Julien Delprato

Arnaud Délias

Pierre Medrel

Denis Barataud

Michel Campovecchio

Guillaume Neveux

Audrey Martin

Philippe Bouysse

Jean-Michel Nebus

C Tolant

Pulsed gate bias control of GaN HEMTs to improve pulse-to-pulse stability in radar applications

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Pulsed gate bias control of GaN HEMTs to improve pulse-to-pulse stability in radar applications J. Delprato, A. Delias, P. Medrel, D. Barataud, M. Campovecchio ✉ , G. Neveux, A. Martin, P. Bouysse, J.M. Nebus, C. Tolant and P. Eudeline A significant improvement is demonstrated in the measured pulse-to-pulse stability of an S-band 6 W GaN high electron mobility transistor (HEMT) power amplifier by generating an appropriate pulse of the gate bias and thus a warm-up drain current just before each radiofrequency (RF) pulse of a periodic and coherent radar burst. The amplitude and the width of this gate bias pulse preceding each periodic RF pulse of the burst are experimentally varied to investigate the trade-off between the improvement of pulse-to-pulse stability and the total power-added efficiency. Finally, this technique of synchronised warm-up gate bias pulse demonstrated a 10 dB improvement of measured amplitude pulse-to-pulse stabilities to meet the critical stability requirement below -55 dB for the RF power amplifier.

Introduction: Pulse-to-pulse (P2P) stability is a critical figure of merit of radar transmitters, which quantifies the variations of envelope amplitude and phase over time between consecutive radio-frequency (RF) pulses of a radar burst [START_REF] Delprato | Measured and simulated impact of irregular radar pulse trains on the pulse-to-pulse stability of microwave power amplifiers[END_REF][START_REF] Cicolani | Pulse-to-pulse stability characteristics of robust design centered high performances/low cost T/R module[END_REF]. The P2P stability is a key requirement for the suppression of unwanted radar echoes, called clutter [START_REF] Delprato | Measured and simulated impact of irregular radar pulse trains on the pulse-to-pulse stability of microwave power amplifiers[END_REF]. For GaN high electron mobility transistor (HEMT) power amplifiers (PAs) of radar transmitters, it has already been observed that the P2P stability is mainly affected by low-frequency dispersive effects such as selfheating and traps [START_REF] Delprato | Measured and simulated impact of irregular radar pulse trains on the pulse-to-pulse stability of microwave power amplifiers[END_REF]. Indeed, typical radar sequences may integrate long silence durations between the successive bursts of RF pulses, which lead to exacerbating the thermal effects [START_REF] King | A comprehensive electrothermal GaN HEMT model including nonlinear thermal effects[END_REF] across the pulsed RF signals. Therefore, to enhance the P2P stability of a PA, this Letter investigates a pulsed bias control that consists in applying a short gate bias pulse before each RF pulse of the periodic burst to generate a warm-up drain current in the GaN HEMT. The idea is to impose a quasi-steady-state operation of the device during all the RF pulses. In this Letter, the trade-off between P2P stability and efficiency is experimentally characterised for different widths and amplitudes of the warm-up gate bias pulse.

Definition of P2P stability: Typical radar applications make use of irregular RF pulse trains. However, the P2P stability of these complex pulse trains can be analysed through a periodic test signal [START_REF] Delprato | Measured and simulated impact of irregular radar pulse trains on the pulse-to-pulse stability of microwave power amplifiers[END_REF] shown in Fig. 1a, which consists of N periodic RF pulses followed by a long silence duration T OFF (i.e. off-time). Each of the N periodic RF pulses has a pulse width PW RF and a pulse-repetition-period PRP RF , thus defining the on-time T ON of the periodic RF burst T BURST . The P2P stability is defined at each time t k along PW RF by the variations between the envelopes at t k of the N consecutive RF pulses of the burst. The sampling point t k corresponds to the same time position within each RF pulse. In the case of M successive bursts, the root mean square method is used to calculate the amplitude or phase P2P stability at t k , which is defined as

Stab RMS (t k ) = 10 × log 1 N -1 N -1 i=1 xi+1 (t k ) -xi (t k ) 2 (1) with xi (t k ) = 1 M M m=1 xim (t k ); 0 , t k , PW RF (2)
where x stands for the envelope amplitude or phase in the case of amplitude or phase P2P stability, respectively. The indices of xim refer to the ith pulse of the mth burst. For the sake of clarity, the stability levels reported in this Letter correspond to the worst-case values measured in the first 10 μs of RF pulses (0 < t k < 10 μs) [START_REF] Delprato | Measured and simulated impact of irregular radar pulse trains on the pulse-to-pulse stability of microwave power amplifiers[END_REF].

Stability measurement setup for pulsed gate bias control: With the aim to mitigate thermal and trapping effects that degrade the P2P stability after long silence duration [START_REF] Delprato | Measured and simulated impact of irregular radar pulse trains on the pulse-to-pulse stability of microwave power amplifiers[END_REF], a gate bias pulse is applied to warm up the GaN HEMT prior to each RF pulse. Fig. 1a shows the ideal (i.e. without instabilities) time-domain envelope of the RF burst with warm-up gate bias pulses. Eight periodic RF pulses (N = 8) of 50 μs pulse width PW RF and 250 μs pulse period PRP RF constitute the RF pulse train with a total duration T ON of 2 ms. The silence T OFF between each RF pulse train is 8 ms long. Fig. 1b shows the ideal envelope of the resulting drain bias current with its warm-up value I warm-up due to the gate bias pulse and its operation value I DS0 due to the RF level. The total power-added efficiency PAE TOT of the PA including the gate bias pulses can be expressed as a function of its initial PAE, as follows:

PAE TOT = P RFout -P RFin V DS0 I DS0 1 + bg = PAE (1 + bg) with g = PW Gate PW RF ; b = I warm-up I DS0 (3)
where γ is the pulse-width ratio and β is the drain-current ratio between the gate bias pulse (PW Gate , I warm-up ) and the RF pulse (PW RF , I DS0 ). A heterodyne measurement setup of time-domain envelopes was previously developed to characterise the P2P stability of a PA [START_REF] Delprato | Measured and simulated impact of irregular radar pulse trains on the pulse-to-pulse stability of microwave power amplifiers[END_REF] under constant bias conditions. As shown in Fig. 2, the coherent generation and capture of time-domain envelopes for periodic pulsed RF signals are performed by a pulsed RF generator with an internal IQ modulator and an RF heterodyne receiver, respectively. The signal coherence is ensured by a 10 MHz reference clock and the envelope trigger synchronises signal generation and data acquisition. To synthesise the pulsed gate bias proposed in this Letter, the initial setup was modified by connecting an arbitrary waveform generator (AWG) to the gate access of the PA under test. All P2P measurements reported here were performed on a 6 W GaN HEMT test board (CGH4006P-TB) [START_REF] Inc | CGH40006P: 6 W, RF Power GaN HEMT[END_REF] at an RF frequency of 3 GHz. The value of PW Gate was limited to 10 μs (γ = 0.2) because of the AWG capability while the minimum value of I warm-up was set to I DS0 (β = 1) below which the stability requirement of -55 dB was never achieved.

Figs. 3a andb show that the gate bias pulse enables achievement of the required level of amplitude P2P stability below -55 dB for different combinations of γ and β ratios. However, due to the impact of gate bias pulses on the total PAE, the best combination of PW Gate and I warm-up that meets the critical P2P stability requirement while also limiting the decrease of PAE corresponds to γ and β ratios of 0.2 and 1.7, respectively. The resulting stability performances are given in Table 1. To illustrate the relative impact of gate bias pulses along the RF burst, the measured time-domain envelopes of the first and eighth RF pulses with or without gate bias control are superimposed in Fig. 4.

First, it can be observed that the gate bias pulses limit the envelope variation along PW RF for both RF pulses, but much more for the first RF pulse. This leads to a more constant level of amplitude P2P stability along PW RF , but the maximum value of stability levels along PW RF remains the main issue to meet the P2P stability requirement.

However, the level of P2P stability at t k (1) is directly linked to the envelope variation between RF pulses at the same time t k of PW RF . In this regard, Fig. 4 shows that the most positive impact of the gate bias pulse on P2P stability levels is observed for the eighth RF pulse. Indeed, in the first 10 μs of PW RF which correspond to the worst case of stability levels [START_REF] Delprato | Measured and simulated impact of irregular radar pulse trains on the pulse-to-pulse stability of microwave power amplifiers[END_REF], the comparison between envelope amplitudes of the first RF pulse (with or without gate bias pulse) and the eighth RF pulse with the gate bias pulse shows that the stability level would have been equally improved without the gate bias pulse preceding the first RF pulses, and thus without its consumption. Accordingly, an adaptive control using non-systematic gate bias pulses can be implemented for actual radar signals to meet the required stability level while limiting the impact on PAE predicted by (3) in the worst case of systematic gate bias pulses.

Conclusion:

This Letter investigates the impact of a gate bias pulse which is applied before each RF pulse of a periodic radar burst to improve its P2P stability degraded by a long silence. All measurements were performed on a 6 W GaN HEMT test board at an RF carrier frequency of 3 GHz for a worst-case radar burst of 10 ms integrating a silence of 8 ms. This control technique of systematic gate bias pulses demonstrated a 10 dB improvement of amplitude P2P stabilities for this worst-case radar burst to meet the critical stability requirement below -55 dB without any degradation of phase stability. Therefore, depending on the required level of P2P stability, and on the timedomain characteristics of the radar signal, these results open the way to adaptive bias control techniques of an RF PA for improving their P2P stabilities.

