Thierry Bréchet
email: thierry.brechet@uclouvain.be

Carmen Camacho

Vladimir M Veliov
email: veliov@tuwien.ac.at.

Documents de

The purpose of this paper is to extend the use of integrated assessment models by defining rational policies based on predictive control and adaptive behavior. The paper begins with an review of the main IAMs and their use. Then the concept of Model Predictive Nash Equilibrium (MPNE) is introduced within a general model involving heterogeneous economic agents operating in (and interfering with) a common environment. This concept captures the fact that agents do not have a perfect foresight for several ingredients of the model, including that of the environment. A version of the canonical IAM (DICE) is developed as a benchmark case. The concept of MPNE is then enhanced with adaptive learning about the environmental dynamics and the damages caused by global warming.

The approach is illustrated by some numerical experiments in a two-region setting for several scenarios.

Introduction

Once upon a time there was a world in which people were refusing to see that their world was changing -let's say because of global warming. The main wish of these many people was to keep on doing their business-as-usual. For sure, the best strategy in the changing world would be for them to learn as much as possible about the expected changes (all of them) and to adopt the optimal behavior with respect to this large set of knowledge. But implementing this optimal strategy was beyond their force or skill. The question we raise in this article is not to define what would be the optimal strategy from the whole society's standpoint (which is already widely explored in the literature) but to highlight possible alternative trajectories, considering that agents are always rational, but sometimes stubborn, lazy or myopic. Stubborn because they always refuse to change their view. Lazy because they revise their view but only after a while (or after some evidence). Myopic because they are more or less short-sighted about how the world will look like in the future. The objective of this article is to explore the consequences of such behaviors in the context of global warming. In this purpose, we develop an innovative theoretical framework to redefine more realistic trajectories of the economies that are fully rational, in contrast with the usual "business-as-usual" scenario defined in integrated assessment models in the current literature.

Our contribution relies on the integrated assessment modeling of the economy and the climate. Integrated means that feedbacks in both ways are considered: economic activity generates greenhouse gases emissions that cause global warming, and global warming a↵ects the economy with productivity and welfare losses. Integrated assessment models (IAMs) allow to implement a dynamic cost-benefit analysis and to determine the optimal policies. Basically, policies in IAMs consist in choosing the path for productive investment and emission abatement that maximize some objective function, like country's welfare. It is important to stress that welfare, in IAMs, is expressed as consumption net of climate damages. It is indeed green consumption that is maximized.

In order to develop this new framework we use the concepts of model predictive control and adaptive behavior, and we combine them into the IAM framework. Ideas from the model predictive control (see e.g. [START_REF] Grüne | Nonlinear model predictive control[END_REF] are employed due to the uncertainties about the future environment and its impact on the economy that the agents persistently face. Adaptive learning or is involved in order to take into account the improvement (with time and/or experience) in the measurements quality and in the agents' knowledge about the environmental-economic dynamics.

The article is organized as follows. In Section 2, integrated assessment modeling and how it is used for the climate change analysis is presented with some details. This section gathers a condensed explanation of the very concept of IAM, a benchmark model, and a survey of the many uses of IMAs in the literature. In Section 3 we present the general model describing the dynamics of a multi-agent economic-environmental system that will be used in the paper. The concepts of "model predictive rational behavior" and "adaptive behavior" are presented in Sections 4 and 5. The adaptive behaviors considered in this article will concern the knowledge about climate damages (a better knowledge with evidence for climate change) and the discount factor (a decrease in the discount factor as wealth increases). Some numerical experiments are provided in Section 6 with a two-country setting (the world is roughly divided into OECD and non-OECD countries).

The purpose of this first section is threefold: (i) to provide the reader with some elements on the history of applied integrated assessment models (IAMs) and its economic rational, (ii) to sketch a benchmark model, (iii) to survey the wide variety of uses of applied integrated assessment models in the literature. This will allow us to better gauge the importance of each contribution we shall introduce later in the paper.

What is Integrated Assessment?

Although the IPCC reports (1990,1995,2001,2007) had been repeatedly calling for sharp cuts in greenhouse gases emissions (minus 50 to 80 percent at the world level, immediately), they never attempted to balance the costs and benefits of such policies, as it was initially suggested by Nordhaus in 1984.1 Nonetheless, balancing costs and benefit has been a prominent methodological and normative contribution of economics for many years. Why not for climate change? Although cost-benefit raises several methodological and theoretical challenges (and it is far beyond the scope of this paper to tackle them; see [START_REF] Pearce | Cost-Benefit Analysis and the Environment: Recent Developments OECD[END_REF] for a comprehensive analysis of CBA analysis and policy applications) it remains a comprehensive framework to understand what should be done, and what could be achieved.

Starting in the early 90s, some aggregative models were developed to analyze the consequences of economic activity on greenhouse gases (GHGs) concentration and how this concentration may harm the economy (see [START_REF] Rotmans | IMAGE: an integrated model to assess the greenhouse e↵ect[END_REF][START_REF] Nordhaus | An Optimal Transition Path for Controlling Greenhouse Gases[END_REF]and 1993a[START_REF] Gaskins | Model Comparisons of the Costs of Reducing CO2 Emissions[END_REF], Manne and Richels, 1992[START_REF] Yang | Essays on the International Aspects of Resource and Environmental Economics[END_REF]. These are the very first integrated assessment models (IAMs), so-called because they model the economy and its interplay with climate. Economic activity generates greenhouse gases that cause global warming, and global warming provokes physical damages that have an economic cost. IAMs seek at maximizing intertemporal welfare by taking these two components into account. Indeed, it boils down to a standard cost-benefit analysis, but applied to a worldwide and long-term issue.

Basically, the economic part of IAMs is made of a dynamic general equilibrium model of the economy. A policy-maker is assumed to optimally choose consumption/saving path that maximizes the discounted sum of the utility, taking into account how physical capital evolves with time and taking the adverse impacts of climate change into account. In this purpose, IAMs make use of damage functions that translate temperature increase into economic losses.

Besides, the policy maker knows the flow of GHGs emissions due to economic activity, how they convert into concentration in the atmosphere and how this concentration a↵ects the average temperature of earth.

In sum, there exists a closed loop between polluting economic human activities, how these a↵ect the climate, and how climate change impacts on the economy. What causes global warming is not the flow of GHGs but their accumulation in the atmosphere at a stock. So IAMs are necessarily intertermporal optimisation models. They endogenously determine not only the flow of GHGs but also emission abatement e↵orts and the path of productive investment.

A benchmark Integrated Assessment Model

The benchmark IAM is based on the DICE model (Dynamic Integrated Climate-Economy model) built up by Nordhaus (1993aNordhaus ( , 1993b)). DICE is a stylized cost-benefit analysis framework to optimally decide on the trajectory of GHG emissions and capital accumulation at the world level. The model represents a central-planner problem which maximizes the discounted utility taking into account economic and climatic constraints and their interconnection. The economic constraints are those of the Ramsey model.2 Output is given by a Cobb-Douglas production function, with the peculiarity that a damage function enters multiplying the formulation:

Q(t) = ⌦(⌧ )A(t)K(t) P (t) 1 ,
where A is a technology index, K physical capital and P population. is the elasticity of output to capital and ⌦ is the afore mentioned damage function. Damage is a function of average temperature ⌧ , and 1 ⌦ is the percentage of foregone production.

Emissions of GHGs flow from the global economic activity (Q) with an exogenous emission factor intensity ( (t)), but taking into account emission abatement e↵orts, denoted by µ 2 (0, 1). Actual emissions are thus given by:

E(t) = (1 µ(t)) (t)Q(t),
The concentration of GHGs in the atmosphere (M ) is given by past concentration plus new emissions net of the natural decay rate:

M (t) = E(t) + (1 M )M (t 1),
with the atmospheric retention ratio, and M the rate of GHGs absorbed in deep ocean.

Then, an equation is added to give the temperature increase. Nordhaus considers three di↵erent layers: the atmosphere, the mixed layer of the oceans, and the deep oceans. The main link is the damage function, with makes the retroaction between climate and the economy. The damage function represents the economic losses for a given a temperature increase. It is an increasing convex function of global temperature increase:

D(t) = ↵ 1 (T (t)/3) ↵ 2 , with ↵ 1 , ↵ 2 2 R + .
The last piece of the model is the abatement cost function. Abatement costs have been extensively studied. This function is thus deemed as more reliable. A 50% decrease in GHGs intensity would cost 1% of the world output. The total abatement cost function is:

C(t) = 1 µ(t) 2 Q(t).
where µ(t) 2 (0, 1) is the abatement rate and 1 , 2 are positive constants. Nordhaus uses the DICE model to compare Business as Usual (defined as µ(t) = 08t) with di↵erent emission stabilization scenarios and the optimal policy. The optimal policy leads to a 10% reduction of carbon emissions from 2005, inducing a temperature decrease of 0.2 0 C by 2100 with respect to the Business as Usual scenario.

On the many uses of applied Integrated Assessment Models

Starting from Nordhaus, IAMs have evolved introducing more realistic economic behaviors or outcomes, trying to escape from the basic comparison between BaU (no climate policy, myopic agent) and the socially optimal solution (perfect foresight), because none of them is realistic.

In this section, our objective is not to provide an exhaustive survey of IAMs but to review some examples of interesting extensions such as the inclusion of the regional dimension, models with a better description of the power sector, R&D behaviors, and coalition formation issues.

A recent survey of these approaches is provided by [START_REF] Ackerman | Inside the integrated assessment models: Four issues in climate economics[END_REF].

A direction along which IAMs were developed was geography, and depending on the paper, geography is understood as space or as the union of economic regions. Let us first mention the Integrated Model to Assess the Greenhouse E↵ect (IMAGE) by [START_REF] Rotmans | IMAGE: an integrated model to assess the greenhouse e↵ect[END_REF]. In its first version, IMAGE was a model integrating three clusters: the energy system, the terrestrial environment system and the atmosphere-ocean system. The second version of the model included a geographical scale, rare at that time. Geography was a grid of 0.5 by 0.5 degrees, making possible the biophysical modeling of land cover, its history, carbon cycle, nutrients, climate, etc. Still, all macroeconomic drivers were exogenous inputs to the model.

