Adnand Bitri

Kevin Samyn

Thomas Jacob

Thomas Jacob Cavity

Cavity Investigation by Surface Wave Tomography and Microgravity

Introduction

Shallow cavities, such as karstic caves in carbonate bedrock and near-surface underground mine workings and tunnels, constitute serious hazards for persons and existing constructions due to the risk of collapse and subsidence involving damages to buildings and increased urban development costs. Detection of karstic features, voids, surficial dissolution, fissuring, alteration and unconsolidated material, is thus a major challenge for geophysical methods. Density and resistivity are known to be good indicators of the alteration and fissuring of draining or saturated epikarst, and electric behaviour may be amplified by the presence of shallow groundwater and strong water circulation. Ground penetrating radar (GPR) can be also suitable where soils or argillaceous cover, that absorbs radar waves, are absent or highly discontinuous. Recent investigations have reported good results for cavity imaging using surface seismic methods [START_REF] Samyn | Imaging a near-surface feature using crosscorrelation analysis of multi-channel surface wave data[END_REF][START_REF] Leparoux | Underground cavity detections: a new method based on seismic Rayleigh waves[END_REF]Vogelaar, 2001, Gucunski et al., 1996;Park et al. 1996). Surface wave have several properties that make them useful for near surface characterization. They are usually the largest amplitude waves generated by a surface impact, and their dispersive property allows for quantification of shear wave velocity with depth. Based on the relative advantages and disadvantages of the different geophysical techniques the most prudent imaging approach is the use of two or more complementary techniques. For example, the exploration of a karst network in the Swiss Jura was conducted by combining GPR which provides a precise image of the most surficial voids and structures, with gravimetry which is sensitive to the presence of deeper and larger heterogeneities (Beres et al., 2001). [START_REF] Debeglia | Karst investigations using microgravity and MASW;Application to Orléans, France[END_REF] proposed microgravity and Multi-channel Analysis of Surface Waves (MASW) coupling to detect and characterise karst structure in an urban environment. This study evaluates the potential of using surface wave tomography (Long, L.T., andKocaoglu, A.H., 1999, Abbott et al. 2006) with microgravity to image complex near surface cavities. The study area is located in Souzay-Champigny, France, and consists of an abandoned underground quarry network in cretaceous chalky limestone. The cavities are accessible and were partly mapped. The study focuses on the largest mapped cavity.

Data acquisition and processing

Gravity measurements at Souzay-Champigny site, France, were conducted using two Scintrex CG-5 relative gravimeters. A total of 462 gravity stations were measured, over 80 m by 82 m area using 4 by 4 meter grid, and the stations were tied to a local base station that was repeated at an interval of approximately one hour, to estimate and minimise instrumental drift correction. Station elevation was acquired by direct levelling with centimetre accuracy and sensor height was measured to allow gravity reduction to ground level. To obtain the Bouguer anomaly, readings were corrected for instrumental drift, earth tides, elevation, latitude, and terrain effects using BRGM proprietary software. The uncertainty of Bouguer anomaly estimation mainly depends on the errors in the gravity readings. About 20% of the stations were reoccupied during the survey in order to evaluate the repeatability of gravity measurements. The repeat measurements are in the range of 0 to 15 µGal, with 68 % of the repeats being lower than 7 µgals. Taking into account levelling and terrain correction errors, the combined standard uncertainty on Bouguer anomaly is estimated at 8 µGal.