Fig. 1

 1 Fig. 1 Ideal time-domain envelopes generated for P2P stability control a Periodic coherent burst made of silence T OFF that follows N periodic RF pulses (PW RF ), each one being preceded by its gate bias pulse (PW Gate ) b Ideal envelope of drain bias current during warm-up and RF pulses

Fig. 2 Fig. 3

 23 Fig. 2 Heterodyne measurement setup of time-domain envelopes and P2P stability with capability of pulsed gate bias control

Table 1 :Fig. 4

 14 Fig. 4 Normalised envelope amplitudes of first and eighth RF pulses along PW RF that were measured with and without gate bias pulses

©

  The Institution of Engineering and Technology 2015 Submitted: 27 March 2015 E-first: 10 June 2015 doi: 10.1049/el.2015.1052 One or more of the Figures in this Letter are available in colour online. J. Delprato, A. Delias, P. Medrel, D. Barataud, M. Campovecchio, G. Neveux, A. Martin, P. Bouysse and J.M. Nebus (XLIM, Université de Limoges/CNRS, 123 Ave. A. Thomas, Limoges 87060, France) ✉ E-mail: michel.campovecchio@xlim.fr C. Tolant and P. Eudeline (Thalès Air Systems, Ymare 76520, France)


Jean Auriol

Florent Di Meglio

Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time

We solve the problem of stabilizing two coupled linear hyperbolic PDEs using actuation at both boundary of the spatial domain in minimum time. We design a novel Fredholm transformation similarly to backstepping approaches. This yields an explicit full-state feedback law that achieves the theoretical lower bound for convergence time to zero.

I. INTRODUCTION

This article solves the problem of boundary stabilization of two coupled heterodirectional linear first-order hyperbolic Partial Differential Equations (PDEs) in minimum time with one PDE in each direction and with actuation applied on both boundaries.

First-order hyperbolic PDEs are predominant in modeling of traffic flow [START_REF] Amin | On stability of switched linear hyperbolic conservation laws with reflecting boundaries[END_REF], heat exchanger [START_REF] Xu | Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems[END_REF], open channel flow [START_REF] Coron | A lyapunov approach to control irrigation canals modeled by saintvenant equations[END_REF], [START_REF] De Halleux | Boundary feedback control in networks of open channels[END_REF] or multiphase flow [START_REF] Di | Dynamics and control of slugging in oil production[END_REF], [START_REF] Djordjevic | Boundary actuation structure of linearized two-phase flow[END_REF], [START_REF] Dudret | Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models[END_REF]. Research on controllability and stability of hyperbolic systems have first focused on explicit computation of the solution along the characteristic curves in the framework of the C1 norm [START_REF] James | The effect of boundary damping for the quasilinear wave equation[END_REF], [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF], [START_REF] Hu | Global smooth solutions of dissipative boundary-value problems for 1st order quasilinear hyperbolic systems[END_REF]. Later, Control Lyapunov Functions methods emerged, enabling the design of dissipative boundary conditions for nonlinear hyperbolic systems [START_REF] Coron | Control and nonlinearity[END_REF], [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]. In [START_REF] Coron | Local exponential hˆ2 stabilization of a 2\times2 quasilinear hyperbolic system using backstepping[END_REF] control laws for a system of two coupled nonlinear PDEs are derived, whereas in [START_REF] Castillo Buenaventura | Dynamic boundary stabilization of hyperbolic systems[END_REF], [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], [START_REF] Prieur | Iss-lyapunov functions for time-varying hyperbolic systems of balance laws[END_REF], [START_REF] Prieur | Robust boundary control of systems of conservation laws[END_REF], [START_REF] Valérie | Boundary control of open channels with numerical and experimental validations[END_REF] sufficient conditions for exponential stability are given for various classes of quasilinear first-order hyperbolic system. These conditions typically impose restrictions on the magnitude of the coupling coefficients.

More recently, the backstepping approach has enabled the design of stabilizing full-state feedback laws. These controllers are explicit, in the sense that they are expressed as a linear functional of the distributed state at each instant. The (distributed) gains can be computed offline.

Comparing results obtained via backstepping design with existence results for stabilizing controllers reveals a gap. In [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF], an extensive review of controllability results for linear hyperbolic systems is given, along with the theoretical lower bounds for convergence times. These bounds vary according, mainly, to the number and location of available actuators. Backstepping results have, until now, focused on single-boundary actuation, see e.g. [START_REF] Coron | Local exponential hˆ2 stabilization of a 2\times2 quasilinear hyperbolic system using backstepping[END_REF] for the case of two coupled PDEs, [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF] for an arbitrary number of PDEs or [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic pdes[END_REF] for a minimum-time result in the general (single boundary actuation) case.

When actuation is applied at both boundaries, the literature usually focuses on design dissipative boundary conditions to stabilize the system. This not guarantee stabilization in the minimum theoretical time, and is only possible for small coupling terms between PDEs, but can generally be achieved using static boundary output feedback, which is much less computationally intensive.

In this paper, we partially bridge the gap between the existence results of [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] and the explicit control design results. More precisely, the main contribution of this paper is a minimum-time stabilizing controller in the case of two heterodirectional hyperbolic PDEs with actuation at both boundaries. A proposed boundary feedback law ensures finite-time convergence of the two states to zero in minimum time. The minimum time defined [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] is the largest time between the two transport times in each direction.

Similarly to recent approaches [START_REF] Bribiesca | Backsteppingforwarding control and observation for hyperbolic pdes with fredholm integrals[END_REF], [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], using a Fredholm transformation, the system is mapped to a target system with desirable stability properties. This target system is a copy of the original dynamics from which the coupling terms are removed. The well-posedness of the Fredholm transformation is a consequence of a clever choice of the domain on which the kernels are defined. The proof of the invertibility of this transformation is non-trivial and uses an operator-approach inspired by the one developed in [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF].

The paper is organized as follow. In Section II we introduce the model equations and the notations. In Section III we present the stabilization result: the target system and its properties are presented in Section III-A. In Section III-B we derive the integral transformation and we present the domains on which the kernels are defined. Some arguments about the well-posedness of the kernels are given in Section III-C. Section IV contains the proof of the invertibility of the Fredholm transformation. In Section V we present the control feedback law and its properties. Finally in Section VI we give some simulation results.

II. PROBLEM DESCRIPTION

A. System under consideration

We consider the following 2-states linear hyperbolic system

u t (t, x) + λu x (t, x) = σ +-v(t, x) (1) v t (t, x) -µv x (t, x) = σ -+ u(t, x) (2)
evolving in {(t, x)| t > 0, x ∈ [0, 1]}, with the following linear boundary conditions

u(t, 0) = U (t), v(t, 1) = V (t) (3)
with constant coupling terms and constant speeds

0 < λ ≤ µ (4)
The initial conditions denoted u 0 and v 0 are assumed to belong to L 2 ([0, 1]).

Remark 1: The coupling terms are assumed constant here but the results of this paper can be adjusted for spatiallyvarying coupling terms.

Remark 2: System (1)-( 2) is equivalent to the following system

u t (t, x) + λu x (t, x) = σ ++ u(t, x) + σ +-v(t, x) (5) v t (t, x) -µv x (t, x) = σ -+ u(t, x) + σ --v(t, x) (6)
This can straightforwardly be proved using a variable change.

Remark 3: There is no loss of generality in assuming that (4) holds.

B. Well-posedness

To study the invertibility of the Fredholm transformation used for control design, it is necessary to introduce elementary concepts of operator theory. Thus, taking the scalar product of ( 1)-( 2) with a smooth test function Φ T = (φ 1 , φ 2 ) and integrating by parts leads to the following definition of a solution.

Definition 1: Consider system (1)-( 2) with initial conditions u 0 , v 0 ∈ L 2 and control laws U (t) and V (t). We say that u v is a (weak) solution if for every τ ≥ 0 and every

function Φ = (φ 1 , φ 2 ) T ∈ (C 1 ([0, τ ] × [0, 1])) 2 such that φ 1 (•, 1) = φ 1 (•, 0) = 0 we have 0 = τ 0 1 0 -(φ 1t (t, x) + λφ 1x (t, x)+ σ -+ φ 2 (t, x))u(t, x) -(φ 2t (t, x) -µφ 2x (t, x) + σ +-φ 1 (t, x))v(t, x)dxdt + 1 0 (u(τ, x)φ 1 (τ, x) -u(0, x)φ 1 (0, x) + v(τ, x)φ 2 (τ, x) -v(0, x)φ 2 (0, x))dx - τ 0 [λU (t)φ 1 (t, 0) + µV (t)φ 2 (t, 1)] dt (7)
We can consequently rewrite the system in the abstract form

d dt u v = A u v + B U V (8)
where the operators A and B can be identified through their adjoints. The operator A is thus defined by

A :D(A) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 u v -→ -λu x + σ +-v µv x + σ -+ u (9)
with

D(A) = {(u, v) ∈ (H 1 (0, 1)) 2 |u(0) = v(1) = 0} (10)
A is well defined and its adjoint A * is

A * :D(A * ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 u v -→ λu x + σ -+ v -µv x + σ +-u (11)
with

D(A * ) = {(u, v) ∈ (H 1 (0, 1)) 2 |u(1) = v(0) = 0} (12)
The operator B is defined by

< B U V , z 1 z 2 >= λU z 1 (0) + µV z 2 (1) (13)
Its adjoint is

B * z 1 z 2 = λz 1 (0) µz 2 (1) (14)

C. Control problem and previous results

The goal is to design feedback control inputs U (t) and V (t) such that the zero equilibrium is reached in minimum time t = t F , where

t F = max 1 µ , 1 λ = 1 λ (15)
This "minimum time" is the time needed for the slowest characteristic to travel the entire length of the spatial domain.

The existence of a control law reaching the null equilibrium in time t F is proved in [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] using a method of characteristics. To the best of our knowledge, no explicit feedback law has been designed to achieve this goal. Previous approaches yield

• exponential stability for small coupling terms when twosided static output feedback is used [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]. • finite-time stability in time 1 λ + 1 µ > 1 λ when one-sided backstepping design is used, i.e with one controlled boundary only. In the latter case, the system is mapped to a target system that has a cascade structure, which is natural for backstepping but does not enable stabilization in minimum time t F .

III. CONTROL DESIGN

The control design is based on a modified backstepping approach: using a specific transformation, we map the system (1)-( 3) to a target system with desirable properties of stability. However, unlike the classical backstepping approach where a Volterra transformation is used, we use a Fredholm transformation here.