In 1996, Nordhaus and Yang extended the DICE model by producing a regional model, RICE (Regional Integrated Climate-economic Model). In this model, the decision is taken at the national level, and the authors consider di↵erent levels of coordination among nations. They propose three di↵erent scenarios to study how nations could deal with climate change: market policies (i.e. no-control on emission), cooperative policies (countries act as a unique decision maker), and non-cooperative policies (in which countries act in their own interests ignoring the externality create on the other countries). These scenarios were labelled "Business-as Usual", "Cooperative" and "Nash equilibrium" scenarios, respectively. This terminology will be widely used later on in the literature.

Taking DICE or RICE as benchmarks, many authors searched to refine their modeling by incorporating detailed descriptions of the energy sector, allowing for a plethora of mitigation policies, etc. [START_REF] Edenhofer | The impact of technological change on climate protection and welfare: Insights from the model MIND[END_REF] introduce learning by doing in R&D, allowing for investment in R&D in di↵erent sectors. In the long term, improving energy e ciency of existing technologies becomes too costly to be kept as the major mitigation policy. Instead, they find that a backstop technology with the potential of learning by doings the best option to protect climate at a lesser cost. They put forward Carbon Capturing and Sequestration (CCS) techniques to reduce the cost of the transition from a fossil-fuel based system to a system based on renewable resources. [START_REF] Bosetti | WITCH: A World Induced Technical Change Hybrid Model[END_REF] build the WITCH model (World Induced Technical Change Hybrid Model). WITCH is a multiregional neoclassical growth model in which technological progress is endogenous that is, the price of new vintage of capital and R&D investment are endogenous.

The model is hybrid because the energy sector (a key sector) is modeled in great detail, separating electric and non-electric uses of energy, with seven power generation technologies and allowing the use of multiple fuels. The authors account for seven channels for regional interaction. Let us mention amongst them the first, which is the fact that both R&D and consumption decisions are a↵ected by energy prices worldwide. Other interactions are learning by doing, R&D spillovers, international trade of oil and gas, and emissions trading.

The MARKAL-TIMES family of models aims at better describing the technology options, in particular in the power and industry sectors. They are technico-economic models.

The modeler needs to introduce technology characterizations and costs, resource availability, environmental constraints, services demands and macro-economic indicators. In this sense, MARKAL-TIMES is much more detailed than other IAMs. In TIMES, the quantities and prices of the various commodities are in equilibrium, i.e. their prices and quantities in each time period are such that the suppliers produce exactly the quantities demanded by the consumers. This equilibrium has the property that the total surplus (consumers plus producers surpluses) is maximized. There also exists a World multi-regional Markal-Times model [START_REF] Kanudia | The World-Markal Model and Its Application to Cost-E↵ectiveness, Permit Sharing, and Cost-Benefit Analyses[END_REF]. Notice that MARKAL models can be developed at all decision levels from wide regions of the world with several countries, to single countries, regions, counties, cities or even villages.

Another kind of model is MERGE [START_REF] Manne | Merge: An Integrated Assessment Model for Global Climate Change[END_REF]. MERGE is a model for estimating the regional and global e↵ects of GHGs reductions. The model is flexible enough to explore alternative views on a wide range of contentious issues, such as costs of abatement, damages from climate change, valuation and discounting. The model covers the domestic and international economy, energy-related emissions of greenhouse gases, non-energy emissions of GHGs, global climate change, and market and non-market damages. Each region's domestic economy is viewed as a Ramsey-Solow model of optimal long-term economic growth. Priceresponsiveness is introduced through a top-down production function where output depends upon the inputs of capital, labor and energy bundle. Separate technologies are defined for each source of electric and nonelectric energy.

Two specific problems in the climate issue are that, first, there exists no supranational authority entitled to implement the optimal policy and, second, emission reduction must be worldwide to be e↵ective against global warming. As a result, a wide international agreement among the countries is required, and such an agreement can only be found on a voluntary basis. This is the issue of coalition formation raised by [START_REF] Eyckmans | Simulating coalitionally stable burden sharing agreements for the climate change problem[END_REF] with the CWS integrated assessment model: which international agreements are feasible, and how to implement them? In other words, between Nash and the socially optimal solution, what international agreement could be achieved? Bréchet et al. (2011) extend this analysis by comparing the policy implications of the two competing theoretical streams available to date, namely, the cooperative and non-cooperative settings.3

3 The dynamics of a multi-agent economic-environmental system Let the global economy consists of n agents (regions, countries or groups of countries) and let

x i (t) denote the economic state of the i-th agent at time t (this may include physical capital, human capital and other dynamic stock-variables, so that x i is a single or multi-dimensional vector). Let v i (t) be the policy vector of the i-th agent at time t, which may include investments, abatement and other components. The economic agents operate in a common environment, the state of which may influence the productivity or the utility of the agents. The state of the environment at time t will be represented by a vector y(t), whose components can be the concentrations of GHG in di↵erent sectors of the environment and the average world temperature. Let the economy of the i-th agent be driven by the equation

ẋi (t) = f i (t, x i (t), v i (t), y(t)),
where v i (•) is the chosen by this agent policy (control) function. (Everywhere in this paper ẋ denotes the derivative with respect to the time.) Then the overall dynamics of the world economy is described by the equation

ẋ(t) = f (t, x(t), v(t), y(t)), where x = (x 1 , . . . , x n ), v = (v 1 , . . . , v m ), f = (f 1 , . . . , f n ).
On the other hand, the economic activities have impact on the evolution of the environment, say due to emission of GHGs. Let e(t, x, v, y) represents the instantaneous impact vector resulting from global economic state x, control v and environmental state y at time t.

Because GHGs are fungible (they melt in the upper atmosphere irrespective to the country of origin), the impact vector is represented by the aggregate emissions

e(t, x, v, y) = n X i=1 e i (t, x i , v i , y), (1)
where e i (t, x i , v i , y) is the emission of agent i at time t, determined by her economic state, control, and the environmental state at this time.

We assume that the dynamics of the environment can be represented by an equation of the form ẏ(t) = h(t, e(t, x(t), v(t), y(t)), y(t)).

Thus, given the control function v(•) chosen by the agents, the overall economicenvironment system is described by the equations

ẋ(t) = f (t, x(t), y(t), v(t)), x(0) = x 0 , (2)
ẏ(t) = h(t, e(t, x(t), v(t), y(t)), y(t)), y(0) = y 0 , (3)
where x 0 and y 0 are initial data.

In the numerical analysis in this paper we use one simple version of the IAM as described below.

The benchmark model. We specify x i (t) = k i (t) -the physical capital stock of the i-th agent, v i (t) = (u i (t), a i (t)) -the investment intensity and the abatement e↵ort, y(t) = (⌧ (t), m(t)) -the average atmospheric temperature at the earth surface and the concentration of GHG (measured in the CO 2 equivalent units in the warming context). Equations ( 2), (3) are specified as

ki (t) = i k i (t) + [1 u i (t) c i (a i (t))] ⇡ i (t) ' i (⌧ (t)) (k i (t)) i (l i (t)) 1 i , k i (0) = k 0 i , (4) ⌧ (t) = (m(t)) ⌧ (t) + d(m(t)), ⌧ (0) = ⌧ 0 , (5)
ṁ(t) = µm(t) + n X i=1 e i (t, k i (t), a i (t), ⌧ (t)) + ⌫(t, ⌧ (t)), m(0) = m 0 , (6)
with

e i (t, k i , a i , ⌧ ) = (1 a i ) ⌘ i (t)⇡ i (t)' i (⌧ )k i i l 1 i i .
Since versions of the above most simple IAM are widely used in the literature (see the literature review in Section 2.3) we only shortly explain the appearing notations.

Physical capital accumulation is described by equation ( 4). The depreciation rate of the physical capital of agent i is i > 0. The labor supply of agent i is l i (t) and the production function is of Cobb-Douglas type with elasticity of substitution i 2 (0, 1). The productivity of the i-th agent is ⇡ i (t) and ' i (⌧ ) is a correction factor for the productivity depending on the A fraction u i (t) of the output is consumed and another fraction, c(a i ), is devoted to CO 2 abatement at rate a i , the rest is invested, as seen in equation ( 4). Abatement at rate a i reduces the emission by a factor a i : e i = (1 a i )⌘ i Y i and costs a fraction c i (a i ) of the total product.

current temperature ⌧ . Thus Y i (t) = ⇡ i (t) ' i (⌧ (t)) (k i (t)) i (l i (t))
The evolution of the CO 2 concentration is described by equation ( 6), where µ is the natural absorption rate, ⌫(t, ⌧ ) is the non-industrial emission at temperature ⌧ . Finally, (5) establishes the link between CO 2 concentration and temperature change. The CO 2 concentration increases the atmospheric temperature through d but also may a↵ect the cooling rate . The initial values k 0 , ⌧ 0 , m 0 are given.

The control functions v i = (u i , a i ) chosen by the agents should satisfy the constraints

u i (t), a i (t) 0, u i (t) + a i (t)  1. ( 7
)
These inequalities define a constraining set V for (u i , a i ), which imply in particular, that no transformation of existing capital into consumption or abatement is possible.

The particular specifications used in the numerical simulations will be given below. The main trouble with the above model and its extensions is that most of the model components are actually not known with certainty. In fact this applies to all of the involved in the benchmark model exogenous functions.

Since the economic agents have to make their policy decisions, v i (t), in conditions of uncertainty about the future changes of the data, these decisions have to be made on the basis of predictions. Therefore in the next section we develop the concept of prediction-based rational behavior of an individual agent.

In the following two sections we define the two concepts we shall introduce in the IA framework: model predictive Nash equilibrium and adaptive behavior. The methodology presented below is not restricted to a specific model. Therefore the exposition is carried out for the general model ( 2), (3), while we refer to the benchmark case ( 4)-( 6) only for clarification, and in Section 6 -for numerical simulations.

