. The vector field f : R3 → R 3 defined as

f (x) = x 2 2 +x 2 3
x 2 1 (cos(ln |x1|)+sin(ln |x1|)) x 2 1 (cos(ln |x1|)-sin(ln |x1|))

and the function h : R 3 → R given by h

= x 3 1 + (x 2 2 + x 2 3 ) Definition 6 An open nonempty set Ω ⊂ R n is said to be d-homogeneous cone if d(s)Ω ⊆ Ω.
The name"homogeneous cone" is very natural for the set Ω, since each point x belongs to Ω together with the homogeneous curve {z :

z = d(s)x, s ∈ R}. If the dilation is uniform d(s) = e s I n then Ω becomes the conventional cone in R n . Let F d (Ω) (resp. H d (Ω)) be the set of d-homogeneous vector fields Ω ⊆ R n → R n (resp. functions R n → R), which are continuous on Ω ⊆ R n \{0}. Here Ω is d- homogeneous cone. Let deg d (h) (resp. deg d (f )) denote the homogeneity degree of h ∈ H d (Ω) (resp. f ∈ F d (Ω)).
The homogeneity allows local properties (e.g. smoothness) of vector fields (functions) to be extended globally [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF]. Moreover, if the function (resp. vector field) is smooth then homogeneity is inherited by its derivatives.

Corollary 1 If h ∈ H d (Ω) ∩ C 1 (Ω\{0}, R) and f ∈ F d (Ω) ∩ C 1 (Ω\{0}, R n ) then e deg(h)s ∂h(x) ∂x = ∂h(x) ∂z z=d(s)x d(s), (15)
∂h(x) ∂x G d x = deg d (h) h(u), (16)
e deg(f )s d(s) ∂f (x) ∂x = ∂f (z) ∂z z=d(s)x d(s), (17)
∂f (x)

∂x G d x = (deg d (f )I n + G d ) f (x), (18)
for x ∈ Ω\{0} and s ∈ R.

The proofs of the identities ( 15) and ( 16) are given in [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF]. The identities ( 17) and ( 18) can be proven similarly.

The next corollary shows that homogeneity degree specifies some properties of homogeneous functions. Proof. Since the homogeneous norm • is continuous at the origin, then using the homogeneous identity h

Corollary 2 If h : Ω ⊂ R n → R is a d-
(x) = h(d(ln x d )z) = x deg(h) d h(z) with z = d(-ln x )
x ∈ S ∩ Ω we trivially complete the proof.

The similar results can be obtained for d-homogeneous vector fields. Proof. Similarly for the vector field f we derive

Corollary 3 Let f : Ω ⊂ R n → R n is
f (x) = x ν d d(ln x d )f (z) and x ν d d(ln x d ) A f (z) ≤ f (x) ≤ x ν d d(ln x d ) A f (z) . Hence, using Theorem 2 for u < 1 we have x ν+α d f (z) ≤ f (x) ≤ x ν+β d f (z) and x ν+β d f (z) ≤ f (x) ≤ x ν+α d f (z) if x > 1.

Homogeneous approximation

Local homogeneity and homogeneous approximations has been studied in [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF] in order to cover a wider class of non-linear systems.

Definition 7 Let a function

h : Ω ⊂ R n → R be continuous on d-homogeneous cone Ω. A d-homogeneous function h 0 : Ω ⊂ R n → R (resp. h ∞ : Ω ⊂ R n → R) of degree ν is said to be homogeneous approximation of h at 0 (resp. at ∞) if lim s→-∞ h(d(s)x) e νs = h 0 (x) lim s→+∞ h(d(s)x) e νs = h ∞ (x) (19)
In [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] these limits are assumed to be uniform on the unit sphere S in R n . In our case, such a condition cannot be posed since the functions h, h 0 , h ∞ may be unbounded on S ∩ Ω.

Homogeneous Optimal Control Problem

Admissible solutions and maximal horizon

Homogeneity (symmetry) of ODE implies symmetry of its solutions. The similar facts can be proved for optimal/admissible pairs of HOCP.

Definition 8 A pair (x,u) ∈ W 1,1 loc ((0,T ),R n )×L ∞ ((0,T ),R m
) is said to be admissible for HOCP with T ≤ +∞ and x 0 ∈ R n \{0} if it is satisfies (1) almost everywhere on (0, T ), (2), ( 4), (5) and J(x, u, T ) < +∞.

Let the set of admissible pairs (the admissible set) of HOCP with T ≤ +∞ and x 0 ∈ R n \{0} be denoted as

P(x 0 , T ) ⊂ W 1,1 loc ((0, T ), R n ) × L ∞ ((0, T ), R m ). (20)
The next theorem discovers homogeneous relations between admissible sets.

Theorem 3 If one has (x, u) ∈ P(x 0 , T ) then

(x s , u s ) ∈ P(d(s)x 0 , e -ν f s T ),
Proof. Homogeneity of the function χ immediately follows from Theorem 3. The claims 1)-4) are straightforward consequences of Corollary 2. Let us prove the claim 5) using the same corollary. Due to homogeneity one has 1) is forward complete. Finally, taking into account ν L -ν f > 0 and homogeneity of the Lagrangian we conclude L(0, u) = 0 and continuity of L with respect to the first variable. So, P (x 0 , T ) is bounded for any x 0 ∈ R n and any T ∈ (0, +∞).

f (x, u) ≤ x ν f d d(ln x d ) A f (z, u) with z = d(-ln x )x ∈ S and d(ln x d ) A ≤ x β if x ≤ 1, but d(ln x d ) A ≤ x α if x ≥ 1. Taking into account ν f + α ≤ β and x β d ≤ x for x ≥ 1 we derive f (x, u) ≤ C 2 x if x ≥ 1, where C 2 > 0. On the other hand, ν f + β ≥ 0 implies that f (x, u) ≤ C 1 if x ≤ 1, where C 1 > 0. Therefore, for any u ∈ L ∞ ((0, +∞), R n ) : u(t) ∈ U we have f (x, u) ≤ C 1 + C 2 x and the system (
The function ξ defined above may be discontinuous in the general case. However, for the HOCP without terminal constraints (i.e. g i ≡ 0, i = 1, .., l) the continuity of χ on R n \{0} is granted by the continuous dependence of solution x and functional J on parameters x 0 ∈ R n \{0} and T ∈ (0, +∞).

Homogeneous systems with positive degree (ν f > 0) may have solutions that blow up in a finite time, e.g. ẋ(t) = x 3 (t)u(t). In the latter case the HOCP may remain meaningful, e.g. if ν L < 0 then L(x, u) → 0 as x → +∞ and an infinum of the cost functional J may correspond to fastest blow up of the solution.

Definition 9 An admissible pair (x * , u * ) ∈ P (x 0 , T ) is said to be optimal if J(x * , u * , T ) ≤ J(x, u, T ) for all admissible pairs (x, u) ∈ P(x 0 , T ).

In general, the optimal pair may be non-unique. Let O(x 0 , T ) denote the set of optimal pairs. Obviously, O(x 0 , T ) ⊂ P (x 0 , T ) and the next corollary immediately follows from Theorem 3. 1), ( 2) may have non-unique solution in a forward time for a given x 0 . So, an optimal control u * may generate a family of solutions

Corollary 5 If one has (x * , u * ) ∈ O(x 0 , T ) then (x * s , u * s ) ∈ O(d(s)x 0 , e -ν f s T ) for any s ∈ R, where x * s (t) = d(s)x * (e ν f s t), u * s (t) = u * (e ν f s t), t ∈ (0, e -ν f s T ). The system (
X x0,u * ⊂ W 1,1
loc ((0, T ), R n ) and, probably, just one of them forms an optimal pair (x * , u * ) ∈ O : J(x * , u * , T ) ≤ J(x, u * , T ) for all x ∈ X x0,u * . The results proven above hold for such a weak statement of HOCP. Definitely, additional assumption about uniqueness of solutions to [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF] in forward time makes them correct in a strong (conventional) sense.

Assumption 2 For any x 0 ∈ R n \{0} it holds: if (x 1 , u 1 ) ∈ P(x 0 , T ) and (x 2 , u 2 ) ∈ P(x 0 , T ) then the identity

u 1 -u 2 L ∞ = 0 implies x 1 -x 2 W 1,1 ((0,T ),R n ) = 0 for all T ∈ (0, T ).
Under this assumption the specific relations between optimal/admissible solutions to HOCP with different optimization horizons imply homogeneity of necessary and/or sufficient optimality conditions. In fact, to fulfill Assumption 2 is it sufficient to consider the regular case, i.e. f, L, g i ∈ C 1 (R n \{0}). By default below we assume that Assumption 2 holds.