A. Target system design

We map the system (1)-(3) to the following system

α t (t, x) + λα x (t, x) = Ω(x)β(t, x)h [ λ λ+µ ,1] (x) (16)
β t (t, x) -µβ x (t, x) = 0 (17)
with the following boundary conditions

α(t, 0) = 0 β(t, 1) = 0 (18)
h I (x) (I is an interval) is defined by

h I (x) = 1 if x ∈ I 0 else (19)
while Ω ∈ L ∞ (0, 1) is a function that will be defined later. This system is designed as a copy of the original dynamics, from which most of the coupling terms of (2) are removed. Lemma 1: The zero equilibrium of ( 16)-( 17) with boundary conditions [START_REF] John | Continuous dependence on data for solutions of partial differential equations with a prescribed bound[END_REF] and initial conditions (α 0 , β 0 ) ∈ L 2 ([0, 1]) is exponentially stable in the L 2 sense.

Proof: The proof, using a Lyapunov function, is quite classical and is omitted here. Besides, the following lemma assesses the finite-time convergence of the target system.

Lemma 2: The system ( 16)-( 17) reaches its zero equilibrium in finite-time t F = max{ 1 λ , 1 µ } = 1 λ . Proof: Using the same arguments than the ones presented in [17, Lemma 3.1] (i.e the characteristic method), we can easily prove that for t ≥ 1 λ+µ

β(t, x) = 0 if x ≥ λ λ + µ (20) α(t, x) = 0 if x ≤ λ λ + µ (21)
Consequently, for t ≥ 1 λ+µ , the system ( 16)-( 17) can be rewritten

α t (t, x) + λα x (t, x) = 0 (22) β t (t, x) -µβ x (t, x) = 0 (23)
with the additional conditions

α(t, λ λ + µ ) = 0 β(t, λ λ + µ ) = 0 (24)
Once again, using the method of characteristics, we can prove that ∀x ∈ [0, 1], α(t, x) = 0 for t ≥ λ λ+µ +

1-λ λ+µ λ = 1 λ
and that β(t, x) = 0 for t ≥ λ λ+µ + λ λ+µ µ = 1 λ . Therefore ( 16)-( 17) reaches its zero equilibrium in finite-time t F = 1 λ Using an operator framework, system ( 16)-( 17) rewrites as

d dt α β = A 0 α β (25)
The operator A 0 is defined by

A 0 : D(A 0 ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 α β -→ -λα x + Ωβh [ λ λ+µ ,1] µβ x (26)
with

D(A 0 ) = {(α, β) ∈ (H 1 (0, 1)) 2 |α(0) = β(1) = 0} (27)
A 0 is well defined and its adjoint

A * 0 is A * 0 : D(A * 0 ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 α β -→ λα x -µβ x + Ωαh [ λ λ+µ ,1] (28) with D(A * 0 ) = {(α, β) ∈ (H 1 (0, 1)) 2 |α(1) = β(0) = 0} (29)
B. Fredholm transformation 1) Definition of the transformation: Without any loss of generality we recall that λ ≤ µ. In order to map the original system (1)-(3) to the target system ( 16)-( 18), we use the following transformation

α(t, x) = u(t, x) + h [0, λ µ+λ ] (x) -µ λ x+1 x (K(x, ξ)u(t, ξ) + L(x, ξ)v(t, ξ))dξ + h ] λ µ+λ ,1] (x) x λ µ (1-x) (M (x, ξ)u(t, ξ) + N (x, ξ)v(t, ξ))dξ (30) β(t, x) = v(t, x) + h [0, λ µ+λ ] (x) λ µ (1-x) x ( K(x, ξ)u(t, ξ) + L(x, ξ)v(t, ξ))dξ + h ] λ µ+λ ,1] (x) x λ µ (1-x) ( M (x, ξ)u(t, ξ) + N (x, ξ)v(t, ξ))dξ (31)
where, for any interval I, h I (x) is defined by

h I (x) = 1 if x ∈ I 0 else (32)
We define the following triangular domains, depicted in Figure 1 and Figure 2: The kernels K, L are defined on T 0 , M, N are defined on T1 . The kernels K, L are defined on T0 and M , N are defined on T1 . They are continuous in their domains of assumed definition. They all have yet to be defined. Remark 4: One may think that due to the use of the hfunctions, the transformation presents a discontinuity in x = λ µ+λ . Nevertheless, one can check that the right and left limits are equal since the integral vanishes and that consequently we do not have any discontinuity.

Remark 5: This transformation is a Fredholm transformation and can be rewritten using integrals between 0 and 1 as follows α(t, x) = u(t, x) -

T 0 =

 0 {(x, ξ)| x ∈ [0, λ λ + µ ], x ≤ ξ ≤ µ λ (1 -x)} (33) T0 = {(x, ξ)| x ∈ [0, λ λ + µ ], x ≤ ξ < λ µ (1 -x)} (34) T1 = {(x, ξ)| x ∈] λ λ + µ , 1], λ µ (1 -x) < ξ ≤ x} (35)

1 0Q

 1 11 (x, ξ)u(t, ξ) + Q 12 (x, ξ)v(t, ξ)dξ

1 0Q

 1 (36)β(t, x) = v(t, x) -21 (x, ξ)u(t, ξ) + Q 22 (x, ξ)v(t, ξ)dξ (37)

Jean Auriol is with MINES ParisTech, PSL Research University, CAS -Centre automatique et systèmes, 60 bd St Michel 75006 Paris, France.jean.auriol@mines-paristech.fr

Florent Di Meglio is with MINES ParisTech, PSL Research University, CAS -Centre automatique et systèmes, 60 bd St Michel 75006 Paris, France.florent.di meglio@mines-paristech.fr

with

) Remark 6: Since α(0) = β(1) = 0 the two control laws U and V can be computed as functions of (u, v).

2) Kernel equations: We now differentiate the Fredholm transformation (30)-(31) with respect to time and space to compute the equations satisfied by the kernels. We start with the β-transformation (31) if x ≥ λ µ+λ : Differentiating (31) with respect to space and using the Leibniz rule yields

Differentiating (31) with respect to time, using (1), (2) and integrating by parts yields

Plugging these expressions into the target system ( 16)-( 17) yields the following system of kernel equations

The corresponding domains, characteristic lines and boundary conditions in Figure 1 We now focus on the alpha- if x ≤ λ µ+λ : As above, differentiating (30) with respect to space and time and then plugging into the target system ( 16)-(17) yields the following system of kernel equations

if x > λ µ+λ : Similarly we get

In order to have a well-posed system, we add the following artificial boundary condition

The function Ω(x) is defined by

The corresponding domains, characteristic lines and boundary conditions in Figure 2.

. Representation of the alpha-kernels

Remark 7: The artificial boundary condition we add for the kernel N is not a degree of freedom since it has no impact on the control law and on the stability of the target system.

C. Well-posedness of the kernel equations

Theorem 1: Consider systems (44)-( 47), ( 48)-( 51), ( 52)-( 55), ( 56)-(60). There exists a unique solution K, L (defined on L ∞ (T )), M, N (defined on L ∞ ( T1 )), K, L (defined on L ∞ ( T0 )), M , N (defined on L ∞ ( T1 )). Classically (see [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF], [START_REF] John | Continuous dependence on data for solutions of partial differential equations with a prescribed bound[END_REF] and [START_REF] Beresford | Linear and nonlinear waves[END_REF]) the proof of this theorem consists in transforming the kernel equations into integral equations using the method of the characteristics. These integral equations are then solved using the method of successive approximations. We start with the systems (44)-( 47), ( 48)-( 51),( 52)-( 55) and finish with the system (56)-(60) since for this last one we need to use the fact that M (x, ξ) and N (x, ξ) are bounded.

IV. INVERTIBILITY OF THE FREDHOLM

TRANSFORMATION Unlike the Volterra transformation, the Fredholm transformation is not always invertible. In [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], the authors prove the invertibility of such a transformation in the case of a first-order integro-differential hyperbolic equation. In this section we use similar arguments (in particular we rely on the Fredholm alternative) to prove the invertibility of our transformation.

1) Operator formulation of the Fredholm transformation and properties: In this subsection we rewrite the previous Fredholm transformation using operators. This will lead to some relations verified by the adjoint operators. The Fredholm transformation (36)-(37) can be written as an operator P acting on u v . More precisely we have

where Q : (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 is the integral operator defined by

Its adjoint is:

One can easily check that:

The control U V can also be rewritten using operators

with

Using ( 25) and (63) yields

Moreover using ( 8) and (63) we get

Consequently P and Γ satisfy the following relation:

Taking the adjoints, this is equivalent to

2) The Fredholm alternative: We give first the following useful lemmas:

Lemma 3: ker P * ⊂ D(A * 0 ) = D(A * ) Proof: Let us consider z ∈ ker P * . Consequently we have P * z = 0. We can rewrite it

If we evaluate the first line for x = 1 and the second one for x = 0, using the fact that

Consequently z ∈ D(A * 0 ) and we can write

Lemma 4: ker P * ⊂ ker B * Proof: Let us consider z ∈ ker P * . Consequently we have P * z = 0. We can rewrite it

If we evaluate the first line for x = 0 and the second one for x = 1, using the fact that

with the boundary conditions

Consequently we have z = 0 0 We can now state the following theorem Theorem 2: The map

Proof: Since Q * is a compact operator we can use the Fredholm alternative (e.g [START_REF] Brezis | Functional analysis, sobolev spaces and partial differential equations[END_REF]): Id 2 -Q * is either noninjective or surjective. Consequently it suffices to prove that P * is injective. In addition, the Fredholm alternative also gives [START_REF] Brezis | Functional analysis, sobolev spaces and partial differential equations[END_REF] dim ker(Id

By contradiction we assume that ker P * = {0}. We first prove that ker P * is stable by A * 0 . We have ker P * ⊂ D(A * 0 ). Let then consider z ∈ ker P * . Using (72) we can obtain

We thus have A * 0 z ∈ ker P * . Consequently the restriction A * 0 |kerP * of A * 0 to ker P * is a linear operator from ker P * to ker P * . Since the dimension of ker P * is finite we can find at least one eigenvalue ν. Let e ∈ ker P * be a corresponding eigenvector (by definition e = 0). We have e ∈ ker P * and so e ∈ ker B * . Moreover we have A * 0 e = νe. Consequently

which contradicts Lemma 5 and concludes the proof.