Model predictive rational behavior

Even if the model ( 2), (3) provides a reasonable description of the dynamics of the real global economic-environment system, it is not exactly known to the agents due to imperfection of the modeling and due to uncertainties in its parameters. That is, at time s agent j chooses her future control policy based on a model that may di↵er from the "true" one. At any time instant s agent j models her economy in her own way, including the impact of the environment on the economy and her own input to the environment. Moreover, the performance criterion of each individual agent may depend on the time at which the control decision has to be made.

Thus at any time s agent j maximizes an individual objective function representing the total (possibly discounted) utility Z 1 s g s j (t, x j (t), v j (t), y(t)) dt ( 8)

subject to the controlled dynamics ẋj (t) = f s j (t, x j (t), v j (t), y(t)), x j (s) = x s -known at time s, t s,

ẏ(t) = h s (t, e j (t, x j (t), v j (t), y(t)) + ēj (t), y(t)) y(s) = y s -known at time s (10)

and the control constraint (see ( 7) for the benchmark constraints)

v j (t) 2 V. ( 11
)
Here g s j is the function that agent j uses at time s for evaluation of the future (discounted) utility, f s j represents the model that agent j employs at time s for predicting the evolution of her economic state x j (t) (for any given future control policy v j (t) and future environmental state y(t), t s), h s is the model that all agents use at time s for predicting the evolution of the environmental state y(t), t s (given the future total emission e(t)). From the point of view of agent j the total emission e(t) consists of own emission e j (t, x j (t), v j (t), y(t)) depending on the own control and economic state and on the environmental state y(t), and of the emission of the rest of the agents, ēj (t), which is not a priori known to agent j. The environmental dynamics h s employed at time s is the same for all agents. 4As it will be argued at the end of this and in the next section, this assumption is not too restrictive, since the agents may use the predictions obtained by the environmental model in diverse ways, varying between total ignorance and complete trust.

The interconnected problems ( 8 A specific feature of this context is the information pattern. In solving her optimization problem agent j is not necessarily aware of the models f s i that agents i 6 = j use at time s (as we will see below these models may change with s due to agent-specific adaptive learning).

Instead, it is assumed that agent j solves the problem (8)-( 11) if the emission ēj (t) of the rest of the agent is given. Let (x s j [ē j ](t), v s j [ē j ](t), y s j [ē j ](t))), t s, be a solution of problem ( 8)-( 11) for the given function ēj (t), t s. 5 The resulting emission of agent j is

e s j [ē j ](t) := e j t, x s j [ē j ](t), v s j [ē j ](t), y s j [ē j ](t) .
In the definition of Nash equilibrium it is enough to assume that (instead of the dynamics f s i of all other agents) the emission functional

ēj (•) ! e s j [ē j ](•) (12)
of each agent is known to the rest of the agents. That is, each agent gives a correct information about her future emissions, given any scenario for the cumulated future emission of the rest of the agents. (This information would be automatically available if the models f s i on which the agent's decisions are based were known to all agents.) Then the Nash solution consists of an n-tuple of control policies {v s i (t)}, trajectories {x s i (t)} of the economies, and emissions {e s j (t)}, t s, such that following equilibrium conditions hold for j = 1, . . . , n and t s:

v s j [ē s j ](t) = v s j (t) with ēs j (•) := X i6 =j e s i (t), ( 13
)
x s j [ē s j ](t) = x s j (t), (14)
e s j (t, x s j (t), v s j (t), y s (t)) = e s j (t), (15)
where y s is the solution of the equation ẏ(t) = h s (t, e s (t), y(t)), y(s) = y s , with e s (t) := n X i=1 e s i (t). ( 16)

The meaning of the above equalities is the following. Equations ( 13) and ( 14) mean that for the cumulated emission ēs j (t) of the rest of the agents, agent j will have (x s j , v s j ) as an optimal solution. Equations (15) mean that the optimal emission of each agent j would equal e s j (t), provided that the trajectory of the environmental state is y s (•).

It remains to notice that due to (15) the equalities y j [ē s j ](t) = y s (t), j = 1, . . . , n are automatically fulfilled. That is, at the Nash equilibrium solution each agent evaluates the future environmental state in the same way. 6

5 In this paper we ignore the issues of existence and uniqueness. Some comments on these issues in a slightly di↵erent framework are given in [START_REF] Bréchet | Coalition Theory and Integrated Assessment Modeling: Lessons for Climate Governance[END_REF]. In the economic-environmental model in [START_REF] Greiner | Growth and climate change: threshold and multiple equilibria[END_REF], for example, the solution does not need to be unique.

6 This is the place where the assumption that all agents use the same environmental model h s at any given instant s plays a role. If each agent uses its own model h s j for the environment, then the definition of the Nash equilibrium solution becomes more complicated and depends on the information pattern: do the rest of the agents know what is the environmental model used by the agent j, or they know only the emission mapping (12). In the later case only a slight modification of the above definition of Nash equilibrium is needed. In the former case the definition of a Nash solution is more complicated. The numerical calculation consists of iterating the fixed point system (13)-( 16). Between 3 and 7 iterations give enough accuracy in the numerical investigation in Section 6. Now we continue with the definition of the model predictive rational behavior of the economic agents. Let us fix a time-step ✏ > 0 and set s k := i✏, k = 0, 1, . . .. At time s = s 0 = 0 the agents determine the Nash equilibrium controls {v s 0 i (t)} resulting from the models f s 0 j , g s 0 j , h s 0 that the agents use at time s 0 , and from the measurements x s 0 j , y s 0 of the states. The so obtained controls are implemented, however, only in the time-interval [s 0 , s 1 ]. Then the agents update their models and measure the actual state x s 1 j , y s 1 . The agents determine the Nash equilibrium controls {v s 1 i (t)} resulting from the updated models f s 1 j , g s 1 j , h s 1 and implement them in the time interval [s 1 , s 2 ]. The same procedure repeats further on. The resulting control policies are

v✏ j (t) := v s k j (t) for t 2 [s k , s k+1 ], k = 0, 1 . . . .
The time-step ✏ can be viewed as the length of the commitment periods defined under a legally binding international agreement, like the Kyoto protocol. However, both for mathematical convenience and due to the continual and non-synchronized adjustment of the policies of the agents at micro level, we eliminate the dependence of the control policies on the choice of ✏ letting it tend to zero.

Definition 1. Every limit point of any sequence v✏ j defined as above in the space L loc 1 (0, 1) when ✏ ! 0 (if such exists) will be called Model Predictive Nash Equilibrium (MPNE) policy. 7 In practice the time-step at which the actual state of the environment is updated may be many years long due to the slow change of the environment and the relatively high fluctuations from the trend. However, the model updates may take place more frequently due to the relatively faster change of the economic states and the progress in the modeling methodologies.

We outline the particular case in which an agent j chooses her model f s j , g s j independent of the environmental state y. That is, in her current control policy agent j does not take into account the influence of the future changes in the environmental state on her economy. Accordingly, such an agent disregards the impact of her economic activities on the environment, hence the environmental component ( 10) is irrelevant for her decisions. In [START_REF] Bréchet | Coalition Theory and Integrated Assessment Modeling: Lessons for Climate Governance[END_REF] we interpret such an agent as doing Business as Usual (BaU). A BaU agent disregards her interconnections with the environment and, consequently does not abate emissions. We stress that the above notion of BaU di↵ers from the one used in the literature (see e.g. [START_REF] Nordhaus | A Regional Dynamic General-Equilibrium Model of Alternative Climate-Change Strategies[END_REF], where BaU is an agent who does not abate, although having a foresight about the influence of the future environmental changes on the economy.

7 A sequence v ✏ converges to v in L loc 1 (0, 1) if R T 0 |v ✏ (t) v(t)
| dt converges to zero for every T > 0.

In the above consideration the models of the individual economic dynamics and objectives of the agents, as well as the model of the environment, are considered as given, although changing with the time in a non-anticipative way (the future changes in the models are not known, hence not involved in the formation of the current control policies). In the next section we partly endogenize the evolution of the models that agents employ by using a simple version of adaptive learning.

Adaptive behavior

At any time s the agents use models (given by the triplet (f s j , g s j , h s )) to determine their policy for some period of time after s, as described in the previous subsection.

In this subsection we address the following question: how the agents change the models that they use, depending on the newly available measurements of the actual economic and environmental states?

A variety of data assimilation techniques can employed for this purpose, out of which a simple adaptive learning is chosen in this paper.

For the sake of clarity and for numerical simulations we consider here only the benchmark model. The environment is assumed to be relevantly described by equations ( 5), ( 6), thus in this case h s = h for all s. Moreover, we apply adaptive learning to only two crucial uncertain factors that may vary with the time at which the control decisions are taken and that essentially influence the behavior of the agents: the damage function which represents the e↵ect of the climate change on the economy, and the discount rate used by the agents in the formulation of their future objectives. It is reasonable to apply adaptive learning to many other economic factors, such as future productivity, ⇡ i (t), future labour l i (t), future emission per output, ⌘ j (t), future natural emission, ⌫(t), etc., but here we assume a perfect knowledge for their evolution.

In the benchmark case the model ( 9)-( 11) that agent j uses at time s reads as

kj = j k j + [1 u j c j (a j )] ⇡ j ' s j (⌧ ) k j j l 1 j i , k j (s) = k s j , (17)
⌧ = (m) ⌧ + d(m), ⌧ (s) = ⌧ s , ( 18
) ṁ = µ m + (1 a j ) ⌘ j ⇡ j ' s j (⌧ ) k j j l 1 j i + ēj + ⌫(⌧ ), m(s) = m s , ( 19
)
u j (t), a j (t) 0, u j (t) + a j (t)  1. ( 20
)
To complete the benchmark agent's model we consider a particular objective function g s j in (8) defined as

Z 1 s e r s j t ⇥ u j (t) ⇡ j (t) ' s j (⌧ (t)) (k j (t)) j (l j (t)) 1 j ⇤ 1 ↵ dt, ( 21
)
where r s j is the discount rate used by agent j at time s. As already said, the model components that the agent j updates at time s (based on the available measurements for k j (t) and ⌧ (t) till time s) are the damage function ' s j (⌧ ) the discount rate r s j .