Homogeneous Bellman Function

The cost functional J is additive, so HOCP, obviously, satisfies the Bellman principle of optimality (see e.g. [START_REF] Poznyak | Advanced Mathematical Tools for Automatic Control Engineers[END_REF]) saying that "any tail of optimal trajectory is optimal too". The sufficient condition of optimality can be derived using on the so-called Bellman (value) function V : R n × R + → R + , which in our case can be defined as follows

V (x 0 , T ) = J(x * , u * , T ),
where (x * , u * ) is an optimal pair for HOCP with a given x 0 ∈ R n \{0} and a given T ≤ 1 χ(x0) . For time invariant systems (as we consider) Bellman function usually depends only of the space argument x ∈ R n (see e.g. [START_REF] Poznyak | Advanced Mathematical Tools for Automatic Control Engineers[END_REF]). We have added the second argument to V in order to study also its dependence on the optimization horizon. Obviously, that V (x 0 , T ) → 0 as T → 0 + for any fixed x 0 ∈ R n \{0}.

Theorem 4 Let the function ξ defined by (21) be continuous on R n \{0}. Then the set D ⊂ R n × R + given by D = (x 0 , T ) :

x 0 ∈ R n \{0} and 0 < T < 1 χ(x0)
is nonempty, open and connected and the Bellman function V is continuous on D. Moreover, if V is continuously differentiable on D then V satisfies the Hamilton-Jacobi-Bellman(HJB) equation of the form

min u∈U ∂V ∂x f (x, u)+L(x, u) = ∂V ∂T , (x, T ) ∈ D (22)
and the identity

V (d(s)x, e -ν f s T ) = e (ν L -ν f )s V (x, T ) holds for (x, T ) ∈ D\{0}.
Proof. The set D is non-empty and open by construction. Let us show that it is connected due to continuity and homogeneity of the function χ defined by [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]. Indeed, if (x 1 , T 1 ) ∈ D and (x 2 , T 2 ) ∈ D then the homogeneous curves

Γ 1 = {(x, T ) : x = d(s)x 1 , T = e -νs T 1 , s ∈ R} and Γ 2 = {(x, T ) : x = d(s)x 2 , T = e -νs T 2 , s ∈ R} belong to D.
Since d is a dilation then there exists s 1 ∈ R and s 2 ∈ R such that d(s 1 )x 1 ∈ S and d(s 2 )x 2 ∈ S, where S is a unit sphere. This means that any point from D can be connected by continuous curve with the cylinder Π = S × (0, +∞). Let (x 1 , T1 ) ∈ D ∩ Π and (x 2 , T2 ) ∈ D ∩ Π are the corresponding crossing points of Γ 1 and Γ 2 , respectively. Since the function χ is non-negative and continuous on R n \{0} then there exists a continuous function T : S → R + such that 0 < T (x) < 1/χ(x) for x ∈ S and T (x 1 ) = T1 , T (x 2 ) = T2 . Let {x = φ(s) : s ∈ [0, 1]} ∈ S be an arbitrary continuous curve that connects the points x1 ∈ S and x2 ∈ S. In this case, the continuous curve Γ = {(x, T ) :

x = φ(s), T = T (φ(s)), s ∈ [0, 1]} ∈ D ∩ Π connects the points x1 , x2 , i.e. the set D is connected.
The rest part of the proof can be done in the standard way (see, e.g. [START_REF] Poznyak | Advanced Mathematical Tools for Automatic Control Engineers[END_REF]Theorem 22.18]) taking into account that the conventional Bellman function Ṽ : (0, T ) × R n → R + in our case admits the representation Ṽ (t, x) = V (x, T -t). Homogeneity of V follows from the identity J(x, ũ) = e (ν L -ν f )s J(x, u) given by Theorem 3. More constructive representation of the homogeneous Bellman function V can be obtained in the case of non-zero degree of homogeneity of the vector field f . Corollary 6 If ν f = 0 then under conditions of Theorem 4 the Bellman function V admits the representation

V (x, T ) = T ν f -ν L ν f Φ d ln T ν f x ( 23
)
where Φ : Ω 1 → R + is a non-negative continuously differentiable function defined on Ω 1 = {x ∈ R n \{0} : χ(x) < 1} as a solution to the modified HJB equation

min u∈U ∂Φ ∂y f (y, u)+L(y, u) = (ν f -ν L )Φ+ ∂Φ ∂y G d y ν f . (24)
Moreover, for

x ∈ Ω 0 = {x ∈ R n \{0} : χ(x) = 0} one has lim T →+∞ V (x, T ) = Φ ∞ (x) if ν f > 0 lim T →+∞ V (x, T ) = Φ 0 (x) if ν f < 0, (25)
where the functions Φ 0 : Ω 0 → R + and Φ ∞ : Ω 0 → R + are homogeneous approximations of the function Φ at zero and at infinity, respectively.

Proof. The formula [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF] follows from the identity

V (d(s)x, e -ν f s T ) = e (ν L -ν f )s V (x, T ) if Φ(y) = V (y, 1), y = d(s)
x and s = ln T ν f . The modified HJB equation ( 24) is obtained from [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF] using the identity

∂V ∂x G d x -ν f T ∂V ∂T = (ν L -ν f )V,
which holds due to [START_REF]Mathemtical Control Theory[END_REF]. Finally, the identities [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF] immediately follow from the definition of homogeneous approximations.

Example 2 For f (x, u) = -ux 1/3 , L(x, u) = x 2/3 and U = [0, 1] the optimal control problem (1) -( 4) is an HOCP with d(s) = e s , ν f = -2/3 and ν L = 2/3. Its Bellman function V can be found analytically

V (x, T ) = 3x 4/3 /4 if 3x 2/3 /2≤T, T x 2/3 -T 2 /3 if 3x 2/3 /2>T.
Obviously, it admits the representation [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF] with

Φ(y) = 3/4y 4/3 if y≤(2/3) 3/2 , y 2/3 -1/3 if y>(2/3) 3/2 .
Note that χ ≡ 0 (i.e. any horizon is admissible), so Ω 1 = R n \{0}. The function Φ is continuously differentiable on Ω 1 and it satisfies the modified HJB equation [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF]. Moreover, V (x, +∞) = Φ 0 (x) = 3x 4/3 /4, where Φ 0 is the homogeneous approximation of Φ at zero.

Remark 1 The Bellman function

V : Ω 0 ⊂ R n → R + of HOCP on infinite horizon (T = +∞) is d-homogeneous, i.e. V (d(s)x) = e (ν L -ν f )s V (x), x ∈ Ω 0 and s ∈ R + .
To construct V on the whole domain Ω one can be defined on the unit sphere (or its part) only.

Homogeneous Maximum Principle

The necessary condition of optimality of HOCP given by Pontryagin Maximum Principle also admits a homogeneous representation. In particular, homogeneity of the vector-field f and Lagrangian L implies certain homogeneity of the Hamiltonian:

H(ψ, x, u) = ψ f (x, u) -µL(x, u), where x, ψ ∈ R n , u ∈ R m , µ ∈ R + . Indeed, one has H(e (ν L -ν f )s d (-s)ψ, d(s)x, u) = e ν L s H(ψ, x, u), s ∈ R.
Theorem 5 If the functions f , L and g = (g 1 , g 2 , ..., g l ) are differentiable on R n \{0} and (x * , u * ) ∈ O(x 0 , T ) is an optimal pair with x * (t) = 0 for t ∈ [0, T ] then there exist a number µ ∈ R + , a vector v ∈ R l + and a function ψ * : [0, T ] → R n such that a) ψ * (T ) + µ + v > 0 and |g(x * (T ))| v = 0; b) the tuple (ψ * , x * , u * ) solves the boundary value problem

ψ * (t) ẋ * (t) = F (ψ * (t), x * (t), u * (t)), F = -( ∂H ∂x ) ( ∂H ∂ψ ) , x * (0) = x 0 , ψ * (T ) = -∂g(x(T )) ∂x v, (26)
where the vector field

F : R n × R n \{0} × R m → R 2n satisfies the homogeneous relation F ( d(s)z, u) = e ν f s d(s)F (z, u), z ∈ R 2n , u ∈ R m , s ∈ R, (27)
for the uniformly continuous group d given by

d(s) = e (ν L -ν f )s d (-s) 0 0 d(s) ; c) H(ϕ * (t), x * (t), u * (t)) = max u∈U H(ϕ * (t), x * (t), u) = C for all t ∈ [0, T ],
where

C ∈ R is a constant; d) d[(ψ * (t)) G d x * (t)] dt = -ν f (ψ * (t)) f (x * (t), u * (t)) + µν L L(x * (t), u * (t)
) for all t ∈ (0, T ).