V. CONTROL LAW AND MAIN RESULTS

We now state the main stabilization result as follows: Theorem 3: System (1)-( 2) with the following feedback control laws

where K, L and M , N are defined by ( 52)-( 55) and ( 44)-(47), reaches its zero equilibrium in finite time t F , where t F is given by [START_REF] Dudret | Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models[END_REF]. The zero equilibrium is exponentially stable in the L 2 -sense.

Proof: Notice that evaluating (30) at x = 0 yields (84) and evaluating (31) at x = 1 yields (85). Since the kernels are invertible, there exists a unique operator S such that

Applying Lemma 2 implies that (α, β) go to zero in finite time t F , therefore (u, v) converge to zero in finite time t F

VI. SIMULATION RESULTS

In this section we illustrate our results with simulations on a toy problem. The numerical values of the parameters are as follow. λ = 0.5, µ = 1, σ +-= 0.5, σ -+ = 1 (87) Using the backstepping approach we have presented a stabilizating boundary feedback law for a system of firstorder hyperbolic linear PDEs controlled in both boundary. The zero equilibrium of the system is reached in minimum time t F which is the largest time between the two transport times in each direction. This result is a first step towards completely bridging the gap between the theoretical results of [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] and explicit control design. By combining the presented approach with the result of [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic pdes[END_REF], we believe it is possible to design a minimum-time stabilizing controller for general heterodirectional hyperbolic systems. The dual observer problem, crucial to envision application of this method on an industrial problem in a potential observer-controller structure, will also be the topic of future contributions.


P Sizun

M Cassé

S Schanne

B Cordier

CONSTRAINTS ON THE INJECTION ENERGY OF POSITRONS IN THE GALACTIC CENTRE REGION

Recent observations of the 511 keV positron-electron annihilation line in the Galactic centre region by the INTE-GRAL/SPI spectrometer have stirred up new speculations about the origin of the large corresponding positron injection rate. Beyond astrophysical candidates, new mechanisms have been put forward. We focus on the annihilation of light dark matter particles and review the various gamma-ray radiation components related to such a source of mono-energetic positrons in addition to the 511 keV line itself. We study the influence of the degree of ionisation of the bulge on this radiation, and its possible effects on the observational constraints on the mass of the hypothetical light dark matter particle or the injection energy of a mono-energetic source of positrons in general.

INTRODUCTION

Four years of observations with the INTEGRAL/SPI spectrometer have permitted to precisely ascertain the flux -(1.07 ± 0.03) × 10 -3 ph cm -2 s -1 [1]-and the morphology -∼ 8 ˚FWHM [2]-of the 511 keV gamma-ray line in the Galactic centre region discovered in 1978 by Leventhal et al. [3], caused by the direct annihilation of positrons with electrons and the decay of para-positronium. Studies of the associated three gamma continuum from ortho-positronium decay revealed a similar morphology [4] and lead to a positronium fraction of f Ps = 0.967 ± 0.022 [1]. These results motivated new reflections on the origin of the positrons responsible for this emission. Because astrophysical candidates fell short explaining the ∼ 1.4 × 10 43 e + /s positron injection rate deduced from the 511 keV line flux, non nuclear candidates were considered, among which dark matter. However, it was found that the gamma-ray radiation that would be produced if "heavy" dark matter particles such as the neutralino (> 50 GeV/c 2 ) were the source of the positrons injected in the bulge, both immediately during the annihilation of dark matter particles and subsequently during the thermalization of the positrons, would be too high compared to CGRO/EGRET measurements. This is During the thermalization phase (ii), these positrons and electrons lose energy through their interactions with the ISM and Bremsstrahlung radiation is emitted; a fraction of the positrons annihilate in flight into two photons of variable energy. However, most of the positrons annihilate at rest (iii), either directly into a pair of 511 keV photons, or through the formation and decay of positronium. why an alternative possibility, i.e. the self-annihilation of light dark matter (hereafter LDM) particles, with a mass m χ below ∼ 200 MeV/c 2 , was proposed [5,6]. However, even such a scenario implies, besides the 511 keV line and the ortho-positronium continuum, additional gamma-ray radiation, as illustrated in Fig. 1. It includes Internal Bremsstrahlung radiation during the annihilation of LDM particles [7], radiation from Bremsstrahlung interactions of the leptons with the interstellar medium and radiation from the in-flight annihilation of a small fraction of the positrons [8]. Confronting this radiation with observations enables us to set constraints on the mass of the hypothetical LDM particle. We developed [9, and references therein] a simple model of this radiation and derived upper masses based on CGRO/COMPTEL data. We summarize here the results, focusing on the influence on our constraints of the ionization fraction of the medium in which positrons thermalize.

MODEL

To model the gamma-ray radiation components associated to LDM annihilation, the first step consists in computing the steady-state spectrum of the positrons -and electrons-which are injected at a unique energy. It involves taking into account energy losses and in-flight annihilation through interactions with the inter-stellar medium.

Positron and electron spectra

Q e ± inj (E) = R inj δ(E -E inj ). (1)
We do not consider the morphology of the source, i.e. the spatial distribution of dark matter particles, or the spatial propagation of these leptons within the bulge; we simply assume they are confined to the bulge. As they travel through the interstellar medium, they loose energy to the ambient medium and positrons can undergo in-flight annihilation.

The spectral distribution N e ± (E) of these leptons is obtained by solving the steady-state diffusion-loss equation

d dE -N e ± dE dt = -Q e ± (E) , (2)
where the key ingredients are the energy loss rate dE dt and the source term Q e ± . For electrons, the latter simply consists of the monoenergetic injection term, which yields the following distribution:

N e -(E) = Rinj -dE/dt (mc 2 < E < E inj ). (3)
For positrons, the source term also includes the in-flight annihilation (i.e. before full thermalization) of a fraction (Fig. 2) of the positrons injected:

Q e + (E) = Q e + inj (E) -n e σ IA (E)v(E)N e + (E) , (4)
where σ IA is the cross-section associated to in-flight annihilation. It results in a somewhat lower (Fig. 3) steady state spectral distribution:

N e + (E) = R inj -dE/dt e - E inj E ne σ IA (E ′ )v(E ′ ) -dE/dt(E ′ ) dE ′ . (5)

Radiation spectra

The total rate of positron annihilation at rest depends on the initial positron injection rate R inj and the in-flight annihilation fraction R IA /R inj given by our model. The respective fluxes of the 511 keV 2γ line and the 3γ orthopositronium continuum are then determined using the positronium fraction f P s = 0.967 ± 0.022 [1]. As we change the mass of the LDM particle and other parameters of the model such as the ionization fraction x i of the ISM, we tune the injection rate R inj so that the 511 keV line flux remains equal to the observed value. The other radiation components result from internal Bremsstrahlung during the LDM annihilation process, from Bremsstrahlung interactions of both positrons and electrons with the ISM during their thermalization and from the in-flight annihilation of some positrons into two photons. They are computed using the steady state spectra above and the appropriate cross-sections. More details are given in [9]. Fig. 4 displays the total and individual LDM radiation spectra for an arbitrary injection energy. Above the 511 keV line, in-flight annihilation dominates; Bremsstrahlung emission is negligible at high energies but, around a few tens of keV, it is of the same order as internal Bremsstrahlung

Influence of ionization

The ionization of the medium through which positrons propagate during their thermalization phase influences the in-flight annihilation and Bremsstrahlung radiation

components. As 51 +3

-2 % of the rest annihilation takes place in the warm ionized phase of the ISM [1], we considered two cases for the ionization fraction x i = n HII /n H of the propagation region: a completely neutral medium and a medium made of 51% of ionized phase. Energy losses occur mainly through Coulomb and Bremsstrahlung interactions and Coulomb cross-sections are greater in the ionized phase than in a completely neutral environment. Consequently, the lepton spectra bump at lower energies (Fig. 3) and a smaller fraction of positrons annihilate in flight (Fig. 2). In order to reach the same 511 keV line flux, a smaller injection rate is required which results in less Bremsstrahlung and in-flight annihilation radiation (Fig. 45).

OBSERVATIONAL CONSTRAINTS

The CGRO/COMPTEL measurements of diffuse emission in the 1-30 MeV energy range [10] provide the most stringent constraints, even though they shall be assigned systematic uncertainties of ∼ 30%. Assuming the morphology of LDM gamma-ray continuum emission is similar to that of the 511 keV line, i.e. is compatible with an 8 ˚FWHM gaussian, we consider a circular region of 5 or 8 ˚of diameter around the Galactic center and compute the flux expected from such a region (Fig. 6) for various test injection energies. As no excess is observed in the COMPTEL skymaps in the Galactic bulge region with respect to the surrounding inner disk, we set our constraints by superposing our calculation to the mean COMPTEL flux and requiring that the total diffuse flux remains within data uncertainties (Fig. 7). Resulting 2σ upper limits on the positron injection energy are given in Tab. 1, both for the specific case of light dark matter and for the more general case of a source of mono-energetic positrons, i.e. without the internal Bremsstrahlung component. As expected, the larger ratio of positrons surviving the thermalization phase within a partially ionized medium leads to slightly less constraining limits.

CONCLUSION

By combining in-flight annihilation and internal Bremsstrahlung radiation, we conclude on a maximum mass of 3-4 MeV/c 2 for the hypothetical LDM particle propagating within a neutral medium, in agreement with [8]. Taking into account a possible ionization of the propagation medium enables us to release this constraint by a factor two. A scenario without internal Bremsstrahlung could even allow a source of 10 MeV mono-energetic positrons. A refined appraisal of the flux, morphology and spectral shape of diffuse emission in the 20 keV-20 MeV range, with an improved resolution and removal of point sources will be useful to fine-tune these upper limits. However, the mass range still allowed for the hypothetical light dark matter particle remains narrow.

2 Figure 1 .

 21 Figure 1. Diagram illustrating the components of the gamma-ray radiation related to light dark matter. During the annihilation phase (i) of dark matter particles χ, energetic positron-electron pairs are produced, and the Internal Bremsstrahlung radiation component is emitted.During the thermalization phase (ii), these positrons and electrons lose energy through their interactions with the ISM and Bremsstrahlung radiation is emitted; a fraction of the positrons annihilate in flight into two photons of variable energy. However, most of the positrons annihilate at rest (iii), either directly into a pair of 511 keV photons, or through the formation and decay of positronium.

Figure 2 .Figure 3 .