Updating the damage function

In the next paragraphs we analyze how agents with diverse level of knowledge and concerns or with diverse opinion about the reliability of the presently used environmental models and monitoring (in our case ( 18), ( 19)) may build their formal estimates about the influence of the global warming on their regional economic e ciency.

Below it will be assumed that the true damage on the productivity in the region of agent j caused by a temperature increase ⌧ above the pre-industrial level is given by the formula

'(⌧ ) = 1 1 + ✓ ⇤ ⌧  , ( 22
)
where  is a known constant and the value of ✓ ⇤ is not known to the agent (considering also  or more parameters as unknown does not bring principal di↵erence). The above constants are agent-specific, but we skip the index j in the notations since the considerations below apply to an individual agent.

As a specification of the benchmark model we assume that at any time s instead of the "true" damage function ( 22) for her region agent j uses the following one:

' s (⌧ ) = 1 1 + ✓ s ( s ⌧ s + (1 s )⌧ )  , ( 23
)
where ✓ s represents the current estimate of the true value ✓ ⇤ , ⌧ s is the measured average temperature at time s, and s 2 [0, 1] is an additional parameter chosen by the agent at the current time s. As argued below, this parameter reflects the level of confidence in the environmental model ( 18), (19). Notice that at the current time s the temperature is ⌧ (s) = ⌧ s , hence the evaluation of the damage function gives

' s (⌧ (s)) = 1 1 + ✓ s ⌧ (s)  . ( 24
)
Since the true value of the damage is measurable, the agent may calculate the value ✓s that fits to the current measurements of ⌧ (s) and '(⌧ (s)). Due to the uncertainties in the measurements the agent evaluates the parameter ✓ s to be used in her current model as

✓ s = ✓ s + "⇢ s ( ✓s ✓ s ),
where ✓ s is the agent's estimation of ✓ prior to time s (which depends on past measurements).

The parameter ⇢ s 2 [0, 1] reflects the agent's uncertainty about the currently estimated factors: temperature, capital stock, economic output: the lower is the confidence of the agent, the smaller is ⇢ s . The parameter " > 0 is the time step for updating the damage function, as in the preceding subsection. In the limit case with " ! 0 and ⇢ s = ⇢ the value ⇢ can be interpreted as the exponential decay rate of the error:

✓ ⇤ ✓ s = e ⇢s (✓ ⇤ ✓ 0
). On the other hand, at time s the agent employs the damage function in her long-run investment/abatement planning model ( 17)-( 21), as described in the previous subsection. Since the temperature ⌧ may change in the future and the agent uses the environmental model ( 18), ( 19) to predict this change, the anticipated error in ⌧ produced by the model may distort the predicted damage rate. To take into account these uncertainties the agent modifies the damage function ( 24) in the way given by ( 23). The argument for choosing a damage function in the form of ( 23) is shortly explained below, taking for simplicity the value  = 2.

At time s the agent has updated her damage function ' s by choosing the new parameter ✓ s in (24) as described above. However, the agent realizes that the true temperature at time t > s may be di↵erent from the one resulting from the model ( 18), ( 19), with an error ⇠ = ⇠(t).

Then the anticipated value of the damage function, when using (23) at time t and with the true temperature ⌧ = ⌧ (t) will be

1 1 + ✓ s ( ⌧ s + (1 )(⌧ + ⇠) 2 ) .
Since ⇠ can be viewed as a random variable (although its distribution is unknown), the rational agent would try to minimize by choosing the parameter 2 [0, 1] the expectation

E ✓ 1 1 + ✓ s ⌧ 2 1 1 + ✓ s ( ⌧ s + (1 )(⌧ + ⇠) 2 ) ◆ 2 ! .
Having in mind that ✓ ⇤ , hence also ✓, is a small number (✓ ⇤ = 0.0054 according to [START_REF] Nordhaus | A Review of the Stern Review on the Economics of Climate Change[END_REF], one can reasonably represent the above expression as

✓ 2 s E ⇣ ( + 2⌧ ) + 2( + ⌧ )(1 )⇠ + (1 ) 2 ⇠ 2 2 ⌘ + O(✓ 3 s ), (25)
where = ⌧ s ⌧ and O(")/" is bounded when " ! 0. From here one can make the following essential observation:

(i) if the agent trusts the employed environmental model (hence ⇠ = 0), then this agent would choose = 0, which gives value zero in the error-expectation formula (25);

(ii) if the agent does not believe in the long run trend of the temperature change (assuming = 0) and anticipates an error ⇠ 6 = 0 of the model-base prediction for the temperature, then this agent would choose = 1.

The analysis may be continued by considering the optimal choice of the parameter , that is, choosing the that minimizes the expression in (25). We skip this technical (and not very precise) consideration, which suggests that an agent whose opinion about the accuracy of the environmental model ( 18), ( 19) measured by P 4 n=1 |E(⇠ n )| is large relative to the expected by the agent temperature change = ⌧ ⌧ s (a "skeptical" agent), then this agent would choose closer to 1, while an agent who trusts more the model and anticipates a temperature increase would choose closer to zero. Of course, the same learning procedure as for ✓ can be applied also for .

Summarizing, one may say that the choice of the predictor for the future damage caused by the global warming is a subjective decision of the agent, which can be formally characterized by the following statement in terms of the parameters ⇢ s and s in the update of the damage function: agents who are sceptic about the measurements (and the model) of the economy and about the measurement of the current temperature choose lower value of ⇢ s ; agents who are sceptic about the credibility of the employed environmental model choose a higher value for the parameter s 2 [0, 1].

We note that an agent choosing = 1 makes no use of the environmental model at all, since this agent always takes the current temperature as a proxy for the future one. Thus = 1 represents a BaU-agent. In the present framework the agents are distinguished by their parameters ⇢ s and s and the BaU agent is an extreme case in a continuous variety of agents.

Updating the discount rate

The discount rate r s j chosen by agent j at time s in its utility function represents her opinion about the value of the present utility relative to a future one. Of course, the discount rate r s j depends on the agent's view on the future uncertainty, but in the context of the environmental concerns it depends also on the agent's per capita wealth. As one can clearly see from the daily practice, a rich country tends to be more far-sighted than a poor country in which the political and the economic policies are more myopic. Similar suggestions are given in the economic literature (see e.g. [START_REF] Lawrence | Poverty and the rate of time preference: evidence from panel data[END_REF][START_REF] Samwick | Discount rate homogeneity and social security re-form[END_REF]. Therefore, it is reasonable to assume that the discount rate of agent j depends on her per capita wealth measured at time s by k j (s)/l j (s) = k s j /l j (s) (where labor is proportional to the population size). Thus, at time s the agent would have an endogenous discount rate r s j = R(k s j /l j (s)). The particular specification of the function R used in the subsequent numerical analyses is given in Section 6.

Some simulation results

The agents in the benchmark model ( 17)-( 21) are heterogeneous with respect to all involved parameters. However, we shall focus on the heterogeneity with respect to their behavioral features, represented by the damage function ' s j (⌧ ) and the discount rate r s j used at time s by agent j.8 (of course it is not assumed to be known to the agents).

Finally, the e↵ect of CO 2 concentration on the average temperature increase is captured by the standard function d(m) = 0.5915 ln(m/m ⇤ 0 ), where m ⇤ 0 = 596.4 GtC is the preindustrial CO 2 concentration in the atmosphere. All the specifications are within usually suggested ranges.9

Scenarios description

For the behavioral parameters ' s j (•) in ( 23) and the discount rate r s j we consider several scenarios as described below.

Scenarios 1 and 2. These are two "extreme" scenarios. In Scenario 1 both agents are myopic and totally neglect the environmental dynamics in their decisions, adapting to the temperature change only post factum (this is what we called BaU agent in the end of Section 4). Precisely, agent R has the parameters r s 1 = r 1 = 0.02 (myopic), ✓ s = ✓ ⇤ (evaluates correctly her damage at the current temperature ⌧ s ), = 1 (ignores the prediction for future change of temperature), ⇢ is irrelevant. Agent P has the same parameters with the only di↵erence that ✓ s = ✓ ⇤ /6 and ⇢ = 0 (underestimates the damage of the current temperature and does not learn).

In Scenario 2 we consider that both agents as far-sighted, perfectly informed about the environmental dynamics and the damage function. Precisely, for both agents r s i = 0.005, ✓ s = ✓ ⇤ , s = 0. In fact, in this case the MPNE coincides with the usual Nash equilibrium due to the perfect foresight of both agents. Scenario 3. In the third scenario we take into account that agent P may learn and may become less sceptic with experience, still remaining myopic. The role of this scenario is to exhibit the e↵ect of learning. Formally, agent R is exactly as in Scenario 2, while agent P has

r s 2 = 0.02, ✓ s = ✓ ⇤ e ⇢s (✓ ⇤ ✓ 0 )
, with ✓ 0 = ✓ ⇤ /6, ⇢ = 0.03465, and s = e ⇢s . The chosen value of ⇢ means that agent P reduces the error in ✓ by half and the value of the distrust parameter from 1 to 0.5 in 20 years.

Scenario 4. This last scenario involves endogenous discount for agent P, all the rest is as Scenario 3. Agent P is initially myopic (r 0 2 = 0.02), but her discount rate decreases quadratically with the per capita stock of capital till the value r = 0.005 is achieved when the capital stock reaches the initial value of the initial capital stock of agent R. That is, at this point agent P starts discounting as low as agent R, who has r 1 = 0.005 all the time.

In all scenarios the agents use the investment and the abatement rates, u i (t) and a i (t), as policy instruments. Of course, in Scenario 1 the agents have no reasons to abate (hence a j (t) = 0).

Simulation results

The four scenarios are graphically depicted in figures1-3. Our results support the intuitive idea that the more you know the better you do. Indeed, Scenario 2 represents the economy with the best informed individuals who care the most about the future. On the other side, agents in Scenario 1 are myopic about global warming and do not care much about the future. On top of this, they are unable to learn. This implies that they never revise their vision about global warming nor do they increase their concern about the future. They are stubborn and short-sighted, which are rather common psychological features in the the real word. As a result, in short, Scenario 2 provides the highest consumption and GDP per capita in the long-term, incurring in the lowest increase in temperature while Scenario 1 provides the worst results.