Proof. Since x * (t) = 0 for all t ∈ [0, T ] then the proof of the properties a)-c) can be completed using conventional arguments based on concept of Lagrange multipliers (see, e.g. [START_REF] Poznyak | Advanced Mathematical Tools for Automatic Control Engineers[END_REF]Theorem 22.11 Finally, let us prove the claim d). One has

d[(ψ * (t)) Gdx * (t)] dt = ( ψ * (t)) G d x * (t) + (ψ * (t)) G d ẋ * (t) = -(ψ * (t)) (ν f I n + G d )f (x * (t), u * (t)) + µν L L(x * (t), u * (t))+ + (ψ * (t)) G d f (x * (t), u * (t)),
where the identities ( 16), [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF] are utilized on the last step. The presented theorem holds for both fixed and non-fixed horizon (see, e.g. [START_REF] Poznyak | Advanced Mathematical Tools for Automatic Control Engineers[END_REF]Theorem 22.13]). Evidently, the restriction x * (t) = 0 can be omitted in some cases, e.g. if f , L and g can be smoothly prolonged to the origin. This claim immediately follows from the property d) proven in Theorem 5 and the identity H(ϕ * (t), x * (t), u * (t)) = 0, t ∈ [0, T ] that holds for the case of non-fixed horizon (see, e.g. [START_REF] Poznyak | Advanced Mathematical Tools for Automatic Control Engineers[END_REF]Corollary 22.6]).

Conclusions

In this paper the problem of optimal control design for nonlinear plant is considered under assumption that the plant model is described by ordinary differential equation with a generalized homogeneous right-hand side. The conventional theorems of Dynamic Programming and Maximum Principle are refined under assumption cost function and the terminal constraints are also homogeneous in a generalized sense. There are several features of homogeneity, which can be useful for optimal control design.

• Homogeneity reduces the dimension of the control problem since optimal solutions for initial states x 0 belonging to the unit sphere uniquely define optimal solutions for all other initial conditions. This feature may essentially simplify the computational algorithms for optimal control design based on gradient methods/Maximum Principle.

• Homogeneity of the Bellman function implies interrelation between optimal solutions with different time horizon. This may allow to design optimal control for infinite time horizon (T = +∞) using an optimal control solution constructed for a finite time horizon (T < +∞). Such a property may be useful for model predictive control (MPC).

Two problems mentioned above are promising directions of future research of homogeneous control problems.

  s→-∞ d(s)u = 0 and lim s→+∞ d(s)u = +∞ uniformly on the unit sphere S = {u ∈ R n : u = 1}.

3 Preliminaries: Generalized homogeneity 3 . 1

 31 Monotone dilationsThe matrix G d ∈ R n×n defined as G d = lim s→0 d(s)-In s is known (see, e.g. [21, Ch. 1]) as the generator of the group d(s). It satisfies the following properties d ds d(s) = G d d(s) = d(s)G d and d(s) = e Gds := +∞ i=0 s i G i d i! . Denote also A A = inf x∈R n Ax

  homogeneous function and sup x∈S∩Ω |h(x)| < +∞ then a) for deg(h) > 0 one has h(x) → 0 as x → 0 and h is radially unbounded 1 provided that h(x) = 0 on S ∩ Ω; b) for deg(h) = 0 the function h is bounded in Ω and continuity of h at the origin implies that h(x) ≡ const; c) for deg(h) < 0 one has that h is discontinuous at the origin, unbounded in any neighborhood of the origin and |h(x)| → 0 as x → ∞.

  a d-homogeneous vector field and sup x∈S∩Ω f (x) < +∞. If the numbers α, β ∈ R + defined by[START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] for some P > 0 satisfying[START_REF] Coron | Adding an integrator for the stabilization problem[END_REF] are such that a) deg(f ) + β > 0 then f is continuous at the origin, f (x) → 0 as x → 0 and f is radially unbounded if f (x) = 0 on S ∩ Ω; b) deg(f ) + β = 0 (resp. deg(f ) + α = 0 ) then f is bounded on B d (r) ∩ Ω (resp. on Ω\B d (r)) for any fixed r > 0; c) deg(f ) + α = 0 then f is bounded on Ω\B d (r) for any fixed r > 0; d) deg(f ) + β = deg(f ) + α = 0 then f is globally bounded on R n ;e) deg(f ) + α < 0 then f is discontinuous at the origin, unbounded in any neighbourhood of the origin and f (x) → 0 as x → ∞;

  , Corollaries 22.4 and 22.5]). The only identity (27) must be proven. Since ∂H ∂ψ = f is d-homogeneous vector field of degree ν f then we just need to study the term ∂H ∂x = ∂f ∂x ψ -µ ∂L ∂x . The required homogeneous relation follows from the identities e ν L s ∂L(x,u) ∂z = ∂L(y,u) ∂y y=d(s)x d(s) and e ν f s d(s) ∂f (x,u) ∂z = ∂f (y,u) ∂y y=d(s)x d(s) proved in Corollary 1.

Corollary 7

 7 The property d) is the specific feature of the homogeneous systems. If ν f = ν L = ν then the right-hand side of the identity given by d) form the Hamiltonian multiplied by -ν, i.e d[(ψ * (t)) Gdx * (t)] dt = -νH(ψ * (t), x * (t), u * (t)). If the time horizon T is non-fixed and ν f = ν L then under conditions of Theorem 5 one has (φ * (t)) G d x * (t) = const for t ∈ [0, T ].

2 are d-homogeneous of degree 1 and 3, respectively.In the general case, homogeneous functions and vector fields can defined on some open unbounded subsets of R n , e.g.h(x 1 , x 2 ) = 1 [x1-x 2 2 ]+ , where x 1 , x 2 ∈ R, but [ρ] + = ρ if ρ > 0 and [ρ] + = 0 if ρ = 0.The domain of such homogeneous function forms a homogeneous cone (set).

A function h (resp. a vector field f ) is radially unbounded if x → ∞ implies h(x) → +∞ (resp. |f (x)| → +∞).

J(x s , u s , e -ν f s T ) = e (ν L -ν f )s J(x, u, T ), for any s ∈ R, where

x s (t) = d(s)x(e ν f s t), u s (t) = u(e ν f s t), t ∈ (0, e -ν f s T ).

The obtained relation completes the proof. Continuity of the vector-field f and the Lagrangian L on R n \{0} × R m imply that for any x 0 ∈ R n \{0} there exists T > 0 such that the admissible set P(x 0 , T ) is non-empty.

Let us introduce the function χ : R n \{0} → R + χ(x 0 ) = inf

that characterizes an admissible horizon of HOCP in the following way:

• if χ(x 0 ) > 0 then P(x 0 , T ) = ∅ for T ∈ (0, 1/χ(x 0 )) and P (x 0 , ε + 1/χ(x 0 )) = ∅ for any ε > 0;

• χ(x 0 ) = 0 implies P(x 0 , T ) = ∅ for any T > 0.

Corollary 4

The function χ is d-homogeneous of degree ν f and 1) if χ(x 0 ) = 0 for all x 0 ∈ S then χ ≡ 0, where S is a unit sphere in R n ;

2) if ν f > 0 then χ(x 0 ) → 0 as x 0 → 0;

5) if 0 ≤ ν f + β ≤ 2β -α and ν L ≥ ν f then χ ≡ 0 and the set P(x 0 , T ) is bounded for any x 0 ∈ R n and any T < +∞, where α, β ∈ R are defined by (12) for some P > 0 satisfying [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF].


Katharina Haude
email: khaude@uni-koeln.de

Relativization, nominalization, or none of the above? Verbs in referring expressions in Movima (isolate, Bolivia)

The problem

How can we do justice to the fact that in Movima, verbs can occur inside a DP ?