Seismic acquisition geometry consisted of 6 parallels lines with 10m spacing. Each line consists of 24 10Hz single component vertical geophone at 5m intervals. Two 48 channels Stratavizor and two 24 channels Geodes from Geometrics were used to record and vertically stack four impacts from 10kg hammer. Data were recorded at a sample rate of 0.5 msec during 1 second. As shown in the figure 2a, the raw traces show clear developed surface waves. The first step in surface wave tomography is to estimate group travel time for each source geophone pair at regular frequency interval. The S-transform was used to identify the arrival times of the surface waves at selected frequencies. The S-transform [START_REF] Stockwell | Localization of the complex spectrum: The S transform[END_REF]) is a time-frequency transform which provides a time depended distribution of the signal. It is very similar to the Gabor transform which utilize a Gaussian for spectral localization. The difference is in the S-transform the Gaussian window is scalable with frequency which enhances a better time-frequency resolution. The selected frequencies were limited by the energy in the recorded traces and by their wavelength. In this study reliable energy was observed for frequencies from 13 to 40 Hz. Figure 2b shows the results of S transform for frequencies of 10Hz. The measured traveltimes for all receivers in the Rayleigh wave fundamental mode time window at a given frequency are used in a tomographic inversion algorithm to reconstruct a 2-D tomogram of group velocity (figure 3). The inversion for each frequency is done separately. Our algorithm uses the finite difference method to calculate travel time to every point on a slowness grid, and solves the inverse problem using a least-squares scheme [START_REF] Grandjean | JaTS: a fully portable seismic tomography software based on Fresnel wavepaths and a probabilistic reconstruction approach[END_REF]. The core program is mainly based on two algorithms, respectively dedicated to the traveltime computation and to the velocity model reconstruction. A second-order Fast Marching Method (FMM) is used to solve the Eikonal equation over a highly discretized numerical grid, therefore enabling a fast and robust computation of seismic traveltimes between sources and receivers. Wavepaths are computed using 3D Fresnel volumes that enter in the regularization strategy and ensure a good convergence. This approach accounts for complex velocity models and has the advantage of considering the effects of the wave frequency in the velocity model resolution [START_REF] Husen | Local earthquake tomography between rays and waves: fast ray tomography[END_REF]. The velocity model reconstruction is based on a simultaneous iterative reconstruction technique (SIRT).

Results

We remove a planar regional trend to the Bouguer anomaly to obtain the residual Bouguer anomaly, in which strong negative anomalies associated to the underground quarry are evident (see Fig 1a). The amplitude of the anomalies reaches some -0.090 mGals at most. As expected, there is a strong spatial correlation between the shape of the residual gravity anomaly and that of the mapped cavities. Note that the cavity has not been mapped in its entirety, as shown by the large negative gravity anomaly to the North, associated to an unmapped part of the quarry (Fig 1).

In figure 3, we plot the group velocity map for the 10Hz frequency. On the plot, we presented also the layout experiment, the negative residual Bouguer anomaly and a part of the mapped quarry. The group velocity ranges from 150 to 450 m/s. The study area included a cavity structure whose roof is located at the depth of 8 to 10m. Significant low velocities anomalies in the center of the plot are probably due to the propagation perturbation of surface wave in the presence of cavities. These anomalies overlap well with gravity anomalies and the part of mapped carry.

Conclusions

Microgravity and surface wave tomography coupling is a promising approach for the detection near surface cavities in an urban environment, the two methods being complementary in depth of investigation and origin of detected anomalies. Microgravity, conducted with low spacing and careful implementation in order to ensure high resolution and accuracy, remains, one of the methods most suited to the detection of voids, even relatively small, in the first 20m of depth. Its weakness lies in its ambiguities of interpretation. Surface wave tomography which is sensible to the soil stiffness , contributes at reducing these ambiguities, by characterising the mechanical behaviour of the gravity anomalous zones, and finally at minimising the volumes and costs of mechanical controls.

Fig. 1 -

 1 Fig. 1a-residual Bouguer anomaly, the thick black lines denote the mapped cavity limits

Fig. 2 -

 2 Fig. 2a-3D raw shot gather b-S-transform for the 10Hz frequency.

Fig. 3 .

 3 Fig. 3. Group velocity map for frequency 10Hz with layout experiment.


Noël Bonneuil
email: bonneuil@ined.fr

Raouf Boucekkine
email: boucekkine@uclouvain.be

Optimal transition to renewable energy with threshold of irreversible pollution

Keywords: Multi-stage optimal control, threshold effects, irreversibility, non-renewable resources, viability JEL classification: Q30, Q53, C61, O33

When cheap fossil energy is polluting and pollutant no longer absorbed beyond a certain concentration, there is a moment when the *