A striking result to be mentioned about these two scenarios is the shape of GDP (or consumption) over time. After an increase in GDP per capita at the beginning of the simulation period, Scenario 1 displays a shrink in the level of the world GDP by 25% in 90 years. This reflects the economic consequences of neglecting climate change. Actually, in Scenario 1 World GDP declines by 0.3% per year between time 50 and 140. In the same period of time world GDP increases by 0.3% per year on average in Scenario 2. This shows how a more realistic definition of a BaU translates into climate costs on the economy. Let us remind that we use the same calibration parameter values as [START_REF] Nordhaus | A Review of the Stern Review on the Economics of Climate Change[END_REF]. What makes the di↵erence is the rational behind the scenarios. Clearly, this result sheds a new light on the potential costs of no-action against global warming. Most people agree that emission abatement is costly but forget that climate change itself is costly to the economy. This simulation reveals that these costs may be much higher than usually appraised with IAMs because of they ill-define what is business-as-usual.

After 100 years, Scenario 2 induces a temperature increase of less than 3 C, providing a GDP of 30,000 USD per capita. Scenario 1 provides higher consumption during the first 90 years, given that agents do not abate. Nevertheless, because they incur in the largest emissions, their productivity is harmed the most. In order to preserve a high level of consumption during the entire period, agents in Scenario 2 abate more than 50% of their emissions, attaining up to 70% after 50 years. Given that agents do not abate at all in Scenario 1, temperature increases by more than 6.5 C after 100 years, which dampens their productivity and hence their income.

Naturally, Scenarios 3 and 4 provide results that lie in between Scenario 1 and 2, getting closer to Scenario 2 as the amount of information and concern about the future increase. Scenarios 3 and 4 are similar during the first 20 years. Di↵erences arise after and one can observe the behavioral di↵erences between agents who update their discount rate, and those who do not. In Scenario 3, agent R knows exactly the damages induced by global warming, and she discounts future at at a higher rate. On the other side, agent P, who does not abate at the beginning, starts soon doing so, adding her e↵ort to agent R's. However, agent P's lack of care about the future impedes larger abatement e↵orts. Consequently, GDP per capita and consumption are larger than in Scenario 1, and temperature increase after 100 years is 3.75 C above the preindustrial level. Therefore, although this economy starts abating a 10% of their emissions on the average, it gradually increases abatement e↵orts until 50% after one century.

Finally, building on Scenario 3, we allow agent P in Scenario 4 to become more patient as she becomes wealthier. Hence, as agent P accumulates wealth with time, she starts caring more about its future and increases her abatement e↵ort accordingly. We can see in figure 2 that the average abatement rate equalizes Scenario 2's after 70 years and then over-reach it for some decades. Agents get very close to Scenario 2 in terms of consumption and GDP as well, but they cannot catch them because of the damages accumulated on productivity during the first 60 years.

It is interesting to notice that, although Scenario 2, 3 and 4 are relatively close in terms of temperature increase, they display contrasting profiles regarding GDP, consumption, and abatement policies. For example, in Scenario 4 abatement e↵orts are stronger than in the "optimistic" Scenario 2 for most time periods because of the delay incurred by the endogenous discounting. It shows that the idea of relying on endogenous discounting (driven by economic development) to cope with global warming is inadequate because it takes too long.

It is interesting to compare the aggregated discounted utility of agent R (given by (21) on a 200 years long horizon) in scenarios 2 and 4. The discount factor of agent R is r = 0.005 in each of these scenarios, therefore the results are comparable: the values are 52,995 in Scenario 2 and 37,022 in Scenario 4. Since agent R has exactly the same parameters in the two scenarios, the reason for the large di↵erence of her welfare is caused by agent P. Notice that due to learning and due to the endogenous discount, after 100 years agent P in Scenario 4 behaves exactly as agent P in Scenario 2: has a perfect knowledge and discounts with r = 0.005. However, due to the delay in the evolution of agent P from a myopic ignorant to a far-sighted knowledgeable agent (as R is from the very beginning), agent R loses about 30% of her 200-years utility. This result shows how important it is for the rich country to help the poor one to develop, because both share the same common good, climate.

Conclusion

The purpose of this article was to extend the standard integrated assessment framework applied to climate change by incorporating model predictive control and adaptive behavior. Model predictive control is employed due to the uncertainties about the future environment. It allows agents to redefine their optimal strategy on a regular basis, on the grounds of the observed changes in the world or in the agents' time preferences (endogenous discounting in our model).

With this setting, agents are rational (they adopt the optimal policy) but revise it (with some inertia) as the word changes. Adaptive behavior (or learning) is involved since the agents gradually improve their knowledge about the world (the interconnection between environment and economy, in our model). These ingredients are particularly relevant in the context of global warming. We provide a generic theoretical model encompassing all elements of an integrated assessment model. In particular, we define an innovative concept of Model Productive Nash Equilibrium (MPNE) to characterize an economy with many countries. Simulations show, among other results, how the trajectory of the economy can be a↵ected by the adaptive configuration. In particular, a pessimistic configuration (pessimistic, but maybe not so far from reality) displays a shrink in the world GDP due to the adverse e↵ects of climate change and the persistent agent's will to disregard them. This new framework would deserve to be extended in several directions. A first natural one would be to split the word in many countries or regions. In this case, strategic interactions among countries would become a new ingredient of the framework.

  )-(11) of the n agents at time s are regarded as defining a di↵erential game in which the players (that is, the agents) implement (an open-loop) Nash equilibrium solution. In the next lines we clarify what is the meaning of the Nash equilibrium solution in the present context.

11

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.29

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19Figure 1 :

 1 Figure 1: World GDP per capita.

20Figure 2 :

 2 Figure 2: Average abatement rate.

21Figure 3 :

 3 Figure 3: Temperature increase.

  1 i is the economic output of agent i. It is assumed that the emission (without any costly abatement) is proportional to the output Y i , namely equals ⌘ i (t)Y i (t), where ⌘ i (t) takes into account the change of emission per output due to technological change.

Table 1 :

 1 Values of stationary parameters

	Economic parameters
	intertemporal elasticity of substitution ↵	0.5
	depreciation rate	i	0.1
	capital elasticity	i	0.75
	initial productivity of R	⇡ 1 (0) 1.75
	initial productivity of P	⇡ 2 (0) 1.17
	initial emission rate	⌘ i (0) 0.0427
	Climate parameters
	temperature stability rate		0.11
	CO 2 absorption rate	µ	0.0054
	natural emission	⌫	3.211

Twenty-three years later, Nordhaus publishes a new book entitled "A question of balance". Is it a new illustration of the slow pace between science and policy?

Documents de Travail du Centre d'Economie de la Sorbonne -2012.29

We reproduce here the DICE equations taken from[START_REF] Nordhaus | An Optimal Transition Path for Controlling Greenhouse Gases[END_REF], respecting notation, calibration and interpretation.

See also[START_REF] Bréchet | Coalition Theory and Integrated Assessment Modeling: Lessons for Climate Governance[END_REF] for a survey on the use of game theory with IAMs, or[START_REF] Yang | Strategic Bargaining and Cooperation in Greenhouse Gas Mitigations -An Integrated assessment Modeling Approach[END_REF] for an application with the RICE model.

The assumption that all agents use the same environmental model is made only for the sake of simplicity. An additional comment on this is given in Footnote 6. 10 Documents de Travail du Centre d'Economie de la Sorbonne -2012.29

The adjective "behavioral" indicates dependence on subjective attitude of the agent. Moreover, the agents have no foresight about the damage function and the interest rate that they will use in the future. 16 Documents de Travail du Centre d'Economie de la Sorbonne -2012.29

See the IPCC (2007),Nordhaus (2008),Stern (2008) andYang (2009).

⇤ This research was initially supported by the Belgian Science Policy under the CLIMNEG III Project (SD/CP/05A). The third author was partly financed by the Austrian Science Foundation (FWF) under grant No I 476-N13. The paper was finalized while Th. Bréchet was visiting research fellow at the Grantham Institute for Climate Change at Imperial College London, and visiting professor at the European University at St

Model calibration

The individual damage function used by an agent at time s is characterized in the previous section by three parameters: the knowledge about the "true" damage function prior to s, denoted by ✓ s , the learning parameter ⇢ s , and the "trust" parameter s . Diverse values of these parameters and of the discount rate r s j allow to cover agents with rather di↵erent behavior, including such doing BaU (see the end of Section 4), myopic versus far-sighted agents, sceptical about the accuracy of the climate predictions versus "believers", etc. Given any configuration of agents, their (interconnected) economic behaviour is defined in Section 4 by the Model Predictive Nash Equilibrium (MPNE).

Although the concept of MPNE applies to any number of agents, in the simulation results below we consider for more transparency only two agents. The first agent, named agent R (from "Rich"), is presumably richer, hence less discounting the future, has better knowledge on the damage function and trusts more the predictions for the climate change. The second one, agent P (from "Poor"), is presumably poorer, hence more myopic, has bad knowledge on the damage function but may learn with experience, and is skeptical about the predictions for the climate change. In the benchmark model agents R and P are indexed by i = 1 and i = 2, respectively.

The prototypes of these two agents in some of the simulations below are the OECD and the non-OECD countries, respectively. Therefore in all simulations 74.7% of the world physical capital stock (estimated as 733.2 trillion USD in 2005) belongs to agent R, while the rest of 25.3% belongs to agent P. Thus the initial data for the economy are k 0 1 = 547.7004 and k 0 2 = 185.4996 trillion USD. On the other hand, the population of agent R is 18.2% of the total, which gives l 1 = 0.184L, l 2 = 0.816L with the total population L = 6464.75 million. The initial data for the environment are m 0 = 808.9 and ⌧ 0 = 0.7307, representing the concentration of CO 2 in GtC and the increase of temperature above the pre-industrial level in 2005, respectively. The constant parameters of the benchmark model are given on Table 1. The calibration year is 2005.