(1)

[kinos kwe :ya] ART.F.AB woman 'the woman'

(2)

[kinos jo 'yaj] ART.F.AB arrive 'the one (f) who arrived'

(1) = zero relativization: 'the female one who is a woman'?

(2) = zero nominalization: 'the female arriver/arriving one'? 2 Movima: overview Language:

-lowland Bolivia; isolate; ~500 adult/elderly speakers; endangered (no L1 learners)

Data:

-fieldwork 2001-2012 (15 months) -annotaded (transcribed and translated) oral corpus of ~26,000 intonation units/clauses (~130,000 words; 30h)

Typological characteristics:

-5 vowels, 20 consonants -syllable structure (C)V(C), (C)V:, (C)V -mainly suffixing, but also reduplication and prosody-based infixation -productive compounding and incorporation with classifier-like elements -weak noun-verb distinction: no TAM on verbs; no case, number, gender on nouns etc.

-neither case nor agreement -detailed reference system (articles, personal pronouns, demonstratives) -largely configurational (predicate initial) -two basic transitive constructions: direct (ergative) and inverse (accusative), governed by hierarchies of person, animay, topicality

Nouns and verbs

Weak noun-verb distinction:

-no case, number, gender on nouns -no tense, aspect, mood, or evidentiality on verbs But:

-verbs need DIRECT or INVERSE marking to receive an "internal enclitic" (see below); possessed nouns, marked by an "internal enclitic", do not contain any particular morphology -verbal and nominal predicates are marked differently when embedded:

Embedded verbal predicate:

(3)

[n-os iloni:-wa=Ø]

Transitive verbal predicates

In transitive clauses, both arguments follow the predicate. Transitive verbs are either marked as DIRECT: the first ("internal") argument after the predicate is the agent and the second ("external") the patient; or they are marked as INVERSE: the first ("internal") argument after the predicate is the patient and the second ("external") the agent.

Direct transitive clauses show ergative alignment: the patient argument has the same properties as the single argument of an intransitive clause (i.e. P=S). The agent argument, attached more closely to the verb ("internal cliticization"), is encoded like a possessor. The inverse is chosen when the patient outranks the agent in person (1>2>3), animacy (human>inanimate) or discourse topicality (topical>less topical).

Cf. possessor encoding, identical to "internal argument" encoding: [is majni =Ø]

(
ART.PL offspring=1SG 'my children'

Antipassive

The verb in an RC is detransitivized to relativize the agent (this is the case when the agent outranks the patient; the inverse, cf. (23), is not possible then). The detransitivation is usually only found with direct-marked verbs, i.e. it functions as an antipassive.

VALDECR kwey/kaw:

A  S P  oblique (26) RC with detransitivation:

a.

[kinos kwe:ya di' kaw vel-na n-isko] ART.F.AB woman REL VALDECR watch-DR OBL-PRO.PL.AB 'the woman who looks after them' b.

[us kaw vel-na n-is waki:ya]

ART.M VALDECR watch-DR OBL-ART.PL calf 'the one who looked after the calves'

Negation

The predicate inside an RC is negated with the particle loy. This is different from main clauses, which are negated with a negative copula, ka (and show systematic nominalization)

Or is it nominalization?

Inserting a verb inside a DP might also be considered a participant nominalization, deriving a noun that denotes S of an intransitive predicate, P of a direct-marked predicate, or A of an inverse-marked predicate. An argument favouring this analysis is that ALL nouns can occur as predicates of RCs, including possessed nouns (see ( 25)), whereas not all nouns can occur as predicates of all basic clause structures (see (20) above). Furthermore, the operations found in RCs also apply to DPs with nouns (productive, but rare):

Conclusion

To sum up:

-The predicate in a "verbal DP" is a / shares the properties of a relative clause -The predicate in a "verbal DP" is a / shares the properties of a noun  The placement of a verb inside a DP can be considered a (zero) relativization or (zero) nominalization.

However:

-The distinction between nouns and verbs is difficult to make; when are we dealing with a simple noun vs. with an RC/a nominalization? -Nouns can also function as main-clause predicates; the difficulty of possessed nouns (and also proper nouns) to occur in all main-clause structures could eventually be due to a functional restriction on specific predicate nominals -As a consequence of the analysis, nouns in main-clause predicate function would also have to be considered (zero) verbalized  Assuming a zero-marked operation, just because in more familiar languages some operation would be necessary to arrive at the constructions seen here, seems like an unnecessary complication of the facts.

A more attractive proposal: All classes of Movima content words (verbs, nouns) can be considered basically predicates. In clause-initial position they are main-clause predicates, in subordinate position (i.e. inserted in a DP or in a relative clause) they are syntactically subordinate predicates that characterize the referent of the preceding referential element/phrase (the head).

  grab-DR=ART.N.PST fox ART.N.PST tree 'The fox grabbed the tree.' Inverse transitive clause show accusative alignment: the agent argument has the same properties as the single argument of an intransitive clause (i.e. A=S). The patient argument, attached more closely to the verb ("internal cliticization"), is encoded like a possessor. (12) a. Transitive direct vel-na=us [is juyeni] watch-DR=3M.AB ART.PL person 'He watched (the) people.' b. Transitive inverse vel-kay-a=us [is juyeni] watch-INV-LV=3M.AB ART.PL person 'He was watched (by the) people.'

4 Main-clause structure and the expression of arguments 4.1 Intransitive verbal predicates

		OBL-ART.N.PST walk-NMZ=1SG
		'when I walked (lit.: at my [former] walking)'
	Embedded nominal predicate:
	(4)	[n-os		tolkos<ya:~>ya=Ø]
		OBL-ART.N.PST girl<NMZ.ST~>=1SG
		'when I was a girl (lit.: at my [former] [being a] girl)'
	Some affixes can only be attached to a specific type of lexical base, for instance:
		-DIRECT marking only on verbs
		-verbalizing suffix -tik 'make/do N' only on nouns
		-incorporation into verbs: only noun roots
	Adjectives share properties of both verbs and nouns. Unlike nouns, they cannot be marked as
	possessed, but like nouns, they undergo reduplication in subordination.
	Ways to express (or not) the single argument (S)
	(5)	jayna jo'yaj	[us	dichi:ye]
		DSC	arrive	ART.M	child
		'The boy arrived already.'
	(6)	jo'yaj--us	neyru
		arrive--3M.AB here
		'He arrived here.'
	(7)	jayna	jo'yaj
		DSC	arrive
		'Then (he) arrived.'

Nominal predicates Unpossessed nouns: S is expressed in the same way as with verbs.

	(14) -Headed RC are introduced with the particle di' [us alwaj-a=[isnos a:na=u]]
	ART.M spouse-LV=ART.F.PST younger_sibling=3M -The relativized argument is gapped
	'his younger sister's husband' -Any content word can form the predicate of a relative clause
	-Only the external argument can be relativized
	(21) 4.3 (15)	RC with intransitive verb a. [is rulrul di' ji<wa:~>wa] ART.PL jaguar REL come<MD~> 'the jaguars that came' itila:kwa [kis majniwa=sne] man ART.PL.AB b. [is ji<wa:~>wa] offspring_of=3F.AB ART.PL come<MD~> 'Her children are boys.' 'the (ones that) came'
	(16) (22)	tolkosya--'ne RC with transitive verb (direct) girl--3F a. [is majni=Ø di' vel-na=us] 'She (is a) girl.' ART.PL offspring=1SG REL watch-DR=3M.AB
	(17)	rulrul ___ jayna 'my children, who he looked after'
		jaguar b. [is	vel-na=us]
			ART.PL	watch-DR=3M.AB
			'the (ones) he looked after'
	(18) (23)	jayna RC with transitive verb (inverse) pekato=sne [os DSC sin=3F.AB a. [kis senyo:ra di' vel-kay-a=sne] jeya=sne] ART.N.PST state_of=3F.AB ART.PL.AB lady REL watch-INV-LV=3F.AB 'Her sin (was) her behavior.' 'the ladies who look after her'
	(19)	jeya=sne b. [kis state_of=3F.AB DSC jayna ART.PL.AB '(That's) how she is now (lit.: "[It is] her state now").' vel-kay-a=sne] watch-INV-LV=3F.AB 'the (ones who) look after her'
	(20) (24)	* pa:ko=us--k-as RC with unpossessed noun dog=3M.AB--OBV-3N.AB a. [os ya:ya=as (intended meaning: "It is his dog.") di' rulrul] ART.N.PST uncle=3N.AB REL jaguar
	(Here, a "cleft" construction with initial pronoun is used) 'his (the fox's) uncle, who (was a) jaguar'
		b.	[os	rulrul]
	ART.N.PST 5 The status of verbs in DPs 'the jaguar'	jaguar
	5.1 Relative clauses?
	(25) Bickel 2011: 428: "Relative constructions turn a propositional expression into a referential RC with possessed noun a. [is dichi:ye di' majni =Ø] one, for example, a clause like he read it into the one he read." ART.PL child REL offspring=1SG
	13) Citko 2004: "light-headed relative clause": RCs headed by "morphologically 'light jay<a>moɬ-a=us [isnos alwaj-a=us] 'my children (lit. "the children who [are] my offspring")'
	call<DR>-LV=3M.AB ART.F.PST spouse-LV=3M.AB elements'", e.g. demonstratives or wh-words  in Movima, a determiner (article) b.
		'He called his wife.'
	Properties of Movima relative clauses (RCs):
	-RCs follow their head