Introduction

Optimal technological regime switching appears during trade-offs between obsolescence, learning costs, and productivity gains [START_REF] Hamilton | Analysis of time series subject to changes in regime[END_REF][START_REF] Parente | Technology adoption, learning-by-doing, and economic growth[END_REF][START_REF] Makris | Necessary conditions for infinite horizon discounted two-stage optimal control problems[END_REF][START_REF] Boucekkine | Technology adoption under embodiment: a two-stage optimal control approach[END_REF]. The determination of the best moment for adopting a renewable energy complicates classical optimization. Boucekkine et al. (2013a, b) resume the exhaustible-resource and stock-pollution model of [START_REF] Tahvonen | Fossil fuels, stock externalities, and backstop technology[END_REF] to derive first-order optimality conditions and characterize the geometry of the shadow prices at optimal switching times (if any). We revisit this problem, but with viabilityoptimality techniques allowing us to overcome the technical difficulties of a large number of variables and of gradual adoption. We also find the optimum while having state variables remain in their respective sets of constraints imposed by economic necessities.

We consider pollution as possibly irreversible, as [START_REF] Tahvonen | Optimality of irreversible pollution accumulation[END_REF] and [START_REF] Prieur | The environmental Kuznets curve in a world of irreversibility[END_REF] do. The non-renewable resource, say fossil fuels, is extracted and consumed, causing emissions of pollutant. Beyond a critical concentration, pollutant is no longer absorbed, as it is the case for carbon dioxide by the oceans. The choice is to adopt a renewable but expensive technology, say wind power, or not. [START_REF] Prieur | Optimal emissionextraction policy in a world of scarcity and irreversibility[END_REF] searched for the optimal management of exhaustible resources under irreversible pol-lution but ignored the adoption of the new technology. In Boucekkine et al. (2013b), the timing of adoption is endogenous but not the size of the investment in the technology of renewable energy. This assumption follows [START_REF] Valente | Endogenous growth, backstop technology adoption, and optimal jumps[END_REF]: when adoption starts, the share of renewable energy is set to a constant, reflecting the maximal level of the renewable energy allowed by the technological capacity of the country. We refer to this type of adoption as "instantaneous adoption." We relax the view of instantaneous adoption, which is mathematically convenient but restrictive, and suggest to endogenize the investment into the renewable energy, within the possibilities left by the technology. We refer to this type of adoption as "gradual adoption." For example, the part of renewable energy in the French production of electricity is forecast to grow from 22.9% in 2013 to 35% in 2020. 1 We suggest to determine not only when the transition to a renewable energy should start, but also at what rate it should be substituted to the former one. The proportion of clean but expensive energy can then be adjusted over time. Boucekkine et al. (2013), using Pontryagin, had to check all possible candidates to optimality (inner solutions, corner solutions, solutions with adoption of renewable energy and those without, solutions with reversibility of pollution and those without), computing the associated value functions for each set of parameters and for each set of initial conditions, to finally select the maximal value. In contrast, viability theory yields optima in a necessary and sufficient way, allowing a systematic exploration of the state space.

Our method here allows one to solve the problem without the round-about of Pontryagin or Hamilton-Jacobi-Bellman, without having to compute each candidate solution to select the best one thereafter. Our econometric analysis, as in [START_REF] Bonneuil | Viable Ramsey economies[END_REF], shall capture the main features of viable optimal decisions and help us interpret them. Besides, thanks to [START_REF] Bonneuil | Computing the viability kernel in large state dimension[END_REF] viability algorithm, the technique is flexible enough to add state dimensions, here gradual versus instantaneous adoption, avoiding to derive first-order conditions each time. The technique works with continuousdiscrete time dynamics. The link with optimal solutions, which economists are fond of, was established theoretically [START_REF] Bonneuil | Maximum under continuous-discrete-time dynamic with target and viability constraints[END_REF]. It finds here a practical application.

With the plausible parameters values used by [START_REF] Prieur | Optimal emissionextraction policy in a world of scarcity and irreversibility[END_REF], we shall determine the factors favoring adoption or not. We shall notably find the importance of initial fossil resource and initial level of pollution in the maximal inter-temporal utility. This is consistent with Boucekkine et al. (2013b) who use a different analysis. In addition, we shall also examine the role of the time horizon and the capacity to afford a gradual adoption. We shall also find that a longer time horizon increases the probability of adoption of renew-able energy and the time for this adoption. It also increases maximal utility and the probability to cross the threshold of irreversible pollution. We shall find that the time for starting exploiting a renewable energy decreases when adoption is gradual, compared with instantaneous adoption: the shorter the time remaining after adoption until the time horizon, the more quickly renewable energy is adopted. Then the theory of optimal regime switching is completed by the realistic feature of gradual adoption, a key component of energy policies.