The functional parameters in the benchmark model are specified as follows.

Due to technological progress the productivity ⇡ i (t) is assumed linearly increasing with time so that technology is 25% more e cient after 175 years, that is, ⇡ i (175) = 1.25⇡ i (0) for i = R, P .

The technological progress reduces the emission per unit of output (without abatement) at an exponential rate 0.00384, which corresponds to a decrease by 25% in 75 years: ⌘ i (t) = e 0.00384t ⌘ i (0). The cost-of-abatement function c i (a) = c(a) is specified as c(a) = 0.01a/(1 a), which implies that reducing emission by 50% incurs cost of 1% of GDP. The true damage function for all agents is assumed to be


Yannick Privat
email: yannick.privat@bretagne.ens-cachan.fr

Emmanuel Trélat
email: emmanuel.trelat@upmc.fr

Enrique Zuazua
email: zuazua@bcamath.org

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

, and has been the object of the talk of the second author at Journées EDP in 2012.

In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set Ω ⊂ IR n , with Dirichlet boundary conditions. The observation is done on a subset ω of Lebesgue measure |ω| = L|Ω|, where L ∈ (0, 1) is fixed. We denote by U L the class of all possible such subsets. Let T > 0. We consider first the benchmark problem of maximizing the observability energy T 0 ω |y(t, x) 2 dx dt over U L , for fixed initial data. There exists at least one optimal set and we provide some results on its regularity properties. In view of practical issues, it is far more interesting to consider then the problem of maximizing the observability constant. But this problem is difficult and we propose a slightly different approach which is actually more relevant for applications. We define the notion of a randomized observability constant, where this

constant is defined as an averaged over all possible randomized initial data. This constant appears as a spectral functional which is an eigenfunction concentration criterion. It can be also interpreted as a time asymptotic observability constant. This maximization problem happens to be intimately related with the ergodicity properties of the domain Ω. We are able to compute the optimal value under strong ergodicity properties on Ω (namely, Quantum Unique Ergodicity). We then provide comments on ergodicity issues, on the existence of an optimal set, and on spectral approximations. [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF] The optimal observability problems Let T > 0 fixed, and let Ω ⊂ IR n be a bounded open connected set. We consider in parallel the wave equation

∂ 2 y ∂t 2 = y, (1)
and the Schrödinger equation

i ∂y ∂t = y, (2)
posed in Ω, with Dirichlet boundary conditions. For any measurable subset ω ⊂ Ω, we observe the restrictions of the solutions of (1) and of (2) to ω.

For the wave equation, the notion of observability is the following. For all (y 0 , y 1 ) ∈ L 2 (Ω, C) × H -1 (Ω, C), there exists a unique solution y ∈ C 0 (0, T ; L 2 (Ω, C)) ∩ C 1 (0, T ; H -1 (Ω, C)) of (1) such that y(0, •) = y 0 (•) and y t (0, •) = y 1 (•). We say that an observability inequality holds whenever there exists C > 0 such that C (y 0 , y 1 ) 2

L 2 (Ω,C)×H -1 (Ω,C) T 0 ω |y(t, x)| 2 dxdt, (3)
for all (y 0 , y 1 ) ∈ L 2 (Ω, C) × H -1 (Ω, C). It is well known that within the class of C ∞ domains Ω, this observability property holds if the pair (ω, T ) satisfies the Geometric Control Condition in Ω (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]), according to which every ray of geometrical optics that propagates in Ω and is reflected on its boundary ∂Ω intersects ω within time T . We define the observability constant by

C (W ) T (χ ω ) = inf T 0 ω |y(t, x)| 2 dx dt (y 0 , y 1 ) 2 L 2 (Ω,C)×H -1 (Ω,C) (y 0 , y 1 ) ∈ L 2 (Ω, C) × H -1 (Ω, C) \ {(0, 0)} . (4)
For the Schrödinger equation, the observability goes as follows. For every y 0 ∈ L 2 (Ω, C), there exists a unique solution y ∈ C 0 (0, T ; L 2 (Ω, C)) of ( 2) such that y(0, •) = y 0 (•). The system is said observable whenever there exists C > 0 such that

C y 0 2 L 2 (Ω,C) T 0 ω |y(t, x)| 2 dxdt, (5)
for every y 0 ∈ L 2 (Ω, C). If there exists T * such that the pair (ω, T * ) satisfies the Geometric Control Condition then the observability inequality (5) holds for every T > 0 (see [START_REF] Lebeau | Contrôle de l'equation de Schrödinger[END_REF]). We define the observability constant by

C (S) T (χ ω ) = inf T 0 ω |y(t, x)| 2 dx dt y 0 2 L 2 (Ω,C) y 0 ∈ L 2 (Ω, C) \ {0} . (6)
In the sequel we fix L ∈ (0, 1), and we define the set

U L = {χ ω | ω is a measurable subset of Ω of Lebesgue measure |ω| = L|Ω|},
where χ ω is the characteristic function of ω. Our objective is to maximize the observability properties of the wave or Schrödinger equation over the class U L , in a certain sense. We consider the two following mathematical problems.

First problem (fixed initial data).

• Wave equation (1): given (y 0 , y 1 ) ∈ L 2 (Ω, C)×H -1 (Ω, C), we investigate the problem of maximizing the functional

G T (χ ω ) = T 0 ω |y(t, x)| 2 dx dt, (7)
over U L , where y ∈ C 0 (0, T ;

L 2 (Ω, C)) ∩ C 1 (0, T ; H -1 (Ω, C))
is the solution of (1) such that y(0, •) = y 0 (•) and ∂y ∂t (0, •) = y 1 (•).

• Schrödinger equation (2): given y 0 ∈ L 2 (Ω, C), we investigate the problem of maximizing the functional G T defined by (7) over U L , where y ∈ C 0 (0, T ;

L 2 (Ω, C)) is the solution of (2) such that y(0, •) = y 0 (•).
This first problem is the simplest possible and is interesting to be analyzed from the mathematical point of view, but in view of practical issues it is necessary to define an optimization problem that is independent on the initial data. Indeed, this kind of problem is motivated e.g. by the application to the optimal placement of sensors, in view of reconstructing some environment by means of local measures. The most natural criterion is then the observability constant itself, defined by (4) or by [START_REF] Chen | Exponential decay of energy of evolution equations with locally distributed damping[END_REF]. But, firstly, the problem of maximizing (4) or (4) over U L happens to be very difficult1 , and secondly, the observability constant, defined by an infimum, is probably a too much pessimistic criterion in practice: indeed when an engineer realizes a large number of measures (and would like these measures to be of high quality as far as possible), it is probable that only a few number of them will be "of low quality", corresponding to the observability constant.

In view of this remark, it makes sense to consider rather an averaged version of the observability inequality over randomized initial data. The procedure that we propose here goes as follows. It is inspired by the works of N. Burq and N. Tzvetkov on nonlinear partial differential equations with random initial data (see [START_REF] Burq | Large-time dynamics for the one-dimensional Schrödinger equation[END_REF][START_REF] Burq | Random data Cauchy theory for supercritical wave equations. I. Local theory[END_REF]).

Let (φ j ) j∈IN * be a Hilbertian basis of L 2 (Ω) consisting of eigenfunctions of the Dirichlet Laplacian operator on Ω, associated with the negative eigenvalues (-λ 2 j ) j∈IN * . For all initial data (y 0 , y 1 ) ∈ L 2 (Ω, C) × H -1 (Ω, C), the corresponding solution y (1) can be expanded as

y(t, x) = +∞ j=1
a j e iλ j t + b j e -iλ j t φ j (x), [START_REF] Gérard | Ergodic properties of eigenfunctions for the Dirichlet problem[END_REF] where

a j = 1 2 Ω y 0 (x)φ j (x) dx - i λ j Ω y 1 (x)φ j (x) dx , b j = 1 2 Ω y 0 (x)φ j (x) dx + i λ j Ω y 1 (x)φ j (x) dx . (9)
for every j ∈ IN * . Note that (a j ) j∈IN * and (b j ) j∈IN * belong to 2 (C) and that (y 0 , y 1 ) 2

L 2 ×H -1 = 2 +∞ j=1 (|a j | 2 + |b j | 2 )
. Following Burq and Tzvetkov, we now randomize some given initial data (y 0 , y 1 ) as follows. Let (β ν 1,j ) j∈IN * and (β ν 2,j ) j∈IN * two sequences of random laws on a probability space (X, A, P) that are independent, identically distributed with a common distribution θ for which there exists c > 0 such that IR e γx dθ(x) e cγ 2 . For example, Bernoulli or gaussian random laws can be considered. For every ν ∈ X, we set

y 0 ν (t, •) = +∞ j=1 β ν 1,j a j + β ν 2,j b j φ j (•), y 1 ν (t, •) = +∞ j=1 iλ j β ν 1,j a j -β ν 2,j b j φ j (•),
so that the corresponding solution of ( 1) is

y ν (t, •) = +∞ j=1 β ν 1,j a j e iλ j t + β ν 2,j b j e -iλ j t φ j (•).
Then, instead of considering the deterministic observability inequality (3), we define the randomized observability inequality

C (W ) T,rand (χ ω ) (y 0 , y 1 ) 2 L 2 (Ω,C)×H -1 (Ω,C) E T 0 ω |y ν (t, x) 2 | dx dt , (10)
where

C (W ) T,rand (χ ω ) = 1 2 inf (a j ),(b j )∈ 2 (C) P +∞ j=1 (|a j | 2 +|b j | 2 )=1 E   T 0 ω +∞ j=1 β ν 1,j a j e iλ j t + β ν 2,j b j e -iλ j t φ j (x) 2 dx dt   , (11)
The same procedure is made for the Schrödinger equation, leading to define the randomized observability constant

C (S) T,rand (χ ω ) = inf (c j )∈ 2 (C) P +∞ j=1 |c j | 2 =1 E   T 0 ω +∞ j=1 β ν j c j e iµ j t φ j (x) 2 dx dt   . (12)
We have the following result.