DSC 'It (was) already (a) jaguar.' (a transformed human) Possessed nouns as predicates (rare):

  ART.N.AB NEG.SUB MOD Movima-NMZ.ST 'the one/someone who is not (a) Movima.'

	"Antipassive" with a noun:
		VALDECR kwey/kaw:
		possessor  possessee
		possessee  oblique
	(29)	a. [us		a:na=is]
		younger_sibling=3PL.AB
		'their younger brother'
		b. [is		kwey	a:na	(n-usko)]
		ART.PL	VALDECR younger_sibling OBL-PRO.M.AB
		'(his) older siblings (or: the ones who have [him as] younger sibling)'
	Nominal negation:
	(30)	[kos	loy	rey mowimaj-ɬe]


François Lenglet
email: francois.lenglet@univ-savoie.fr

Influence of terroir products meaning on consumer's expectations and likings

Keywords:

Given the plethora of terroir products, local food companies could find an advantage in specifying their positioning by taking into account what the terroir represents for consumers. This research contributes to identifying specific attributes of a terroir in tune with consumer expectations, and tests their capacity to ameliorate consumer hedonic assessments and judgments for these products. Results show the poten-tial of a positioning based on the environment, which is respected by the practices of terroir producers, and which also guarantees the quality of the product.

Introduction

The market for ''terroir'' food products has grown remarkably over the past twenty years, notably in France and the rest of Europe, giving rise to numerous research studies (e.g. [START_REF] Aurier | Exploring terroir product meaning for the consumer[END_REF][START_REF] Bertozzi | Designation of origin: Quality and specification[END_REF][START_REF] Bowen | Geographical indication, terroir, and socioeconomic and ecological sustainability: The case of tequila[END_REF][START_REF] Elaydi | Cultivating terroir in subsistence markets: Development of terroir strategy through harmony-with-community framework[END_REF][START_REF] Guerrero | Consumer-driven definition of traditional food products and innovation in traditional foods. A qualitative cross-cultural study[END_REF][START_REF] Lengard Almli | General image and attribute perceptions of traditional food in six European countries[END_REF]. Marketing is exploiting this growth extensively; in France, for example, there are over 1000 registered trademarks which incorporate the term ''terroir'' (http://www.inpi.fr). Given this profusion in supply, the evocation of terroir alone soon will lose its capacity to add value to a product. There is thus a critical need to refine positioning strategies through a better understanding of potential differentiation criteria on this market.

After devoting much attention to conceptualising terroir products from the point of view of supply, research efforts now are focusing on demand concerns, in particular to propose consumer-driven definitions [START_REF] Aurier | Exploring terroir product meaning for the consumer[END_REF][START_REF] Guerrero | Consumer-driven definition of traditional food products and innovation in traditional foods. A qualitative cross-cultural study[END_REF][START_REF] Guerrero | Perception of traditional food products in six European regions using free word association[END_REF] and to identify the attributes of a product that contribute most to its overall image [START_REF] Lengard Almli | General image and attribute perceptions of traditional food in six European countries[END_REF].

However, these studies consider the terroir product as a global entity, while research findings show that important attributes differ markedly, not only according to consumers [START_REF] Lengard Almli | General image and attribute perceptions of traditional food in six European countries[END_REF], but also according to the products and terroirs [START_REF] Aurier | The effects of perceived congruity between origin, brand, and product on the purchase intention of a branded product of origin[END_REF]. This conforms with the PSO paradigm holding that all human behaviour results from interaction between characteristics of the Person, the Situation, and the Object of the behaviour [START_REF] Leigh | A review of situational influence paradigms and research[END_REF]. Likewise, the role of extrinsic attributes of a terroir product should be compared to the role of intrinsic attributes such as the appearance and taste of this product: a recent study re-confirmed the essential role of hedonic motivations in the formation of overall attitudes towards traditional food products, at least for French consumers [START_REF] Pieniak | Association between traditional food consumption and motives for food choice in six European countries[END_REF].

This article intends to explore the reasons for the popularity of terroir products, and to study the relative role of the following three types of factors that may explain the perceived value of these products:

-terroir product meanings for the consumer produced by research focussed on consumer-driven definitions; -extrinsic attributes, which are sources of positioning specific to a terroir product; -intrinsic attributes that are linked to the taste of a product.

We first describe these factors in detail, then propose a theoretical model of the formation of attitudes regarding a terroir product, and lastly test the model using a traditional cheese typical of Abondance valley, an area located in the Savoie region of the French Alps. The methodology followed and analyses of the results are presented. The discussion examines the theoretical and managerial implications of the results, and we close with proposals for new research avenues.

The terroir and the consumer

To pursue the objective of this research, we must first clarify the concept, ''terroir'', and understand consumer's expectations of terroir products.

The concept of terroir

Today, the term, 'terroir', which has no direct translation in English, is conceptualised in several different disciplines: terroir is referred to by historians, geographers, economists, sociologists, anthropologists, and managers. Furthermore, terroir food products are confused with closely related concepts such as product of origin (region or country) and local products, and definitions diverge depending on whether they are based on the supply point of view (sector actors) or on consumer expectations. Despite these differences, the majority of actors appear to agree that there are two general dimensions to ''terroir'': ''Terroir is linked to the unique biophysical properties of particular places. . .and is also associated with the cultural practices that have maintained these biological resources over several generations. . .The cultural concept of terroir is understood as the product of interacting natural and human factors'' [START_REF] Bowen | Geographical indication, terroir, and socioeconomic and ecological sustainability: The case of tequila[END_REF]. The specific attributes of a terroir are shaped by natural factors such as the soil, climate, and adaptation of plant varieties to the environment, and by human factors related to history, culture, know-how and tradition.

The definition of a terroir food product also may be elaborated by making a distinction with other closely related concepts. Literature on the effect of country or region of origin generally is evoked when conceptualising a terroir. Starting from the idea of distinguishing country and region of origin [START_REF] Askegaard | Product-country images: Towards a contextualized approach[END_REF], research has refocused on the region of origin and its effect, which is even more powerful when the territory is homogenous and small [START_REF] Stefani | Consumer expectations, liking and willingness to pay for specialty foods: Do sensory characteristics tell the whole story?[END_REF], and when there is a high level of congruence between the region and product [START_REF] Aurier | Exploring terroir product meaning for the consumer[END_REF][START_REF] Van Ittersum | The influence of the image of a product's region of origin on product evaluation[END_REF]. This two-dimensionality (natural and human factors) seem to be shared by products of origin and terroir products, but the product-terroir congruence is acquired in the eyes of consumers who perceive the specialisation and typicality of terroirs and their legitimacy to produce the product [START_REF] Trognon | Attitudes des consommateurs vis-à-vis des produits alimentaires régionaux -Le cas de l'Auvergne[END_REF].

Definitions of terroir products finally are very close to those of traditional food products, which implicitly suggest the same bi-dimensional structure: ''a product frequently consumed or associated with specific celebrations and/or seasons, normally transmitted from one generation to another, made accurately in a specific way according to the gastronomic heritage, with little or no processing/manipulation, distinguished and known because of its sensory properties and associated with a certain local area, region or country'' [START_REF] Guerrero | Consumer-driven definition of traditional food products and innovation in traditional foods. A qualitative cross-cultural study[END_REF].

The attributes related to natural and human factors form images of a terroir such as proposed in consumer-driven definitions, and consequently should influence the perceived value of a product.