After posing the problem as a maximization under 4 differential equations, we shall present viability theory and the procedure to obtain a maximum under viability constraints, with its associated algorithm. The problem becomes a viability problem with six dimensions. An example of trajectories with gradual or with instantaneous adoption will helps us situate the dynamic. Then we shall proceed to the econometric analysis of 600 simulations, so as to highlight the determinants of adoption and its mode.

The problem

The quantity of fossil resource is x 1 (t) at time t. People are to solve the trade-off between cheap polluting energy against expensive cleaner energy:

max v 1 ,v 2 ,ts T 0 u(v 1 (t)) -D(x 2 (t)) -cx 3 (t) e -δt dt, ( 1
)
where T is the time horizon, the function u represents utility and D the damage function, c is a parameter reflecting the production unit cost, x 2 the level of pollution, x 3 the amount of exploited renewable energy, t s ≥ 0 is the time when this renewable resource begins to be adopted:

               x 3 (t) = 0 if t < t s x 3 (t) = v 2 (t) ∈ V 2 if t ≥ t s (adoption) x 3 ∈ [0, x 3 ], (2)
where V 2 is a closed set, v 2 the investment into or disinvestment from renewable energy, x 3 the maximal renewable energy, limited by the possibilities of the country. Total energy consumption v 1 (t) is the sum of the quantity e(t) ≥ 0 of polluting energy and of the quantity x 3 (t) ≥ 0 of renewable non polluting energy:

v 1 (t) = e(t) + x 3 (t). (3)
The fossil resource decreases as:

x 1 (t) = -e(t) = -v 1 (t) + x 3 (t). (4)
The quantity of pollutant is also taken equal to e(t), such that pollution varies according to:

x 2 (t) = max(0, v 1 (t) -x 3 (t)) -α(t)x 2 (t), (5)
where α(t) is the rate of absorption of pollution by the environment, becoming null over a threshold value x 2 , above which the milieu becomes unable to absorb any quantity of pollutant:

       α(t) = α constant for x 2 (t) ≤ x 2 (reversibility) α(t) = 0 otherwise. (irreversibility). (6)
Equation ( 2) is an impulse equation:

                       x 3 (t) = w(t)v 2 (t) w (t) = 0 , w(0) = 0 w(t + s ) = w(t - s ) + 1 = 1 x 3 ∈ [0, x 3 ], ( 7
)
where w is a Heaviside function.

The dynamic {4, 5, 6, 7} is a differential inclusion with impulse:

       x (t) ∈ F (x(t)) x + = R(x) := {x - 1 , x - 2 , x - 3 , 1)} (8) with F (x) := {(-(v 1 +x 3 ), max(0, v 1 -x 3 )-α1 x 2 ≤x 2 x 2 , v 2 , 0), v 1 ∈ [0, x 1 ], v 2 ∈ V 2 }. (9)
The state variables are x 1 , x 2 , x 3 , and w, and the controls t s , v 1 , and v 2 . The system {4, 5, 6, 7} constitutes a differential system in continuousdiscrete time, also called hybrid dynamic [START_REF] Bensoussan | Hybrid Control and Dynamic Programming[END_REF] under constraints Boucekkine et al. (2013b) considered an instantaneous transition to renewable resource, which amounts to replace (7) by:

K := {x = (x 1 , x 2 , x 3 , w) | x 1 (t) ≥ 0, x 2 (t) ≥ 0, x 3 (t) ∈ [0, x 3 ], w(t) ∈ {0, 1}} ⊂ IR +4 .
               x 3 (t) = 0 x 3 (0) = 0 x 3 (t + s ) = x 3 . ( 10
)
We shall compare the solutions from {4, 5, 6, 7} and from {4, 5, 6, 10}, starting from the same initial conditions.