Theorem 1. [ [START_REF] Privat | Optimal observability of wave and Schrödinger equations in ergodic domains[END_REF]] For every measurable subset ω of Ω, there holds

2 C (W ) T,rand (χ ω ) = C (S) T,rand (χ ω ) = T inf j∈IN * ω φ j (x) 2 dx.
This motivates the consideration of the second problem below.

Second problem. We investigate the problem of maximizing the functional

J(χ ω ) = inf j∈IN * ω φ j (x) 2 dx, (13)
over U L .
This criterion is a spectral energy (de)concentration one. Our main contribution is to provide evidence of the intimate relations between this problem and strong ergodicity properties of the set Ω.

This second problem can be motivated in another way, by considering the following time averaging procedure. First of all, note that, for all initial data (y 0 , y

1 ) ∈ L 2 (Ω, C) × H -1 (Ω, C), the quantity 1 T T 0 ω |y(t, x)| 2 dx dt,
where y is the corresponding solution of the wave equation ( 1), has a limit as T tends to +∞. This leads to define the time asymptotic observability inequality

C (W ) ∞ (χ ω ) (y 0 , y 1 ) 2 L 2 (Ω,C)×H -1 (Ω,C) lim T →+∞ 1 T T 0 ω |y(t, x) 2 | dx dt, (14)
where

C (W ) ∞ (χ ω ) = inf lim T →+∞ 1 T T 0 ω |y(t, x)| 2 dx dt (y 0 , y 1 ) 2 L 2 ×H -1 (y 0 , y 1 ) ∈ L 2 × H -1 \ {(0, 0)} . (15)
Similarly, for the Schrödinger equation, we define

C (S) ∞ (χ ω ) = inf lim T →+∞ 1 T T 0 ω |y(t, x)| 2 dx dt y 0 2 L 2 (Ω,C) y 0 ∈ L 2 (Ω, C) \ {0} .
(16) We have the following result. Theorem 2. [START_REF] Privat | Optimal observability of wave and Schrödinger equations in ergodic domains[END_REF] Assume that the domain Ω is such that every eigenvalue of the Dirichlet Laplacian operator is simple. For every measurable subset ω of Ω, there holds

2 C (W ) ∞ (χ ω ) = C (S) ∞ (χ ω ) = inf j∈IN * ω φ j (x) 2 dx = J(χ ω ).
Note that the assumption of the simplicity of the spectrum of the Dirichlet Laplacian is generic with respect to the domain Ω.

It can be observed that, although such optimal design problems have been widely investigated in the engineering literature (see e.g. [START_REF] Kumar | Optimal location of measurements for distributed parameter estimation[END_REF][START_REF] Morris | Linear-quadratic optimal actuator location[END_REF][START_REF] Van De Wal | A review of methods for input/output selection[END_REF] and references therein), few mathematical works exist that provide a theoretical rigorous analysis. We quote the remarkable articles [START_REF] Hébrard | Optimal shape and position of the actuators for the stabilization of a string[END_REF][START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF], where the problem of optimal domain is studied for the stabilization of one-dimensional wave equations with localized damping. These articles were actually the starting point of our own analysis. They are based on a thorough investigation through spectral considerations. In [START_REF] Privat | Optimal observability of the onedimensional wave equation[END_REF] we investigated the second problem presented previously in the one-dimensional case. We also quote the article [START_REF] Privat | Optimal location of controllers for the one-dimensional wave equation[END_REF] where we study the related problem of finding the optimal location of the support of the control for the one-dimensional wave equation. We refer to [START_REF] Privat | Optimal observability of wave and Schrödinger equations in ergodic domains[END_REF] for other references to related problems.

First problem

In this section, we consider fixed initial data (y 0 , y 1 ) ∈ L 2 (Ω)×H -1 (Ω) (resp., y 0 ∈ L 2 (Ω)) for the wave equation (1) (resp., for the Schrödinger equation ( 2)), whose Fourier coefficients are defined by [START_REF] Hassell | Ergodic billiards that are not quantum unique ergodic[END_REF]. Plugging ( 8) into [START_REF] Faure | Scarred eigenstates for quantum cat maps of minimal periods[END_REF] leads to

G T (χ ω ) = T 0 ω |y(t, x)| 2 dxdt = +∞ j,k=1 α jk ω φ i (x)φ j (x) dx, (17)
where

α jk = T 0 (a j e iλ j t + b j e -iλ j t )(ā k e -iλ k t + bk e iλ k t ) dt. (18)
The coefficients α jk , (j, k) ∈ (IN * ) 2 , depend only on the initial data (y 0 , y 1 ), and can be easily computed as

α jk = 2a j āk λ j -λ k sin (λ j -λ k ) T 2 e i(λ j -λ k ) T 2 + 2a j bk λ j + λ k sin (λ j + λ k ) T 2 e i(λ j +λ k ) T 2 + 2b j āk λ j + λ k sin (λ j + λ k ) T 2 e -i(λ j +λ k ) T 2 + 2b j bk λ j -λ k sin (λ j -λ k ) T 2 e -i(λ j -λ k ) T 2 (19)
whenever λ j = λ k , and

α jk = T (a j āk + b j bk ) + sin(λ j T ) λ j (a j bk e iλ j T + b j āk e -iλ j T ). ( 20
)
whenever λ j = λ k . We define the integrable function on Ω

ϕ(x) = T 0 |y(t, x)| 2 dt = +∞ i,j=1 α ij φ i (x)φ j (x). (21)
Then, from [START_REF] Morris | Linear-quadratic optimal actuator location[END_REF], there holds

G T (χ ω ) = ω ϕ(x) dx, (22)
for every measurable subset ω of Ω. With this expression, solving the first problem, that is the problem of maximizing the functional G T over U L , is quite immediate and we have the following result.

Theorem 3. [20]

There exists at least one measurable subset ω of Ω, solution of the first problem, characterized as follows. There exists a real number λ such that every optimal set ω is contained in the level set {ϕ λ}. Moreover, if Ω has a C ∞ boundary and if there exists R > 0 such that

+∞ j=0 R j j! A j y 0 2 L 2 + A j-1 y 1 2 L 2 1/2 < +∞, ( 23
)
in the case of the wave equation, and

+∞ j=0 R j j! A j y 0 L 2 < +∞, ( 24
)
in the case of the Schrödinger equation, where A = √ -(square root of the Dirichlet-Laplacian), then the first problem has a unique2 solution χ ω ∈ U L , satisfying moreover the following properties:

• there exists η > 0 such that d(ω, ∂Ω) > η, where d denotes the Euclidean distance;

• ω has a finite number of connected components;

• if Ω is symmetric with respect to an hyperplane, then ω enjoys the same symmetry property.

Remark 1. The conditions [START_REF] Zelditch | Note on quantum unique ergodicity[END_REF] or [START_REF] Zelditch | Ergodicity of eigenfunctions for ergodic billiards[END_REF] stipulate that the initial data share analyticity properties (note that they imply that y 0 and y 1 are analytic on Ω).

Remark 2. The optimal solution may not be unique whenever the function ϕ is constant on some subset of Ω of positive measure. If ϕ is constant equal to c on some subset I ⊂ Ω of positive measure, and if |{ϕ c}| < L|Ω| < |{ϕ > c}|, then there exists an infinite number of measurable subsets ω ⊂ Ω maximizing [START_REF] Van De Wal | A review of methods for input/output selection[END_REF], all of them containing the subset {ϕ > c}. The part of ω lying in {ϕ = c} can indeed be chosen arbitrarily.

Remark 3. In the one-dimensional case Ω = [0, π], one has φ j (x) = 2 π sin(jx) and λ j = j for every j ∈ IN * . If moreover T = 2pπ with p ∈ IN * , then α ij = 0 whenever i = j, and α jj = pπ(|a j | 2 +|b j | 2 ) for all (i, j) ∈ (IN * ) 2 , and therefore

G 2pπ (χ ω ) = +∞ j=1 α jj ω sin 2 (jx) dx. ( 25
)
Hence in that case the functional G 2pπ does not involve any crossed terms.

The second problem for this one-dimensional case was studied in details in [START_REF] Privat | Optimal observability of the onedimensional wave equation[END_REF]. In particular, in this case the non-uniqueness phenomenon can be exactly characterized in terms of Fourier series.

Remark 4. If we relax, even slightly, the analyticity assumptions [START_REF] Zelditch | Note on quantum unique ergodicity[END_REF], then the optimal optimal set ω is not necessarily unique and moreover may have an infinite number of connected components. In [START_REF] Privat | Optimal observability of wave and Schrödinger equations in ergodic domains[END_REF] we prove that, in the framework of Remark 3, there exist smooth initial data (y 0 , y 1 ) for the wave equation (resp., y 0 for the Schrödinger equation) defined on [0, π] for which the optimal set ω has a fractal structure (it is of Cantor type) and thus in particular has an infinite number of connected components.

Second problem and quantum ergodicity

The second problem sup

χω∈U L J(χ ω ) (26)
is posed on a set U L that does not have nice compactness properties. It is usual in optimal design problems to consider a convexified version of the problem. Here, the convex closure of U L for the weak star topology of L ∞ is

U L = a ∈ L ∞ (Ω, [0, 1]) Ω a(x) dx = L|Ω| . ( 27
)
Replacing χ ω ∈ U L with a ∈ U L , the convexified version of the second problem ( 26) is sup

a∈U L J(a), (28)
where

J(a) = inf j∈IN * Ω a(x)φ j (x) 2 (29) Remark 5. Note that sup χω∈U L inf j∈IN * Ω χ ω (x)φ j (x) 2 dx sup a∈U L inf j∈IN * Ω a(x)φ j (x) 2 dx, (30)
but it is not obvious to see whether the inequality is strict or not. Indeed, since the functional J under consideration fails to be lower semi-continuous, the usual Γ-convergence theory does not apply to our problem. This difficulty can be made evident in dimension one (see Remark 3). In that case, one has φ j (x) = 2 π sin(jx) for every j ∈ IN * , and it is easy to see that sup a∈U L J(a) = L and that the supremum is reached with the constant function a(•) = L. Besides, the sequence of functions χ ω N , where

ω N = ∪ N k=1 kπ N +1 -Lπ 2N , kπ N +1 + Lπ 2N for every N ∈ IN * , converges to the con- stant function a(•) = L for the weak star topology of L ∞ , but nevertheless, computations show that lim sup N →+∞ J(χ ω N ) < L.
In what follows, we make the following assumption on Ω.