Modelling terroir food product consumption behaviour

The attributes of a terroir food product are perceived through two types of stimuli:

-When buying or first encountering the product, the consumer is subjected to ''informational'' stimuli (e.g. price, brand, ingredients, label, nutritional and environmental claims, etc.; [START_REF] Steenkamp | Product quality[END_REF], which shape expectations. These also result from the consumer's a priori image of the terroir product. -The actual consumption of a product then confronts the consumer with sensory stimuli perceived by the five senses, e.g.

appearance, colour, size, shape, texture, odour, aroma, taste, temperature, which leads the consumer to form an assessment of the product.

The consumer's overall judgement finally will lean towards either the expectations or the assessment produced by tasting alone, depending on the influence of respective types of stimuli and the resulting phenomena of assimilation or contrast (see, for example, [START_REF] Schifferstein | Asymmetry in the disconfirmation of expectations for natural yogurt[END_REF], for a review). Individual or contextual factors outside the product also can influence expectations, the gustatory evaluation, and the overall judgement, and also moderate relations between expectations and assessments on one hand, and the overall judgement on the other. This classic stimuli-response approach in sensory marketing is diagrammed in Fig. 1.

This base model enables one to assess the role of the perceived sources of a terroir product. For economic actors in the terroir sector, informational stimuli represent a marketing action variable: it is possible to focus communication on the human factor, the natural factor, the origin, know-how, history, etc. attributes, depending on the positive impact obtained with regard to expectations and overall judgement. A relevant criteria of differentiation is identified when the stimulus carrying it leads to higher expectations (compared to another criteria), and when overall judgment is better compared to when assessment is based on tasting alone (positive deviation due to information). In contrast, consumer images of terroir impose on marketing strategies, as do sensorial stimuli, at least when these mainly result from a fixed process (label or specifications, for example).

The model then can be validated empirically by testing the role of potential differentiation criteria for a terroir food product, presented in the form of informational stimuli, on consumer expectations and judgement.

Materials and methods

An empirical study allowed the proposed model to be applied to Abondance cheese, a food that is emblematic of a Savoie terroir and which benefits from a protected designation of origin label (AOC: Appelation d'Origine Contrôlée). The informational stimuli tested were scenarios illustrating certain characteristics of the terroir corresponding to two factors (natural and human) identified in the literature. The experimental protocol followed rendered it possible to measure consumer images of terroir for Abondance cheese, submit these images to stimuli, and evaluate the resulting expectations, assessments, and overall judgements.

Material stimuli

Cheeses

The terroir product used in this study is a hard pressed cheese that is allowed to mature for at least 100 days on spruce boards.

The organoleptic characteristics of this cheese are fairly consistent from one producer to another. Two types of Abondance cheese are sold, farm and dairy; the type is determined by whether the milk used to make the cheese comes from a single farm or several farms. For the study, four cheeses of each type were selected by interprofessional organisations in the Abondance sector. One cheese served as a training product. The cheese selected has the typical sensorial profile of Abondance cheese. In conformance with current practices in experimental economics, the three other cheeses selected actually correspond to the terroir characteristics and were produced by the producers described in the informational stimuli.

Informational stimuli

The potential differentiation criteria for Abondance cheese were defined on the basis of a literature review and two focus groups involving 19 adult subjects, buyers and consumers of local cheeses: the goal was to elicit the dimensions and attributes of terroir products, in particular mountain cheese. The differentiation criteria used were formulated in the form of scenarios by a committee composed of cheese sector professionals and researchers specialised in marketing. Two attributes were selected for each factor (human and natural): the human factor was illustrated by the artisanal character, and more precisely the small size of the production unit, as well as a cooperative social organisation; the natural factor was represented by environmental preservation thanks to extensive production practices, and the quality of the environment as a guarantee of the quality of the cheese. Two scenarios corresponding to natural and human factors were the same for the dairy and farm cheese types: ''environmental preservation'' and ''small production/small producer''. Due to its inherent nature, the ''cooperative'' scenario was specific to dairy cheeses, and the ''environment serving quality'' was attributed to farm cheeses.

In order to test the possible superiority of these attributes in relation to other characteristics of supply that are less directly linked to the terroir, other scenarios also were formulated. These involved other potential product selection criteria, such as the reputation of the distributor (well known cheese-maker) or producer (locally known cheese-making business) or the presence of an unofficial mark or sign of quality (first prize in an agricultural contest, store brand with an image of terroir). The wording of each scenario is presented in Table 1. For dairy and farm cheeses, scenarios 1, 3, and 5 illustrated specific terroir attributes, and actually correspond to the cheeses used in the blind tests. All five scenarios were presented to collect expectations. Scenarios 1, 3, and 5 also were used to obtain the overall judgments (tasting with information).

The measurement of expectations resulting from scenarios took the form of a 10 point differential semantic scale ranging from ''I probably will not like at all'' to ''I probably will like very much'' in response to the question: ''For each of these Haute-Savoie AOC Abondance cheeses, and in the light of the information provided, please state how you expect you will like them''.

The experimental protocol

The protocol, carried out in a sensory assessment laboratory, incorporated three types of stimuli (sensory alone, information alone, combined) and began with blind tastings. The next step consisted of assessing the expectations induced by the information. The protocol concluded with the evaluation of the combined stimuli that allows a subject to associate the taste with the product. It avoided the contiguity of two evaluation measurements by interjecting a three-week period between sessions: the subjects participated in a first laboratory phase during which they evaluated blind the products proposed, and declared their expectations. They then returned for a second phase, which allowed their hedonic assessments to be collected with information on the criteria. The cheeses tested were exactly the same for both phases (blind tasting and informed tasting). During the interim period between the two phases, the cheeses were refrigerated to impede the continuation of the ripening process.

During the two phases, the same respondent only tasted the farm Abondance or the dairy Abondance cheese. The sample was constituted by the partner sensory evaluation laboratory. It included 325 Savoie inhabitants, cheese consumers who had consumed Abondance cheese at least once over the preceding 6 months. The average age was 45.6 years and the gender breakdown was balanced (45% men, 55% women). In this study, the total sample was divided into two groups of equal size (dairy cheeses, n = 160; farm cheeses, n = 165).

Concretely, the subjects participating in the experiment underwent the following protocol.

First test phase (session 1)

The protocol for the two types of products (farm and dairy) was exactly the same in terms of the order of operations.

-Blind testing and hedonic evaluation of four portions of Abondance cheese, in sequential monadic order: tasting a training product, then three products corresponding to three terroir attributes (marked a or b in Table 1; scenarios not communicated); the hedonic evaluations were measured with a 10 point semantic differential scale ranging from ''I do not like at all'' to ''I like very much''. -Presentation of a self-administered questionnaire to collect individual characteristics. -Presentation of a self-administered questionnaire to collect expectations in response to the five scenarios illustrating three terroir and two non terroir attributes (Table 1) regarding Abondance cheeses.

Second test phase (session 2, three weeks later)

-Informed testing and hedonic evaluation of four portions of Abondance cheese, in sequential monadic order: tasting a training product, then three products associated with three scenarios illustrating terroir attributes (marked a or b in T-tests comparing means for paired samples were performed on the five expectation scores (based on the scenarios) to determine whether the three scenarios illustrating characteristics specific to the terroir were better appreciated than other, more generic scenarios.

Capacity of specific terroir criteria to improve the overall judgement of a product in relation to blind tasting (hedonic deviation due to positive information)

To evaluate the relative influence of information specific to terroir and taste, the first step was to calculate differential scores. They compared the expectations and blind evaluations (disconfirmation), expectations and overall judgement (shift in response due to taste), the overall judgement and blind evaluations (shift in response due to information). One sample T-test then was carried out on the differential scores to check their significance.

Impact of terroir images on expectations and overall judgement with regard to a product

To represent the terroir images of the products, a Principal Component Analysis (PCA) was performed on the items of the scale proposed by [START_REF] Aurier | Exploring terroir product meaning for the consumer[END_REF]. Factor loadings were saved as new variables, and correlations and regressions were used to test the relationship between the terroir image of the product, expectations and overall judgment.