 23 Figure 2. Fraction of positrons annihilating in flight as a function of the injection energy. The cases of a completely neutral propagation medium (black thick line) and of a medium with 51% ionized phase (red thin line) are shown.We start from a source of mono-energetic positrons and electrons, of total energy E inj = m χ /c 2 injected in the

Figure 4 .

 4 Figure 4. Radiation spectra related to 10 MeV /c 2 light dark matter particles, propagating in a neutral medium (black thick lines) or in 51% of ionized phase (red thin lines). Components: positron annihilation spectra (in-flight: IA; 2γ at rest; 3γ at rest), positron energy loss spectra (internal Bremsstrahlung: IB; external Bremsstrahlung: B e + ), electron Bremsstrahlung spectrum (B e -) and total spectrum (plain).

Figure 5 .

 5 Figure 5. Total dark matter radiation spectrum for increasing injection energies E inj (1, 3, 5, 10, 50 and 100 MeV), for a partially ionized (red dotted curves) or completely neutral (black solid curves) propagation medium.

Figure 6 .

 6 Figure 6. Longitude profile of the 3-10 MeV COMPTEL skymap, in a latitude band of ±5 ˚(histogram). The mean COMPTEL intensity in the inner disk region is used as a baseline and a possible excess in the Galactic bulge from LDM radiation is required to remain within data uncertainties.

Figure 7 .

 7 Figure 7. Total integrated flux in the 1-3, 3-10 and 10-30 MeV COMPTEL energy bands for an 8 ˚diameter region around the Galactic centre. Horizontal lines show the flux -solid lines-derived from the mean COMP-TEL intensity in a wider fraction of the disk (|l| < 30 • , |b| < 5 • ) and its 1σ -dashed-and 2σ -dotted-upper values. The sum of the COMPTEL flux and of the LDM flux is shown as a function of the positron injection energy E inj , for a completely neutral propagation medium -circled solid lines-and for 51% ionized mediumcrossed dot-dashed lines.

Table 1 .

 1 Constraints from COMPTEL data on the maximum positron injection energy E inj [MeV]. 2σ upper limits are given for an integration region of 5 or 8 ˚of diameter, for a neutral or 51% ionized medium, with and without the Internal Bremsstahlung radiation component.


IREGE Elodie Gardet

INSEEC Pierre Laurent Felix

Methods for distributing results in exploratory partnerships

Keywords: Exploratory partnership, distribution of results, innovation

Development partnerships enable companies to develop new products and/or processes while sharing costs. In addition to uncertainty and increased risk, these partnerships necessitate adapted management techniques. In particular, partners must determine the distribution of results and methods of distribution. This article describes different results in development partnerships according to their methods of distribution. A qualitative study of three exploratory projects, developed in the MINATEC IDEAs Laboratory®, offers a better understanding of the chosen methods and influential factors. Before the collaboration, partners preferred egalitarian divisions, because it was difficult to identify the resources and skills necessary in the project. However, this method of distribution provided less incentive than an equitable distribution. The structural dimensions (number of partners, degree of partnership maturity, similarity with core professions) and partners' behaviours (level of implication, interest in the project) also influenced the method of distribution implemented.

In organisational relationships, partners often cannot accurately forecast the risks inherent to alliances, particularly in exploratory partnerships that are, by their very nature, uncertain [START_REF] Gillier | Managing innovation fields in a crossindustry exploratory partnership with C-K Design Theory[END_REF]. Prior to the project start, the partners cannot foresee the likely risks, difficulties and outcomes. In turn, it is difficult to define ex ante how to distribute the results [START_REF] Jap | Pie sharing in complex collaboration contexts[END_REF], though this distribution decision is essential for encouraging partners to collaborate. The rules for sharing responsibilities, resources and quasi-rents must be defined to encourage members to transfer key knowledge to the project and maximise chances of success. This incentive requires a prior definition of the general rules for sharing output, which further ensures the protection of each party's contributions. Yet research rarely details these rules regarding sharing, or else reviews them in only summary fashion [START_REF] Bhaskaran | Effort, revenue and cost sharing in collaborative new product development[END_REF].

Prior research suggests two main distributions though: egalitarian (i.e., each partner obtains an equivalent result) or equitable (i.e., the distribution is proportional to the resources and skills contributed by each partner). To choose between them, partners must identify the influential dimensions; accordingly, this article aims to analyse the situations in which it is preferable to use egalitarian versus equitable distribution. We begin with a summary of prior literature on the specificities of exploratory partnerships, the types of results that can be produced and possible methods for distributing these outputs. Then we present three exploratory innovation projects, developed in partnerships in the MINATEC IDEAs Laboratory, and outline the data we collected to analyse them. The results address the methods of distribution adopted by the partners in these projects, as well as the dimensions that influenced these choices.

From exploration to distribution methods

An exploratory partnership is a particular form of R&D collaboration, which possesses specific significance for the study of result distributions, as well as unique incentives.

Specificities of an exploration partnership

In the past 15 years, various new forms of collaboration between companies have arisen [START_REF] Doz | Alliance Advantage: The Art of Creating Value Through Partnering[END_REF], as evidenced by the number of hybrid forms (e.g. alliances, consortiums, joint ventures, agreements). For this study, we define collaboration as ‗a situation in which two organisations or more share resources and activities in order to follow a strategy' (Johnson et al., 2008, p. 461), designed to respond to or anticipate the needs and evolutions of a market. They also can be motivated by a wish to share risks and costs to develop a new product. This means of development appears well adapted to innovation.

New models of open innovation [START_REF] Chesbrough | Open innovation: how companies actually do it[END_REF]) encourage companies to undertake some R&D in partnership with others, which produce new organisational forms: research consortiums, co-development projects, joint ventures and exploration partnerships [START_REF] Gillier | Managing innovation fields in a crossindustry exploratory partnership with C-K Design Theory[END_REF]. In particular, an exploration partnership involves companies undertaking innovation by pursuing ‗investigation activity and recognition of new fields of action, or more specifically, -fields of innovation‖' (Segrestin 2006, p. 6). The partners simultaneously assess technical possibilities and value spaces [START_REF] Le Masson | Les processus d'innovation: Conception innovante et croissance des entreprises[END_REF]. Thus exploration partnerships go beyond the simple co-development of an innovative product or service. Exploration activity implies that partners lack a concrete idea of what the final result of the collaboration will be. Along the way, they construct the object of their collaboration, or their common purpose [START_REF] Gillier | Managing innovation fields in a crossindustry exploratory partnership with C-K Design Theory[END_REF]. Exploration therefore differs from systems of conception in which identified experts define specifications using known characteristics of the product or service. Instead, there is considerable, inherent uncertainty in exploration.

The exploratory partnership also can take different forms, according to its structure (see Table 1), such as an alliance (or joint venture), consortium or project team hosted by one partner. However, the type of structure determines the mechanisms for coordination and control [START_REF] Segrestin | Innovation et coopération interentreprises. Comment gérer les partenariats d'exploration[END_REF], such that the choice of partner is crucial [START_REF] Emden | Collaborating for new product development: selecting the partner with maximum potential to create value[END_REF][START_REF] Todeva | Strategic alliances & models of collaboration[END_REF][START_REF] Kang | Does partner type matter in R&D collaboration for product innovation?[END_REF]. Is it preferable to choose complementary partners or radically different ones (in terms of skills, markets)? For [START_REF] Bierly | Explaining alliance partner selection: fit, trust and strategic expediency[END_REF], partner selection should reflect the resources and skills needed for the collaboration. [START_REF] Nooteboom | Optimal cognitive distance and absorptive capacity[END_REF] instead suggest the partners should have some form of compatibility, which depends on their degree of similarity. If partners are greatly alike (e.g. equivalent financial resources, skills, similar managerial practices), the alliance has significant action potential but limited exploratory potential [START_REF] Kang | Does partner type matter in R&D collaboration for product innovation?[END_REF].

Conversely, if the partners are radically different, their cognitive distance causes difficulties in understanding, which can undermine the management of the alliance but favour exploration.

Defined a priori

Notes: Adapted from Segrestin (2006, p. 133).

Furthermore, a central characteristic of an exploratory partnership is its intrinsic instability, in connection with its exploration activity. This characteristic causes tensions and may make result distribution methods more complex. The intrinsic instability of an exploratory partnership relates to its flexibility, because to respond to technological and commercial uncertainties, an exploratory partnership must adapt to its environment. The resulting innovation is a key sustainable competitive advantage [START_REF] Nelson | An Evolutionary Theory of Economic Change[END_REF], though researchers still question the best way to develop such competitive advantages.

Various works confirm the importance of organisational variables [START_REF] Griffin | An interim report on measuring product development success and failure[END_REF][START_REF] Yap | Factors influencing new product success and failure in small entrepreneurial high-technology electronics firms[END_REF][START_REF] Malik | How external partnering enhances innovation: evidence from Chinese technology-based SMEs[END_REF], with increasing attention devoted to the internal functioning of alliances [START_REF] Dhanaraj | Orchestrating innovation networks[END_REF][START_REF] Rothaermel | Exploration and exploitation alliances in biotechnology: A system of new product development[END_REF].

Exploratory partnerships may improve the chances of success for innovative projects, because they enable risk sharing and exchanges of information and know-how [START_REF] Gillier | Managing innovation fields in a crossindustry exploratory partnership with C-K Design Theory[END_REF]. They also encourage training and are more conductive to innovation than are individual companies [START_REF] Powell | Interorganizational collaboration and the locus of innovation: Networks of learning in biotechnology[END_REF]. As a strategic tool, an exploratory partnership offers a solution that partners can use to acquire new skills or develop new technology quickly. To these ends, different partners share project risks to greater or lesser extents, using the different tactics available to them and exchanging resources and skills [START_REF] Goes | Interorganizational links and innovation: The case of hospital services[END_REF][START_REF] Powell | Hybrid organizational arrangements: New form or transitional development?[END_REF]. Other potential advantages include increased profits, greater client satisfaction and reduced development time, which implies lower costs as well. Despite these benefits, some research implies that they cannot guarantee success [START_REF] Karlsson | Product development, innovation networks, infrastructure and agglomeration economies[END_REF], especially when the inherent tension of collaboration produces high functional and collaboration costs that endanger innovation activity [START_REF] Cooper | The invisible success factors in product innovation[END_REF]. In this setting, the distribution of results could be an important means to minimise these risks [START_REF] Jap | Pie sharing in complex collaboration contexts[END_REF][START_REF] Leiblein | Unbundling competitive heterogeneity: incentive structures and capability influences on technological innovation[END_REF].