3 Method: Capture-Viability, Optimization, Algorithm 3.1 Capture-Viability

A state x = (x 1 (0), x 2 (0), x 3 (0), w(0)) is said T -viable in K if there exists at least one solution x(.) to (F, R) starting from x(0) = x and remaining in K until T : ∀t ∈ [0, T ], x(t) ∈ K. A set of T -viable

Impulse Dynamical Systems and Capture-Viability

An impulse differential inclusion (F, R) is described by 1. a set-valued map F : X → X, representing the dynamic in continuous time:

x ∈ F (x) = {f (x, u, v), u ∈ U (x), v ∈ V (x)} (11)
where U (x) and V (x) are closed sets;

2. a reset map R : X → X, describing the discrete part of the run, associating a new initial state x i+1 ∈ R(x i ) with some instant t i .

For two closed sets Ω ⊂ K ⊂ IR m , how can a system governed by (F, R) through u stay in K and reach Ω before leaving K, despite uncertainty v. Starting from an initial state

x 0 ∈ K, a solution x(•) to the impulse differential inclusion (F, R) is a solution to the differential inclusion x ∈ F (x) viable in K until a time t 1 ≥ 0 when it reaches R -1 (K). At that point, a second initial state x 1 = x(t + 1 ) ∈ R(x(t - 1 )) ∩ K, with x(t - 1 ) := lim t→t - 1 x(t),
is taken, as a starting state for the continuous-time process, which carries on its course.

A state x is said to be viable in a closed subset K with target Ω under (F, R) if there exists at least one solution x(.) starting from x(0) = x under the dynamic F such that there exists a T > 0 such that x(T ) ∈ Ω and for 1999) proved that when F is Marchaud (its graph and its domain are not empty, the values F (x) are convex, and sup y∈F (x) y is bounded by a linear function of y ), then there exists one largest capture-viable subset of (K, Ω) including all others. This is the hybrid capture-viability kernel, denoted Capt (F,R) (K, Ω).

all t ≥ 0, x(t) ∈ K. A subset K is viable under (F, R) if all states x ∈ K are viable under (F, R). If a closed set K is not viable under (F, R), Aubin (

Optimization

Bonneuil (2012) showed that the solutions of the optimization problem {1, 2, 4, 5, 6} are located on the boundary in direction of high welfare

π(t) := t 0 u(v 1 (τ )) -D(x 2 (τ )) -cx 3 (τ ) e -δτ dτ (12)
of the capture-viability basin of target

IR +2 × [0, x 3 ] × {T } × {0, 1} × {0}
within the state space

IR +2 × [0, x 3 ] × {T } × {0, 1} × IR + (containing states (x 1 , x 2 , x 3 , x 4 , w, π))
under the augmented dynamic:

                                                       π (t) = -(u(v 1 (t)) -D(x 2 (t)) -cx 3 (t))e -δx 4 (t)
x 1 (t) = -v 1 (t) + x 3 (t) (fossil resource)

x 2 (t) = max(0, v 1 (t) -x 3 (t)) -α(t)x 2 (t) (pollution) x 3 (t) = w(t)v 2 (t) (renewable energy) x 4 (t) = 1 (time), w (t) = 0 (Heaviside [t s , T ]) α(t) = α 1 x 2 (t)≤x 2 (absorption); v 1 (t) ∈ [0, x 1 (t)], ( energy consumption)
v 2 (t) ∈ V 2 , (variation of renewable energy); t s ≥ 0, (switching time)

π(0) = π 0 , x 1 (0) = x 10 , x 2 (0) = x 20 , x 3 (0) = 0, w(0) = 0 (initial conditions). (13)
The capture-viability kernel of the augmented dynamic (13) has six state variables: x 1 , x 2 , x 3 , x 4 , w, and π, and three controls: v 1 , v 2 , and t s . The variable x 4 is time, which is now a state variable, in contrast to System {4, 5, 6, 7}. [START_REF] Bonneuil | Computing the viability kernel in large state dimension[END_REF] addressed the computation of viable states and of the capture-viability kernel in large state dimension, based on stochastic optimization (to handle the large dimension of the discretized control trajectory).

The idea is to minimize the distance to the set of constraints of solutions starting from a given state, and to assess the viability status of this state whether or not the minimization of the distance leads to at least one trajectory remaining in the set of constraints. The search for viable states is also achieved by the minimization of a distance to the set of constraints, so that the procedure relies on a double stochastic optimization: one where the initial state under examination is fixed, so as to decide whether it is viable

or not, and one where this initial state is varied. We shall use this algorithm later on.