Weak Quantum Ergodicity (WQE) property. There exists a subsequence of (φ 2 j ) j∈IN * converging to 1 |Ω| in weak star L ∞ topology.

Under this assumption, there holds sup

a∈U L inf j∈IN * Ω a(x)φ j (x) 2 dx = L, (31)
and moreover the supremum is reached with the constant function a = L on Ω. Hence, from now on we have sup

χω∈U L inf j∈IN * Ω χ ω (x)φ j (x) 2 dx L, (32)
and we wonder whether this inequality is strict or not, or in other words, whether there is a gap or not between the second problem and its convexified version.

Remark 6. The WQE assumption is always true in dimension one, as obvious consequence of Lebesgue Lemma. The situation is more complicated in dimension greater than one and we have the following facts.

• In a hypercube, or in the unit ball of IR n , WQE holds true.

• More generally, WQE follows from ergodicity properties. In [START_REF] Gérard | Ergodic properties of eigenfunctions for the Dirichlet problem[END_REF][START_REF] Hassell | Quantum ergodicity of boundary values of eigenfunctions[END_REF][START_REF] Zelditch | Ergodicity of eigenfunctions for ergodic billiards[END_REF] it is proved that, if the domain Ω is an ergodic billiard with piecewise smooth boundary, then the following property QE is satisfied.

Quantum Ergodicity (QE) property. There exists a subsequence of (φ 2 j ) j∈IN * of density one converging to 1 |Ω| in weak star L ∞ topology.

Obviously, QE implies WQE. Actually the result (known as Shnirelman Theorem) is stronger: a density-one subsequence of the probability measures µ j = φ 2 j dx converges weakly to the Liouville measure of Ω. Here the domain Ω is seen as a billiard where the flow moves at unit speed and bounces at the boundary according to the Geometric Optics laws.

To obtain a no-gap result, we need to strengthen WQE into the following assumption.

Quantum Unique Ergodicity (QUE) property. The whole sequence (φ 2 j ) j∈IN * converges to 1 |Ω| in weak star L ∞ topology.

Theorem 4. ( [START_REF] Privat | Optimal observability of wave and Schrödinger equations in ergodic domains[END_REF]) If Ω satisfies QUE, then

sup χω∈U L inf j∈IN * ω φ j (x) 2 dx = L, (33)
for every L ∈ (0, 1). In other words, under QUE there is no gap between the problem (26) and its convexified version (28).

Remark 7. The QUE property is very well-known in quantum physics and mathematical physics. The quantity µ j (ω) = ω φ 2 j (x) dx represents the probability of finding the quantum state in ω.

Obviously, QUE implies QE. Note that the QE property lets open the possibility of having an exceptional subsequence of φ 2 j converging weakly to some other measure. It may indeed happen that a subsequence (necessarily of density zero under QE) converges to an invariant measure like for instance a measure carried by closed geodesics or invariant tori generated by such geodesics. This energy concentration phenomenon is referred to as a (strong) scar and up to now remains a widely unknown issue (see [START_REF] Bonechi | Controlling strong scarring for quantized ergodic toral automorphisms[END_REF][START_REF] Faure | Scarred eigenstates for quantum cat maps of minimal periods[END_REF]).

The QUE assumption postulates that there is no such concentration phenomenon (see [START_REF] Zelditch | Note on quantum unique ergodicity[END_REF]), but very few results are known. In the one-dimensional case, QUE is always true. But in the multi-dimensional case, up to now no example is known where QUE holds (in the context of the Dirichlet-Laplacian). The ergodicity of Ω is however a necessary assumption for QUE to hold. Note that strictly convex billiards whose boundary is C 6 are not ergodic (see [START_REF] Lazutkin | On the asymptotics of the eigenfunctions of the Laplacian[END_REF]), and there are sequences of positive density of eigenfunctions which concentrate on caustics. Rational polygonal billiards are not ergodic, but generic polygonal billiards are ergodic (see [START_REF] Kerckhoff | Ergodicity of billiard flows and quadratic differentials[END_REF]). Recently, Hassell has shown in [START_REF] Hassell | Ergodic billiards that are not quantum unique ergodic[END_REF] that there exist some convex (stadium-shaped) sets satisfying QE but not QUE. Note that it is a longstanding conjecture that every compact negatively curved manifold should satisfy QUE (see [START_REF] Rudnick | The behaviour of eigenstates of arithmetic hyperbolic manifolds[END_REF]). Using a concept of entropy, it is proved in [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF] that on a compact manifold with negative curvature the eigenfunctions cannot concentrate entirely on closed geodesics and at least half of their energy remains chaotic.

It is an open question of knowing whether the statement of Theorem 4 still holds true for domains satisfying QE or only WQE. Note however that we have the following corollary. Remark 8. QUE is a sufficient condition implying the no-gap statement (33), but is not sharp: it is not a necessary condition. For instance the twodimensional square does not satisfy QUE (nor QE), but satisfies WQE, and in this configuration it can be easily shown that the no-gap result (33) holds.

Remark 9. The question of knowing whether the supremum in (33) is reached or not is unexpectedly difficult. We have a clear answer only in dimension one. We gather hereafter the results we have been able to prove.

• In the one-dimensional case Ω = [0, π], for L ∈ (0, 1), the supremum of J over U L (which is equal to L) is reached if and only if L = 1/2.

In that case, it is reached for all measurable subsets ω ⊂ [0, π] of measure π/2 such that ω and its symmetric ω = π -ω are disjoint and complementary in [0, π].

• For the two-dimensional square Ω = [0, π] 2 , the supremum of J over the class of all possible subsets ω = ω 1 × ω 2 of Lebesgue measure Lπ 2 , where ω 1 and ω 2 are measurable subsets of [0, π], is reached if and only if L ∈ {1/4, 1/2, 3/4}. In that case, it is reached for all such sets ω satisfying

1 4 (χ ω (x, y) + χ ω (π -x, y) + χ ω (x, π -y) + χ ω (π -x, π -y)) = L, for almost all (x, y) ∈ [0, π 2 ].
It is by the way interesting to note that there exist optimal sets not satisfying the Geometric Control Condition, and thus in this case the deterministic observability constant is equal to 0, whereas 2

C (W ) T,rand (χ ω ) = C (S) T,rand (χ ω ) = T L.
We conjecture that, if Ω is the unit N -dimensional hypercube, then there exists a finite number of values of L ∈ (0, 1) such that the the supremum in (33) is reached; for generic domains Ω, there exists a finite number of values of L ∈ (0, 1) such that the the supremum in (33) is reached.

Remark 10. As a final remark, let us stress that our main contribution is to put in evidence the intimate relations existing between shape optimization problems and the (strong) ergodicity properties of the domain Ω under consideration. It can be noticed that such a connection was suggested in the early work [START_REF] Chen | Exponential decay of energy of evolution equations with locally distributed damping[END_REF], where the authors were concerned with the question of determining sufficient conditions on dissipative terms added into a conservative equation, ensuring an exponential decay of the solutions. The role of the quantum effects of bouncing balls and of the whispering galleries was underlined in their study. Our results here provide clear statements making such a connection and suggest new issues in the analysis of optimal design problems.

We end the study of the second problem by considering the spectral truncation of the functional J defined by [START_REF] Kerckhoff | Ergodicity of billiard flows and quadratic differentials[END_REF] J N (χ ω ) = min

1 j N ω φ j (x) 2 dx, (34)
for every N ∈ IN * and every measurable subset ω of Ω. The spectral approximation of the second problem is defined as

sup χω∈U L J N (χ ω ). (35)
As before, the functional J N is extended to U L by J N (a) = min 1 j N Ω a(x)φ j (x) 2 dx, for every a ∈ U L . We have the following result.

Theorem 5.

1. For every measurable subset ω of Ω, the sequence (J N (χ ω )) N ∈IN * is nonincreasing and converges to J(χ ω ). Moreover, if (a N ) n∈IN * is a sequence of maximizers of J N in U L , then up to a subsequence (a N ) n∈IN * converges to a maximizer of J in U L in L ∞ weak star topology.

3. For every N ∈ IN * , the problem (35) has a unique solution χ ω N , where ω N ∈ U L . Moreover, ω N has a finite number of connected components.

The theorem allows one to construct a maximizing sequence for the second problem. Several numerical simulations are provided in [START_REF] Privat | Optimal observability of wave and Schrödinger equations in ergodic domains[END_REF].

Remark 11. Note however that the following property is proved in [START_REF] Privat | Optimal observability of the onedimensional wave equation[END_REF]. In the one-dimensional case Ω = [0, π], ω N is the union of at most N intervals, is symmetric with respect to π/2, and moreover there exists L N ∈ (0, 1] such that, for every L ∈ (0, L N ], the optimal domain ω N satisfies

ω N sin 2 x dx = ω N sin 2 (2x) dx = • • • = ω N sin 2 (N x) dx.
This technical property (quite difficult to prove), stated in [12, Theorem 3.2] but whose proof is not completely correct, causes the so-called spillover phenomenon, according to which the optimal domain ω N solution of (35) with the N first modes is the worst possible domain for the problem with the N +1 first modes. Actually, ω N concentrates around the nodes kπ N +1 , k = 1, . . . , N . This is rather a bad news for practical purposes, but is in accordance with the fact that the problem has no solution whenever L = 1/2.

Corollary 1 .

 1 Assume that Ω satisfies QE. There exists a subset I of IN * of density 1 such that sup χω∈U L inf j∈I ω φ j (x) 2 dx = L.

It is similar to the well-known open problem of determining what are the best constants in Ingham's inequalities.

Similarly to the definition of elements of L ∞ , the subset ω is unique within the class of all measurable subsets of Ω quotiented by the set of all measurable subsets of Ω of zero Lebesgue measure.