The high level of uncertainty and increased risk associated with the specificities of exploratory partnerships create questions about the best methods for distributing results between partners. Despite the difficulties associated with choosing these methods from the outset, this choice appears absolutely essential to encourage partners to collaborate [START_REF] Segrestin | Innovation et coopération interentreprises. Comment gérer les partenariats d'exploration[END_REF].

Incentives through distribution

The outcomes of an exploration partnership can vary in nature and be subject to bimodal distribution. For example, the decision to undertake a collective exploratory project implies a strategic logic that pursues a portfolio of real options [START_REF] Mcmillan | The Entrepreneurial Mindset: Strategies for Continuously Creating Opportunity in an Age of Uncertainty[END_REF]. Real options refer to ‗limited investments that generate several opportunities for future development' (Johnson et al. 2008, p. 407), and they consist of three types: positioning options, springboard options and exploration options characterised by their degrees of technical and market uncertainty (see Figure 1).

Figure 1. Categorisation of development options

Notes: Adapted from McMillan and McGrath (2000, p. 176).

For this study, exploration options correspond to the exploration logic of [START_REF] Segrestin | Innovation et coopération interentreprises. Comment gérer les partenariats d'exploration[END_REF]: experiments whose main objective is to define innovative concepts, even without knowledge of the market or technology in advance. They represent entrepreneurial experiments that aim to discover and/or create new markets for products and services by using abilities obtained in other fields of investigation [START_REF] Mcmillan | The Entrepreneurial Mindset: Strategies for Continuously Creating Opportunity in an Age of Uncertainty[END_REF]. By choosing an exploration option, the company commits to launching an innovative project in a dedicated, independent structure, such that it is not subject to the constraints of internal development, which matches the research assertion that ‗Exploring innovative opportunities by creating an external entity such as a partnership is highly recommended … [START_REF] Bower | Disruptive technologies: Catching the wave[END_REF][START_REF] Foss | Selective intervention and internal hybrids: Interpreting and learning from the rise and decline of the Oticon Spaghetti Organisation[END_REF]' (Gillier et al., 2010, p. 885).

In turn, its exploratory activities enable the company to develop new skills, register new patents, construct models and tackle new concepts, such that their ‗form of innovation … moves away from the company's existing central skills based on clients or technology' (Chanal and Mothe 2005, p. 5). Because exploration entails more than an attempt to produce an innovation that answers needs, the related activity generates learning and leads to the development of new skills, depending on the explorer's preceding career path, knowledge of instruments and existing skills [START_REF] Gillier | Managing innovation fields in a crossindustry exploratory partnership with C-K Design Theory[END_REF].

Then the results of this exploration activity (whether financial or not) must be distributed, which provides an essential motivating factor [START_REF] Sakakibara | Heterogeneity of firm capabilities and cooperative research and development: An empirical examination of motives[END_REF]). However, little research proposes appropriate solutions for such distribution [START_REF] Catelin | Architecture organisationnelle et politique d'investissement: une illustration à travers une double méthodologie empirique[END_REF][START_REF] Kumar | Differential learning and interaction in alliance dynamics: A process and outcome discrepancy model[END_REF]. According to sparse literature on distribution sharing [START_REF] Jap | Pie sharing in complex collaboration contexts[END_REF][START_REF] Kabanoff | Equity, equality, power and conflict[END_REF], two sharing principles are most common: equity and equality. The complex distribution of results [START_REF] Blanchot | Alliances et performances: un essai de synthèse[END_REF]) should depend on norms of justice, which vary across groups and cultures.

Equity is not universally considered superior to equality, though ‗in collaborative relations where economic productivity is the first objective, equity rather than equality tends to be the principle directive force in distributive justice…. With the aim of encouraging and rewarding the contributions of those who are most capable, results are distributed according to the inputs' (Kabanoff 1991, p. 417). An equitable distribution of results, chosen prior to the collaboration, should encourage members of the partnership to exert effort and perhaps enhance success likelihoods [START_REF] Jap | Pie sharing in complex collaboration contexts[END_REF][START_REF] Kabanoff | Equity, equality, power and conflict[END_REF]. The ratio of distribution may reflect decision makers' perceptions of the members of the team [START_REF] Adams | Toward an understanding of inequity[END_REF], often based on elements such as:

 Sharing of tasks [START_REF] Arino | Learning from failure: Towards an evolutionary model of cooperative venture[END_REF][START_REF] Inkpen | Knowledge, bargaining power and international joint venture instability[END_REF][START_REF] Kogut | The stability of joint ventures: Reciprocity and competitive rivalry[END_REF][START_REF] Larsson | The interorganizational learning dilemma: collective knowledge development in strategic alliances[END_REF]).

 Sharing of information [START_REF] Borys | Hybrid arrangements as strategic alliances: Theoretical issues in organizational combinations[END_REF][START_REF] Mohr | Characteristics of partnership success: Partnership attributes, communication behavior, and conflict resolution techniques[END_REF].

 Commitment (Anderson and Weitz 1992).

If a distribution seems inequitable to members, the alliance results are affected [START_REF] Kumar | Differential learning and interaction in alliance dynamics: A process and outcome discrepancy model[END_REF]. If members perceive they are being treated unequally, the level of commitment by the disadvantaged partner diminishes. Systems based on equitable distribution also encourage competition among members and risk taking to increase profits. The resulting rivalry can paralyse collaboration. [START_REF] Jap | Pie sharing in complex collaboration contexts[END_REF] asserts that the ability to separate each organization's input to the partnership and the joint creation of idiosyncratic input improves relationship quality in general. But the use of more complex sharing principles, such as the equity principle, when the parties have an understanding of each other's transformation process, has a negative impact on relationship quality. He therefore suggests that the use of an equity principle when an organization can map a reasonable expectation of the payoff ex ante may be excessive. For these reasons, equitable distribution is preferable only in certain cases.

An egalitarian distribution instead implies uniformity and a lack of differentiation of the members of the partnership. Different members obtain an equivalent share of the results, regardless of the resources and skills they contribute (i.e. equal shares, independent of investment) [START_REF] Jap | Pie sharing in complex collaboration contexts[END_REF]). The possible injustice created by this distribution represents a significant risk that even might alter the functioning of the partnership. However, this system offers the advantage of encouraging more collaborative, non-competitive behaviour [START_REF] Brickley | Managerial Economics and Organizational Architecture[END_REF]. 1). Prior literature on distribution methods thus offers some hints about the dimensions that influence this choice, by noting that ‗Much attention has been given to how organizations "expand the pie" of benefits between them; however, there is little that addresses the ensuing issue: how organizations divide the expanded pie' (Jap, 2001: 86). In such a situation, an exploratory study can lead to better comprehension of the main characteristics that determine the choice. Accordingly, we conduct an exploratory analysis of several projects that share some common points but also differ enough to reveal dimensions that influence the choice of distribution method.

Three exploration projects from the MINATEC IDEAs Laboratory® 1

To investigate result distribution methods in the specific context of exploration activity in a collaborative situation, we begin by describing our methodology, then detail the case study we use, which features the MINATEC IDEAs Laboratory and three exploration projects that emerged from it.

Exploratory study

Because we aim for a better understanding of the methods of distribution in exploration partnerships and the dimensions that influence these choices, we undertook an exploratory case study to investigate key phenomena in context. Case studies can help accomplish various 1 MINATEC IDEAs Laboratory and MINATEC are the names registered by the CEA.

aims: to provide description, test theory or generate theory for example [START_REF] Eisenhardt | Building theories from case study research[END_REF].

Our main aim is to describe and analyse a complex phenomenon that has received insufficient attention. In addition, the case study method can provide more in-depth and rich information about a phenomenon than surveys and quantitative methods do [START_REF] Yin | Case Study Research. Design and Methods[END_REF], which allows for a careful observation of the focal phenomenon [START_REF] Eisenhardt | Building theories from case study research[END_REF]. This approach also offers a more complete view of the various causes of each phenomenon and their potential interactions [START_REF] Miles | Analyse de données qualitatives: Recueil de nouvelles méthodes[END_REF]. Finally, we can adapt this approach to the research context, as needed.

We carried out 20 semi-directive interviews that lasted, on average, one and one-half hours. The interviews, with different members of the exploration partnerships (including those legally responsible for the partnerships, or project leaders) (see Table 2), had the objective of understanding the result distribution methods and the dimensions that influenced the choice of methods. Each interview thus concentrated on partnership characteristics, the collaboration between members and incentive mechanisms implemented, as well as the means used to distribute the results.

All respondents agreed to allow a digital audio recording of the interview. We began by explaining how the interview would be conducted and assuring them of the confidentiality of their remarks. We created a summary and complete transcript of each interview immediately after its completion. To supplement the interview data, we gathered secondary internal data (e.g. meeting minutes, partnership contracts, documents describing innovation projects) and external data (e.g. press releases). Through manual coding of each interview and the secondary data, we derived a dictionary of themes (as described in the next section) that constituted the material for the analysis.

Description of exploratory partnership and three projects

The 2). This declared ambition reveals its exploratory nature (i.e. discovering technological fields and new applications), rather than simple technological developments. The stated aim of the partnership is to ‗imagine new frequently used objects and services based on advances in micro and nanotechnologies', so it works with state-of-the-art technology and micro-and nanotechnologies to design, characterise and produce mechanisms and systems that can be controlled on the nanometre scale. Thus MINATEC IDEAs Laboratory can be qualified as an exploratory partnership, because of the nature of its exploration: At the beginning of each project, the partners do not know the outcomes a priori. The only certainty is that the innovation will involve micro-and nanotechnology. The return on investment is measured as new concepts created. well as others that for strategic reasons wish to remain anonymous.