The addition of the auxiliary variable π participates in the elegant procedure leading to locate the viable maximum on the boundary in direction of high π of the capture-viability kernel associated with the augmented dynamic [START_REF] Bonneuil | Maximum under continuous-discrete-time dynamic with target and viability constraints[END_REF]. A variant of [START_REF] Bonneuil | Computing the viability kernel in large state dimension[END_REF] algorithm allows us to compute the boundary of the capture viability basin in the direction of high π without the knowledge of the whole capture-viability kernel, which is very memory-consuming. Firstly, a viable state is found for the auxiliary dynamic; secondly π is maximized for the same x. For each new attempt (x, π), stochastic optimization allows us to find one trajectory remaining in K and reaching the target at time T . For T infinite, an approximation and extrapolation of "reaching the target" is used, a task here made easy by the discounting term exp(-δt).

Results

Simulations

We took the parameter values from Prieur et al. ( 2011): c = 300, u(v 1 ) = 27v 1 (32 -v 1 ), D(x 2 ) = 0.0011x 2 2 , α = 0.0083, x 2 = 300. Figure 1 shows an example of trajectory with gradual adoption remaining in K until T = 30 years and leading to the maximum (equal to π(0) = 96352 in this case), and another with instantaneous adoption also remaining in K until T = 30 years and leading to the maximum (equal to π(0) = 95853 in this case).

Both start from (x 1 (0), x 2 (0), x 3 (0), x 4 (0), w) = (363.9, 247.1, 0, 0, 0). A case where renewable energy is not adopted is qualitatively similar from Figure 1, except that the variable x 3 remains null.

Viable states are distributed over the whole plane (x 1 , x 2 ), because from any state (x 1 , x 2 , 0, 0, 0), there exists a trajectory leading to a maximal utility. This is confirmed by computation. The point is whether or not this trajectory requires adoption.

The 100 points for T = 30 years projected onto the (x 2 , π) plane on

Econometric Analysis

Initial conditions

With these 2 × 100 simulated data for each of the three time horizons (then 600 observations), with variables normalized between 0 and 1 for the sake of comparison, except for the time t s at which adoption starts, we fit the regressions (independently of each other, in the absence of hidden variables which could link perturbations i together): logit(Pr(adoption)) = -1.18 0.34 * -0.25 0.26

x 10 + 0.16 0.25

x 20 + 0.05 where the i s are iid, "Pr" means "probability, the tilde over a variable denotes the mean value of this variable over the time interval [0, T ], the significant coefficients at the 5% level are marked by a star, the standard deviations are put in parentheses below the coefficients, and "instantaneous adoption" is taken as reference for adoption of renewable energy.

0.01 * T + 1 , π(0) = 0.

The time horizon

The logit equation of the probability of adoption of a renewable energy in ( 14) shows that adoption is more likely to occur with a longer time horizon, whatever the couple of initial values x 10 and x 20 . The time discount in the objective function could lead us to believe that the influence of the time horizon T vanishes exponentially. However, when T increases, the stock of resources declines, which favors the transition to renewable energy. Our significant estimate of the role of T then comes from the fact that the initial source of energy is not renewable. It cannot be taken for granted in discounted optimal adoption problems. Consistently with this latter result, the maximal utility π(0) does not depend on adoption and its mode (non significant coefficients for non adoption and for gradual adoption). As expected, this maximal utility is higher with initial abundant fossil resources (coefficient 0.06), which allow more consumption and lower pollution (coefficient -0.04), consistently with (1) (and consistent with Boucekkine et al. (2013b)). A longer time horizon (term 0.014) increases π(0), because, from the first equation of ( 14), it is associated with adoption of renewable energy, which itself relieves the disutility brought by pollution. The more abundant the fossil resource, the later the renewable energy is adopted (coefficient 1.41 in the regression of t s ), because more initial fossil resource leaves more flexibility, or conversely, because shortage of fossil resource hastens adoption of a renewable energy.