Results

Impact of criteria on expectations

For dairy cheeses, all of the expectations based on attributes of the terroir are significantly superior to other expectations (Fig. 2). The highest expectations regard scenario 5 (environmental preservation through agricultural practices) and scenario 1 (small production), followed by scenario 3 (cooperative) also based on the terroir. Scenario 2 (reputation) and scenario 4 (store brand with a terroir positioning) showed the weakest expectations. It thus appears that exposing consumers to precise information regarding the natural and human dimensions of a terroir (scenarios 1, 3, 5) significantly improve their expectations in comparison to more neutral information about the product (p 6 0.007).

In the case of farm cheeses, only expectation 3 (environment serving quality) differed significantly from the others (p 6 0.02). The other expectations were comparable at p = 0.05 (Fig. 3). Here again, it is a specific feature of the terroir that best valorises the product. Nonetheless, neither of the two other terroir attributes tested, whether they corresponded to the natural factor (environmental preservation thanks to extensive agricultural practices) or the human factor (small producer), proved to be superior compared to other more conventional attributes (reputation, awards).

In the end, four out of six terroir criteria permitted a priori expectations regarding the product to be better valorised. There was no evidence that non-specific terroir informational stimuli are more effective for either dairy or farm cheeses. This result suggests the potential utility in communicating certain specific terroir attributes rather than more generic, extrinsic product attributes (reputation of the producer or distributor, overall image of the terroir, quality sign). It was observed that beyond the category (dairy or farm cheese), the two most valued attributes were related to the natural factor.

Impact of the criteria on the hedonic deviations

With the exception of farm cheese F5 (scenario: environmental preservation thanks to agricultural practices), all of the cases correspond to a negative disconfirmation of expectations: (EÀB) is significantly different from zero and the cheeses are not as good as expected (Table 2). Furthermore, the hedonic deviation due to sensory processes, in other words, the difference between the informed hedonic evaluation (or overall judgement) and expectations (IÀE) is negative and significant (p < 0.001). Information about the terroir is not enough to maintain expectation levels at the moment of overall judgement.

However, it is interesting to observe the cases where the hedonic deviation due to information, meaning the difference between the informed hedonic evaluation and the blind hedonic evaluation (IÀB), is significant: this situation reflects the capacity of a scenario to influence consumer opinion and to improve the overall judgement in relation to the initial assessment. This was observed for the following scenarios:

-D3: ''cooperative'' in dairy cheese (p = 0.016). -D5: ''environmental preservation thanks to agricultural practices'' in dairy cheese (p < 0.001). -F3: ''environment serving product quality'' in farm cheese (p < 0.001).

Furthermore, two of these scenarios, specific to the natural dimension of the terroir, also corresponded to the highest expectations (Figs. 2 and3). Finally, three of the four attributes that generated higher expectation levels allowed a significant improvement in the consumer's overall judgement after tasting. Note further than in the case of D1 (small producer), the hedonic deviation due to information is close to the level of significance.

For these four scenarios (including D1), an assimilation effect exists, since (IÀB)/(EÀB) > 0. But this effect is not complete because the differences (IÀE) are significantly different from zero: when consumers are informed of the characteristics of the terroir, the taste evaluation remains important in the overall judgement. This result is consistent with previous research (e.g.: [START_REF] Siret | Traditional process: Influence on sensory properties and on consumer's expectation and liking application to 'pâté de campagne[END_REF].

Impact of product ''terroir'' images on expectations and overall judgement

Concerning the data sample on dairy cheeses, it was possible to conduct a factor analysis on the terroir image scale (KMO index = 0.798). The scree-test revealed a single dimension. The eigenvalue of the first component was 3.635 and accounted for 40.39% of the variance. The scale had a good reliability (Cronbach's alpha = 0.812). It was possible to verify the existence of links between the perceived image of terroir for Abondance cheese and consumer's expectations or overall judgement when submitted to information about the terroir (Table 3). For dairy Abondance cheeses, expectations were linked in a systematic manner to the image of terroir. The more powerful terroir brand equity was, the higher were consumer's expectations. Results were more nuanced for overall judgements. Additionally, one should note that correlations were not significant when expectations were based on scenarios that only mentioned terroir in a general manner without evoking either natural or human attributes (terroir store brand: p = 0.074; reputation of cheese-maker: p = 0.273). This result confirms the added value of a positioning strategy based on specific attributes rather than merely mentioning the terroir.

Concerning farm cheeses, factor analysis had a KMO index of 0.766. Using a Promax type of oblique rotation like the one used by the authors of the scale [START_REF] Aurier | Exploring terroir product meaning for the consumer[END_REF] the analysis revealed a bi-dimensional structure of the terroir construct: one factor corresponded to the natural dimension of a terroir while the second factor regrouped elements related to human dimensions (Table 4). Cronbach's alphas showed a good scale-reliability.

The relatively low correlation factors (r = 0.426) allowed multiple regressions to be performed to explain the expectations and overall judgments by the image of the terroir (Table 5).

For farm cheese evaluations based on a specific human attribute (small producer), terroir brand equity had no influence. In contrast, farm cheese expectations based on a natural attribute (environment) were linked in a quasi-systematic manner to the two dimensions of the image of terroir.

The contribution of the two dimensions of the image of the terroir seem to be comparable. The results of [START_REF] Aurier | Exploring terroir product meaning for the consumer[END_REF] showed an effect of only the natural factor on attitudes towards terroir products. However, this attitude was not measured after exposure to informational and/or sensory stimuli, but simply declared a priori on a like/dislike scale. The results are more nuanced for overall judgements. Additionally, one should note that regressions are not significant for expectations based on scenarios which simply evoke a terroir (expert cheese-maker, and awards). For farm cheeses, the perceived image of terroir positively influence expectations based on the highlighting of natural attributes of the product.

Discussion

This study empirically confirms the idea that the explicit statement of a terroir attribute may influence the assessment of a food product more efficiently than simply mentioning the region of origin [START_REF] Van Ittersum | The influence of the image of a product's region of origin on product evaluation[END_REF]. The observation of differentiated effects for natural and human factors also raises a number of questions. While this result is consistent with previous studies about the effect of country or region of origin [START_REF] Van Ittersum | The influence of the image of a product's region of origin on product evaluation[END_REF][START_REF] Verlegh | Country-of-origin effects on consumer product evaluations[END_REF], the attributes of the natural factor have a particular influence on attitudes towards food products while the attributes of the human factor are more important with regard to industrial products. Nevertheless, even for very similar foods, the specificity (dairy or farm cheese) seems to play an important role: for dairy cheeses, the human factor (scenarios D1 and/or D3) influences the level of expectations, the hedonic deviation, and the impact of the image of terroir on the expectations or the overall judgment. However, for farmer cheeses, no effect of the human factor was observed. One possible explanation is that consumers probably consider farm production as artisanal and small (nonindustrial). Thus, the ''Small Producer'' scenario (F1) does not provide additional information and therefore does not change the perception of the product.

The specific characteristics of a terroir food product may only be relevant differentiation criteria if they respond positively to consumer's motivations: for example, the characteristics related to the human factor can satisfy ideological motivations (e.g.: support the local economy, Lengard Almli et al., 2011) and those related to the natural factor can reflect a need for reassurance in relation to health concerns (e.g.: a correlation of 0.74 between health and naturalness motivations, [START_REF] Pieniak | Association between traditional food consumption and motives for food choice in six European countries[END_REF]. However, the ethical and environmental behaviours stem from various motivations and these behaviours may be motivated by self-interest rather than altruism: a study shows that consumer motivations for purchasing organic dairy products were first taste, food safety and health benefits, far ahead of environmental and ethical motivations [START_REF] Mceachern | Organic purchasing motivations and attitudes: Are they ethical[END_REF]. Thus, for terroir food products, environmental concerns may correspond more to health concerns than ecological concerns. This probably depends on the level of consumer's regional ethnocentrism. Moreover, consumers with a greater sense of regional ethnocentrism could be more sensitive to terroir characteristics related to the human factor because they feel proud when they purchase regional products and support regional businesses (Fernández-Ferrín & Bande-Vilela, 2013).

Conclusion

The main goal of this study was to verify if certain specific characteristics of a terroir could valorise a product originating from it, and therefore explicitly contribute to the positioning of the product. The results show that in the case of Abondance cheese, communicating the natural dimension of a terroir (attributes evoking the environment) achieves the most positive impact on productrelated expectations for both dairy and farm type cheeses. For the dairy type, expectations formed based on information regarding the human dimension of terroir (small producer and cooperative) take second and third position.