The current partners have varying characteristics, so the MINATEC IDEAs Laboratory can be qualified as an asymmetric partnership. The differences among partners (e.g. size, resources) can lead to difficulties, such as a lack of trust between partners [START_REF] Sarkar | The influence of complementarity, compatibility, and relationship capital on alliance performance[END_REF]), a delicate approach to development (Smith and Barclay 1997) or conflicts over how to share the results [START_REF] Das | Trust, control and risk in strategic alliances: an integrated framework[END_REF]. In addition, the need for heterogeneity among the partners pushed the management of MINATEC IDEAs Laboratory to include organisations from different sectors.2 This express decision, instituted at the origin of the partnership, aimed to ensure that the innovations would be situated in different sectors (e.g.

research, optics, sport, telecommunication), which might stimulate the innovation process through sharing, making experience and expertise available and creating common synergies (through joint projects). In this sense, MINATEC IDEAs Laboratory counts cross-fertilisation as a major asset in its activity, re-enforced by a lack of competition between partners.

The partnership also is legally coordinated by contracts among the different partners.

These contractual agreements determine the methods of participation and distinguish three Organised in project modes, research within the partnership relies on different disciplines, whether in social sciences (sociology, anthropology, ergonomics, economics) or engineering (programming, engineering). the partnership often focuses on large domains that encompass various elements and services, such as the automobile sector, health, telecommunications, textiles or leisure. The specific project partnerships tend to be relatively small and are managed by a project manager who answers directly to the management committee. As we show in Table 2, the three projects we investigate-MagicBall, Map

Mobile and Interfaces Visuelles-evolved within the same structure but also vary in their length, number of participating partners, objectives pursued and technologies used. The MagicBall and Map Mobile projects were initiated prior to the official start of the partnership (2002 and 2003), then continued after its creation. They use similar technologies and have parallel objectives (virtual displacement in space). The MagicBall project aimed to construct a prototype of a 3D mouse that could facilitate navigation across three-dimensional images. Its principle was based on intuitive gestures that would make it possible to move in virtual space. The partners all participated in achieving the prototype; several applications were envisaged and tested, including a skill-based navigation game using multimedia data (e.g. IGN® maps). Finally, the Interfaces Visuelles project started in 2006 at the initiative of the Essilor company, followed by the participation of all the partners (CEA, France Telecom, EDF, UPMF, U3, Teamlog, Rossignol and a confidential partner). The objective was to define concepts based on elements in personal spheres, using emerging technologies that enabled new visual interactions. This project lasted three years (2006)(2007)(2008). We consider these project details carefully to analyse the dimensions that influence distribution choices.

Map

Heterogeneous forms of distribution

To describe the three focal projects carried out in the MINATEC IDEAs Laboratory, we first consider the question of incentives, then move to the results.

Result distribution as an incentive

The result distribution methods for the three projects were specified in the contracts and referred to two alternative types of results. First, ideas and knowledge produced in the partnership could be integrated and reused by all partners except two: a (main) partner that could not access the knowledge and ideas before it officially entered into the partnership (i.e.

without specific agreement), and an occasional partner, which only had access to the ideas and knowledge produced in the project on which it was is working, without any right of access to other projects' results. Second, the projects could result in industrial property with a patent. As a partnership, MINATEC IDEAs Laboratory chose to adopt a co-property regulation implying the distribution of ‗The rights to equal shares of the invention, whatever the means that are implemented, and rights to the resulting Patent will belong to each party in equal shares' (standard participation contract, 2004, Article 3: Coproperty).

Furthermore, the distribution of the results depends on whether the new knowledge belongs to one of the partners or all partners. In the former situation, each party can exploit its own results and/or intellectual property, acquired or achieved during the execution of the contract, without providing financial returns to other parties. In the latter situation, all parties can exploit the results without having to compensate other parties. The results obtained in the three projects thus differ in nature, though all of them provided a concept definition. Usability testing was performed for MagicBall and Interfaces Visuelles. Others types of results also emerged, as we describe in Table 3.

Table 3. Results distributed in each project

MagicBall Map Mobile Interfaces Visuelles

Basic results

Concept definition Working prototype Usability testing

Concept definition Working prototype

Concept definition Concept testing Working prototypes Usability testing Commercial agreement

Refined results

Usability scenario

Numeric maps Development of microaccelerometer (MEMS)

Possibility of WiFi triangulation Development of 3D magnetometer

Interactive make-up box Semi-immersive glasses Liquid Crystal Shutter

Mobile hypertext technology

The distributions of these results also differ, as we describe next.

Choosing between equitable and egalitarian distribution

For the MagicBall and Map Mobile projects, the partnership opted for egalitarian distribution. The inventions originating from these partnership activities thus were the equally shared property of the parties that had invested in the project. All these partners were owners of the inventions created, and requests for patents were registered, with shared costs and equal shares. For these projects, the partnership was still in the process of determining its structure, and these two projects were its first collective creations. The partners represented relatively closely linked core professions (i.e. embedded microelectronics and telecommunications) and had similar interests. In turn, they appeared satisfied with the results obtained and validated the choice of an egalitarian distribution.

I found it a very good thing. That's to say that there was a part with a common core, where we shared the industrial property. And then, at a certain moment, there was concrete material, consisting of models and returns on the experiences gained in the experimenting which each company could make use of. Therefore, if there were patents to be registered for the Lab, they would be just basic elements in the sector, [and] after that each company could work on his own part with things that were quite clear (Map Mobile, project leader).

The two projects also included the same partners, so the interviews illustrate a collective work situation and considerable implications across both projects. The choice of an egalitarian distribution also might be justified because the partners represent different markets and thus would not compete as the technology develops.

The means adapted to situations in which the partners learn to work together and in which the different professions encounter one another (sociologists, engineers, etc.)

and different markets (Magicball, manager of legal department).

The dependence of the partners on the technologies proposed by the CEA also motivated the choice of an egalitarian distribution. The CEA actually provides most of the technology used in the three projects, which leaves the other partners with relatively limited margins for manoeuvring with regard to the results.

The only partner that cannot leave is the CEA. We can try it: we can remove any one of them and the system still survives. I think that if we removed the CEA, the system would survive but there wouldn't be the same orientation. From this point of view, the other partners have a limited margin for manoeuvring with regard to the results obtained (Magicball, manager of legal department).

However, the Interfaces Visuelles project featured an evolution in the means of distribution. This case included many partners-nine at the outset. The partnership also was far more mature, in that it already had existed for five years and was not launching its first project. We thus observed an effect of experience. The participating partners had grown accustomed to working together, and the operational team was stable, in both its composition and its organisation. Unlike the MagicBall and Map Mobile projects, the partners opted for an equitable distribution of the results, motivated by their recognition that the fields of application did not involve every partner. The respondents we interviewed expressed only an average level of satisfaction though, because the results obtained were below the level they had hoped for.

I saw the short film on the visual interface models, it was very nice, it was sold quite well, it was well done, but in the end, if we measure the difference between our intentions in the beginning and the results, there was quite a flagrant gap. We have progressed on interesting concepts, but in its current state, it still isn't as prolific as it could do (Interfaces Visuelles, manager of legal department).

Thus for Magic Ball and Map Mobile, the distribution sharing method is egalitarian, because that approach offers more advantages than limitations. Nor did the distribution mode change over the course of the projects. Several factors may explain this stability in distribution mode: The projects were initiated during the creation of the overall partnership, which ensured strong involvement, and the partners' businesses were closely related to the topics addressed in the project. Furthermore, the patent-protected technology already existed and was provided by one partner. Prototyping was facilitated, because the short-term projects only entailed technology combinations, and there was just one project manager for each project.

For Interfaces Visuelles though, an equitable distribution may offer more advantages.

The expectations varied by partners, involving new usages, concepts and models. These different expectations affected how the partners regarded the implications of the project outcomes. Compared with its potential (value created, financial aspects, concept used), the partners reconsidered its actual implications during the project and identified two main difficulties that appeared to be caused by the choice of an equitable division. First, the project was relatively unconnected with the professions and markets that each partner represented.

This difficulty challenged the very significance of the project for each of the partners, which meant that perceived benefits strongly influenced their investments during and after the project. Second, it became a challenge to establish effective performance metrics. Some interviewees specifically evoked the difficulty of establishing performance indicators for exploratory projects in which not all the partners had an evident interest. Without performance metrics, it was more difficult for some partners to identify and quantify results that could be relevant for their respective activities.

In brief, several factors likely explain the change from egalitarian to equitable distribution modes: the inclusion of new partners whose businesses differ, a long-term project timeline (e.g. more than three years), a complex project (e.g. combination of different technologies in a broader concept compared with MagicBall and Map Mobile), a larger project team and several project managers. Table 4 contains a summary of the main advantages and limitations of each distribution method. In summary, the MINATEC IDEAs Laboratory structure opted ex ante for an egalitarian distribution, regardless of the characteristics of the projects and partners. However, the Interfaces Visuelles project-which contained more partners than the other two projects, occurred when the partnership was no longer in a creation phase and entailed more heterogeneity in the core businesses-necessitated a change to the method of distribution.

Moreover, the implications and interests of the partners differed greatly across the organisations. These multiple differences prompted the choice of equitable distribution; in particular, the resources and skills provided by each party were too heterogeneous to envisage a different type of sharing.

Discussion

Literature on exploratory partnerships remains relatively evasive about the best methods to distribute results. Beyond their high degrees of uncertainty and risk, these partnerships can struggle because the methods of distribution are difficult to set up ex ante, especially when the results of the alliance (e.g. identifying a new opportunity, the emergence of new business opportunities) and its quasi-rents3 cannot be calculated in advance. However, in these situations, it is possible:

 To forecast, before the alliance, how property rights, connected with emerging opportunities during the alliance, should be distributed.

 To determine the percentage of remuneration and envisage a specific clause to address unexpected factors. For example, in a collaboration with equivalent proportions (50-50), members can create a reserve for contingencies in equal proportions, used in the case of an unexpected risk.

Therefore, even though result distributions should not be rigidly defined in an alliance proposition (which may have a counter-incentive effect on partners), the principles of collaboration can and should be discussed beforehand [START_REF] Segrestin | Innovation et coopération interentreprises. Comment gérer les partenariats d'exploration[END_REF].

Types of results

In exploratory partnerships, two main types of results will emerge and need to be distributed:

 Property and rights to patents, drawings or models, which generally should not be distributed to a single actor, to avoid tests of strength.