Gradual versus instantaneous

If this adoption is gradual, the new technology is implemented sooner (coefficient -1.81) than if adoption is instantaneous, as expected. Then, gradual adoption requires foresight, in order to be competitive in terms of utility, compared with instantaneous adoption. Both the positive effect of the initial fossil resource and the fact that gradual adoption starts sooner than instantaneous adoption still hold true for the fraction t s /T (coefficients 0.05 and -0.07 in the regression of t s /T ). The time t s , at which to start exploiting renewable energy, is also postponed when the time horizon is longer (coefficient 0.92 in the regression of t s and 0.03 in the regression of t s /T ). This time t s increases more quickly than T (positive coefficient of T in the equation of t s /T of ( 14)). Exceeding the no return pollution threshold is more likely when fossil resources are abundant (coefficient -2.06) and pollution high (coefficient -24.07): the scarcity of fossil resource drives people to limit their pollution, and low initial pollution helps people from reaching irreversible pollution. Adoption or not of a renewable energy plays no role in the reversibility of pollution. The solutions having crossed the irreversibility threshold are associated with adoption, because, over the threshold, it is optimal to attenuate the lack of natural self-cleaning by a cleaner technology, a result also found by Boucekkine et al. (2013b).

The date t first x 2 >x 2 , at which the threshold x 2 is crossed for the first time, increases with the time horizon T (coefficient 0.02), because the probability of adoption also increases with T , and, as expected, decreases with a higher initial level of pollution (coefficient -0.07).

The mean energy consumption ṽ1 on [0, T ] increases with the initial stock of fossil resources (coefficient 0.09), because optimal agents use this energy to consume more, producing more pollution. It also decreases with the time horizon (coefficient -0.0036), because adoption is also more likely to occur.

The significant coefficient of "no adoption" (-0.013) adds the explanation that adoption of a renewable energy, which is associated with a longer time horizon (first equation on the probability of adoption), decreases fossil energy consumption.

The mean level x1 of fossil resources is closely linked to its initial level

x 10 (coefficient 1.012), and decreases only with the time horizon, as expected.

The mean level x2 of pollution, as expected, increases with the initial level

x 10 of fossil resources (coefficient 0.03), with the initial level x 20 of pollution (coefficient 0.69), and with the time horizon (coefficient 0.009): the longer the production, the larger the pollution.

Once renewable energy is adopted, its mean variation ṽ2 no longer depends on initial conditions. It increases with the time horizon, but more interestingly, decreases when more time is left until the time horizon after the beginning of adoption (coefficient -0.03 of T -t s ). Conversely, the shorter the transition, the higher the increase v 2 of renewable energy. In regressing (not shown) the amount of renewable energy reached at time T on x 10 , x 20 , and T , this amount depends neither on x 10 nor on x 20 ; it increases from T = 20 to T = 30, is equal from T = 30 to T = 40: the maximization of utility is then obtained by having more or less the same amount of renewable energy at the end of the period, which is obtained, as we saw, by starting adoption later, at a higher speed v 2 , and for a longer time horizon T .

Conclusion

Our extension of the possibility of adoption to gradual adoption rather than only instantaneous confirms the role of initial conditions in the decision to adopt a renewable energy. A longer time horizon increases the probability of adoption of renewable energy and the time for starting this adoption.

It also increases maximal utility and the probability to cross the threshold of irreversible pollution. Exploiting a renewable energy starts sooner when adoption is gradual rather than instantaneous. The shorter the period remaining after adoption until the time horizon, the higher the investment into renewable energy.

Figure 2 Figure 1 :Figure 2 :

 212 Figure 2 delineate the boundary of the capture-viability kernel, with some

Table 1 :

 1 Table 4.1 gives the simulated distributions between adoption or not and reversibility or not, by time horizon, for a given threshold x 2 = 300. The fact that Table 4.1 counts fewer reversible cases depends on the value taken for x 2 . Distribution of the N=600 simulations, 100 by time horizon and

	type of transition.
		gradual adoption	instantaneous adoption		non adoption
	pollution T=20 T=30 T=40 T=20 T=30 T=40 T=20 T=30 T=40
	irreversible	27	47	18	29	46	17	43	34	12
	reversible	6	1	1	6	1	1	10	1	1

source Réseau de transport d'électricité, published by Bertille Bayart in Le Figaro, 10 December