In addition, it appears that attributes evoking the environment significantly improve overall judgement compared to blind tasting alone. Taking the environment into account thus seems to be interesting to promote both a purchase (impact on expectations) and a re-purchase (improvement of the overall judgement after consumption). Information on human organisation (cooperative) has comparable effects for the dairy type Abondance cheese. Finally, the way by which the consumer perceives the terroir influences significantly his or her expectations, if, to a lesser extent, his or her overall judgment of the product. The more powerful the terroir brand equity is, the more the terroir attributes of a product are valued by the consumer. This is all the more significant when these attributes concern the environment, and this is in agreement with the suggestion of [START_REF] Pieniak | Association between traditional food consumption and motives for food choice in six European countries[END_REF] for whom, ''the natural character or image of traditional foods constitutes one of its major assets vis-à-vis consumers''.

These results suggest new opportunities for marketing terroir products:

-Advertising and packaging should emphasise a terroir's specific attributes, especially those corresponding to its natural dimension, rather than a general terroir image; -The effectiveness of such a position is strengthened when the terroir benefits from a powerful image: companies that manufacture terroir products should highlight not only the product, but also the terroir, in partnership with institutional stakeholders.

This research has some limitations linked to the validity of the scale of terroir images and to the stability of its dimensions. These points therefore should be investigated further. Likewise, future studies using other differentiation criteria and other terroir products should be envisioned.

Moving beyond the results and their implications, it would be useful to pursue research on how a terroir is conceived and perceived and the consequences from the consumer's point of view. New research appears necessary to better understand the links between food motivations, the terroir image of the product, and reactions to terroir-specific stimuli.

With a view of positioning and segmentation, the study of interactions between products, terroirs, and consumers should prove to be fruitful. In particular, the need to differentiate the marketing of terroir products according to the target (terroir residents versus non-residents, or tourists) merits new studies.

Fig. 1 .

 1 Fig. 1. Base model for the evaluation of a terroir food product.

Fig. 2 .

 2 Fig. 2. Dairy cheeses: mean consumer expectation scores according to the scenarios presented. Expectations covered by the same grey line are not significantly different according to the T-test.

Fig. 3 .

 3 Fig. 3. Farm cheeses: mean consumer expectation scores according to the scenarios presented. Expectations covered by the same grey line are not significantly different according to the T-test.

Table 1 ;

 1 scenarios communicated this time). A scenario also was communicated for the training product: ''this is an Abondance dairy/farm cheese made in the Haute-Savoie. It has a French AOC label''. The measurement of the hedonic evaluation was exactly the same as that used in Phase 1.-Presentation of a self-administered questionnaire to collect individual perceptions of the terroir image of an Abondance cheese. The measurement of terroir images corresponding to natural and human factors was adapted to the scale proposed by[START_REF] Aurier | Exploring terroir product meaning for the consumer[END_REF]: a set of 9 items (territory, region, land, mountain, history, ritual, know-how, recipe, tradition), with five-point Likert scales and introduced by the question: ''From your own point of view, what determines Abondance as a terroir product is...''.

	3.3. Statistical treatment

3.3.1. Capacity of specific terroir criteria to better valorise a consumer's expectations in relation to other criteria

Table 1

 1 Information stimuli presented in scenario form. This AOC Abondance cheese is made at a small cheese making facility in Chapelle d'Abondance by Joël Bouvier, an artisanal cheese maker. He transforms the milk of 9 farmers located around the commune 2. Reputation This AOC Abondance cheese is made in the workshops of the Pochat & Fils cheese making facility located near Evian on the heights above Lake Leman. It ripens for a period of 5 months 3. Cooperative a This AOC Abondance cheese is produced by the Gruffy Cooperative: a cooperative of 20 producers who have pooled their financial resources to make and sell their products. In this way, they share in an equitable manner the fruit of their work 4. Store brand This AOC Abondance cheese is sold under the label ''Our regions have talent'' (Leclerc stores). This AOC Abondance cheese is made by Sylvie and Dominique Grillet-Aubert, small producers established in Saint-Paul en Chablais. Every day, they transform the milk of their 20 cow herd into cheese 2. Expert cheese maker This AOC Abondance cheese is selected, ripened, and marketed by Pierre Gay, an expert cheese-maker established in Annecy. The cheese ripens in the centuries-old caves of one of the most beautiful cheese-making facilities in France 3. Environment b serving quality This AOC Abondance cheese is made by the Gallay farm, in Chévenoz, in Abondance valley. The dairy cows graze on prairies rich with mountain flowers. The altitude, absence of pollution and biodiversity make this Abondance cheese a very healthy product 4. Awards This AOC Abondance farm cheese, produced by GAEC le Géant in Châtel, won the first prize at the annual 2009 Abondance cheese contest organised in October during the Abondance farm fair 5. Environment b preserved through agricultural pratices This AOC Abondance farm cheese is produced by the GAEC le Mont Chauffé in Abondance on a mountain farm that, through its environmentally friendly practices (extensive grazing, limited fertilizer), contributes to the conservation of protected areas (Natura 2000 zone)

	Cheeses & Criteria	Scenarios (Abondance Dairy)
	1. Small producer a
		It has a fat content of 33% and
		it ripens for at least 3 months
	5. Environment b preserved through	This AOC Abondance cheese, made at Féternes cheese making facility, is produced through mountain farming, which,
	agricultural pratices	through its environmentally friendly practices (extensive grazing, limited fertilizer), preserves water quality in the Evian
		basin
	Scenarios (Abondance Farm)
	1. Small producer a

a Terroir attributes related to the human factor. b Terroir attributes related to the natural factor.

Table 2

 2 Shifts in judgements between blind tasting, expectations, and informed tasting.

	Cheeses	Shift in response due to:			Disconfirmation
		Information		Taste
		IÀB a	p	IÀE b	p	EÀB c	p
	D1: Small producer	.38	.074	À.87	.000	1.23	.000
	D3: Cooperative	.53	.016	À1.38	.000	1.86	.000
	D5: Environmental preservation	.74	.000	À2.46	.000	3.20	.000
	F1: Small producer	.21	.401	À2.72	.000	2.91	.000
	F3: Environment serving quality	.89	.000	À2.17	.000	3.02	.000
	F5: Environmental preservation	.20	.237	.09	.622	.15	.507
	Values in bold are significantly different from zero (p < 0.05).
	D i = dairy cheeses; F i = farm cheeses, with i = number of the scenario.

a IÀB = Informed tasting scores minus blind tasting scores. b IÀE = Informed tasting scores minus expected scores. c EÀB = Expected scores minus blind tasting scores.

Table 3

 3 Correlations between Terroir image and product assessments (dairy cheeses).

		Expectations			Overall judgment (informed tasting)
	Pearson correlation	D1: Small	D3:	D5: Environmental	D1: Small	D3:	D5: Environmental
		production	Cooperative	preservation	production	Cooperative	preservation
	Terroir Image of	.212 ⁄⁄	.195 ⁄⁄	.222 ⁄	.117 ns	.161 ⁄	.131 ns
	Abondance
	Correlations in bold are significant (p < 0.05).
	ns = not significant.

⁄ p < 0.05. ⁄⁄ p < 0.01.

Table 4

 4 The measurement of Abondance farm cheese terroir image (factor analysis).

		Component
		F1: Natural factor	F2: Human factor
	Region	0.896
	Territory	0.832
	Land	0.561
	Mountain	0.559
	History	0.503	0.369
	Recipe		0.780
	Ritual		0.699
	Know-how		0.692
	Tradition		0.647
	Cronbach's a	0.760	0.723

Table 5

 5 Impact of Terroir image on product assessments (farm cheeses).

			Standardised beta	R 2
	Dependant	Farm cheeses	F1 Natural	F2 Human
	variables		factor	factor
	Expected scores F1 Small producer	0.051 ns	0.138 ns	0.028
		F3 Environment	0.170 ⁄	0.185 ⁄	0.090
		serving quality
		F5 Environmental	0.156 a	0.199 ⁄	0.091
		preservation
	Informed	F1 Small producer	À0.085 ns	0.153 ns	0.020
	tasting	F3 Environment	0.278 ⁄	À0.078 ns	0.065
	scores	serving quality
		F5 Environmental	À0.014 ns	0.212 ⁄	0.043
		preservation
	Standardised beta and R 2 in bold are significant (p < 0.05).
	ns = not significant.

⁄ p < 0.05. a p = 0.067.