 Financial results generated by a market launch, for which, unlike other types of inter-organisational relations such as franchise networks, innovation partners rarely seem to use quasi-rents [START_REF] De Guilloux | How France's potential franchisees reach their decisions: a comparison with franchiser's perceptions[END_REF]).

Incentive nature

The distribution of quasi-rents is often a principal motive for development partnerships-as well as a principle cause of arguments to decide on the appropriate distribution (equitable or egalitarian). There is no universal rule for the distribution of value, especially when resources and skills are difficult to measure [START_REF] Das | Instabilities of strategic alliances: An internal tensions perspective[END_REF]. The contribution of each member is complicated to determine in an innovation project, which requires vast intangible resources (e.g. technique knowledge, market information) that cannot be measured in an objective way (cf. financial investment) and are based on the arbitrage of each project. The partners can assess the financial contributions of each member but cannot objectively determine the value of intangible contributions, such as one partner's brand image [START_REF] Urban | Alliances stratégiques coopératives européennes[END_REF]). Yet despite these challenges and regardless of the uncertainty inherent to partnerships and innovation projects, our research shows that it is better to define ex ante the rules for the distribution of results, without setting them too rigidly (i.e. Interfaces

Visuelles case study). They must be redefined according to advances in the project and precise identifications of necessary resources and skills. Causes for renegotiation might include a weak contribution by certain partners or the advent of an alternative technology.

Equitable versus egalitarian distribution

At the beginning of a partnership, partners determine what each will obtain at the end of the project. In line with Lucas and Piron's (1998, p. 5) summary of the European Tactical Missiles Alliance between Matra and Bae Dynamics, we find that equity seems to be the only means of creating a certain legitimacy when allocating resources. This question is often wrongly reduced to a problem of ‗just returns', involving the financial proportionality between the contribution in market terms and the contribution in terms of tasks. For the prescribers or negotiators of the alliance, the question of just returns can be answered, once and for all, prior to the conclusion of the alliance, in financial terms. However equity remains a question during the structuring of the project, and beyond.

Thus the relation between contributions and the compensation of members and the influence of this relation on members' perception of distributive justice is critical. Yet equitable distribution is difficult to establish. It depends on norms of justice, which vary from one group to another and across cultures. Equity does not always appear preferable to equality; rather, results should be distributed according to outputs, such that the distribution encourages and recompenses those members that have invested the most.

Moreover, when a company makes an investment in an exploratory activity that comes close to the activities of other partners, the company should be more inclined to assign its rights [START_REF] Vassolo | Non-additivity in portfolios of exploration activities: A real options-based analysis of equity alliances in biotechnology[END_REF]. By extension, our research indicates that the closer the exploratory results are to a company's core activities, the more reticent the company is to share its results with partners who are only slightly involved.

Conclusion

This article has focused on different types of results and their associated methods of distribution. These elements are inherent to partnership relations and particularly difficult to implement in an exploratory partnership, which is by nature risky and uncertain. Two main contributions emerge from this study:

 The choice of distribution methods depends on three principal structural dimensions.

For Map Mobile and MagicBall, the methods were defined from the point of creation of the partnership. The number of partners was limited, and the core professions were close and even complementary. For Interfaces Visuelles, the broader partnership already was well established when the project was launched, and there were many stable partners (i.e. few new members and few members leaving). According to these findings, equalitarian distribution seems preferable when a partnership is in the preliminary stages, there are few partners and the fields of activity are close. An equitable distribution instead is preferred when the exploratory partnership includes a large number of partners, and their core professions are very different.

 The methods diverge according to the partners' behaviours. The level of contributions and interest shown in the project influences the distribution methods chosen. A partner that can directly re-appropriate the results of the exploration for its own activities will tend to prefer equitable distribution and consider it, in this case, more just.

Along with these contributions, the current study has several limitations. For example, with regard to the characteristics of the projects studied, this research has focused on an asymmetrical partnership containing partners with varied profiles. It would be interesting to study exploratory partnerships between competitive companies, to analyse the possible repercussions of these similarities on result distributions. In such a context, the partners continue to compete [START_REF] Nalebuff | La co-opétition, une révolution dans la manière de jouer concurrence et coopération[END_REF], and the partnership creates a high risk of opportunism and conflict. In addition, we analysed only one incentive mechanism, namely, the methods used to distribute the results. A simultaneous study of several motivating mechanisms could reveal other dimensions that also influence the type of distribution. For example, the extent of the contract or the degree of confidence among partners likely influences the choice of distribution methods. Intuitively, we might imagine that when partners trust each other more, the risks of opportunism and quasi-rents decline.

Thus, partners should be less reticent about opting for an egalitarian distribution, even if they have heterogeneous levels of commitment.

  MINATEC IDEAs Laboratory is a partnership based in Grenoble, next to the French Center of Research in Micro-nanotechnology (MINATEC). It comprises eight partners: six industrial and two academic members. Since 2003, the partnership also has included several corporate partners that, for strategic reasons, wish to remain anonymous. A memorandum of understanding (MOU) among CEA-Leti, France Telecom, ST Microelectronics and Hewlett-Packard, signed in 2001, prefigured the creation of the MINATEC IDEAs Laboratory partnership in Grenoble, France. The partnership was made public in 2003, when its name was registered with the Institut National de la Propriété Industrielle, which changed the set-up from a simple multilateral agreement to a detailed contractual alliance agreement. The MINATEC IDEAs Laboratory defines itself as a ‗platform whose mission is to imagine new frequently used objects and services based on advances in micro and nanotechnologies' (Table

  status levels: main partners, partners-projects and occasional partners. Main partners are all part of the management committee (CODIR) and have rights to veto admittance of new partners or enlarge the circle of main partners. Partner-projects have the same rights as the main partners but only for a specific project. Finally, occasional partners are admitted for precise, short-term missions. Thus this partnership enables partners to share the risks and costs involved in each project. Finally, occasional partners are admitted into MINATEC IDEAs Laboratory for a precise, short-term mission, such as when Ecole Nationale Supérieure en Création Industrielle helped the laboratory set up design workshops in 2008.Strategic decisions for the partnership are made by the executive authority, namely, the management committee. It determines and validates future research directions and partners; that is, the laboratory is managed not by one partner in particular but collectively, which differentiates it from other structures (not partnerships), such as Medialab at the Massachusetts Institute of Technology, Interuniversity Microelectronics Centre (IMEC), I-Lab or the California Institute for Telecommunications and Information Technology (Cal(it)²). We also distinguish MINATEC IDEAs Laboratory from internal R&D structures in various companies, such as Orange Lab (Orange), e-lab (Bouygues), Philips Home Lab (Philips), Cre@team (EDF), the Palo Alto research center (Xerox Corporation) or Sony CSL (Sony). In effect, MINATEC IDEAs Laboratory represents a common structure linking companies and universities.

  Mobile launched in 2003, with the initiative of the founder partners (CEA, ST Microelectronics, France Telecom and Hewlett-Packard). It proposed a system for interactive localization inside and outside buildings. The project lasted three months, between the definition of the concept in spring 2003 and the construction of the first functional prototype.

Table 1 . Legal forms and contractual arrangements in exploration partnerships

 1

		Joint Venture	PLC	Partnership	Collaboration Agreement
	Control	Joint	Key person	Key person	Co-governance
	Decision Power	Multiple voting rights	Voting rights proportional to shares	Unanimity	Multiple voting rights
	Result Distribution Method	Proportion of the shares	Proportion of the shares	Proportion of the shares

COMMERCIAL UNCERTAINTY TECHNOLOGICAL UNCERTAINTY

	STRONG	Positioning Options	Springboard Options
	MEDIUM	Platform for Launch	Exploration
				Options
	WEAK	Improvements
		WEAK	MEDIUM	STRONG

Table 1 . Comparison of two distributions

 1 More difficult to find a ‗hub firm' to manage the project In summary, the challenges of both means of distribution suggest it is not possible to retain one instead of the other (see Table

		Equitable Distribution	Egalitarian Distribution
		 Each partner's outcomes are a	 Each partner receives an equal
	Definition	function contributions (tangible and of its resource	share of the outcomes. It is a 50/50 split
		intangible) to the exploratory

Table 2 . Descriptive summary of MINATEC IDEAs Laboratory partnership (2009) MINATEC IDEAs Laboratory®

 2 EDF, the CEA, Isere County Council and two Grenoble-area universities (Pierre Mendès France: Human and Social Sciences and Stendhal: Literature, Languages and Communication). Since 2003, the partnership also has included corporate partners such as Hewlett-Packard, Essilor, France Telecom, Rossignol, STMicroelectronics and Teamlog, as

	Statute	Partnership of companies and universities, implying the creation of an ad hoc structure
	Year of creation	2001

Table 2 . Nature of focal exploratory projects

 2

		Characteristics	Magicball	Map Mobile	Interfaces Visuelles
		Reception structure	MINATEC IDEAs Laboratory
		Maturity of partnership	Creation	Maturity
		Expected results	Models, studies of usage	Models, studies of usage	Concepts, models
		Number of partners	6	4	9
	Project	Basic technology Length of project Heterogeneity of core professions	Accelerometer 2 years Similar	Accelerometer 3 months Similar	Photochromic glasses 3 years Heterogeneous
		Interest of partners in the project	Strong	Strong	Medium
		Partner providing technology	CEA	Essilor
		Number of interviews	6	4	10
			Prescribers of	Prescribers of	Prescribers of
	collected	People interviewed	project (1), legally responsible members (5)	project (1), legally responsible members (3)	project (1), legally responsible members (8)
	Data	Internal data	Participation contracts, project specifications, meeting minutes, summary of activity
		External data	Internet sites, specialised and general press articles

Table 4 . Main advantages and limitations of distribution choice

 4

		MagicBall	Map Mobile	Interfaces Visuelles
	Type of distribution	Egalitarian		Equitable

A direct industrial competitor of one of the partners can apply to enter though. Admission requires a unanimous vote by the representatives of the partners who sit on the management committee.

Quasi-rents derive from cooperation and refer to the surplus profit made by members because of their partnership. Quasi-rent is therefore the spread between two profits: that which would have been earned without the alliance and that made due to it.