Régis Y Chenavaz

Marta Leocata

Malgorzata Ogonowska

Dominique Torre

Sustainable Tourism

Keywords: sustainable tourism, World Heritage Sites, dynamic pricing, conservation policies, optimal control 1

Current approach to the management of World Heritage sites and the associated tourism is not always effective. Indeed, operators of Chteau de Versailles, Alhambra, and Vatican Palace trade-off visitors' revenues against damage to these sites. We study the effective dynamic behavior of a firm managing a heritage site. In an optimal control model, the operator sets pricing and heritage conservation policies. Heritage sites benefit from spending on conservation but are deteriorated by large numbers of visitors. Visitor numbers increase with the attractiveness of the site but decrease with the entrance fee. Our results characterize optimal pricing and heritage conservation policies over time and provide a stronger foundation for sustainable tourism policies.

Introduction

The media claim that mass tourism is causing irreversible damage to cultural heritage. For instance, the tomb of Tutankhamun in Egypt (Reuters-ABC, 2019), and Buddhist pagodas in Myanmar [START_REF] Ban | Myanmar stops tourists scaling prized pagodas[END_REF] have suffered severe damage due to excessive numbers of visitors to these sites, and following an initial period of intensive exploitation, the authorities have discontinued visitor access. In other, more numerous cases, the deterioration has been less spectacular, with the result that either access has continued or measures to reduce this heritage damage came too late to prevent permanent depreciation of the heritage. Take for instance the case of Venice over which the Editorial Board of The Economist [START_REF] Board | Venice cruise crash is a sign of the risks of tourism[END_REF] recently highlighted concern. For the most part, myopic foresight is at the roots of poor management of heritage sites. Over time, it is possible to observe two periods when heritage is destroyed or severely damaged. In the first period, operators with a myopic long-term view exploit the resource intensively, as in the case of the city of Pompeii [START_REF] Lyons | The paradox of Pompeii[END_REF]. In the second period, sudden awareness of the depreciation of the heritage results in limitations to the number of tourists allowed to visit the site. The Financial Times highlights this limitation policy in relation to China's heritage sites, like the Forbidden City or the Great Wall (Chaguan, 2019). However, this sequence is ineffective: it is insufficiently proactive in the first period; in the second period, it deprives the public or private authorities responsible for the site of the funding necessary for its preservation and overly restricts visitors' access to the heritage. This situation requires a deeper consideration of the strategies needed to achieve a sustainable balance between economic exploitation and long-term preservation of heritage.

Sustainable management of heritage is of interest to practitioners and scholars. For instance, [START_REF] Hansen | The economic evaluation of cultural and heritage projects: Conflicting methodologies[END_REF] and [START_REF] Kim | Assessing the economic value of a world heritage site and willingness-to-pay determinants: A case of Changdeok Palace[END_REF] investigate the different methodologies to evaluate heritage projects; [START_REF] Garrod | Managing heritage tourism[END_REF] and [START_REF] Choi | Economic valuation of cultural heritage sites: A choice modeling approach[END_REF] look at the management of heritage tourism; [START_REF] Emerton | Sustainable Financing of Protected Areas: A Global Review of Challenges and Options[END_REF] consider the sustainable financing of protected heritage; [START_REF] Voltaire | Pricing future nature reserves through contingent valuation data[END_REF] analyses the pricing of natural heritage sites. The literature shows the interest in heritage site management. Despite its societal relevance, the pricing over time of sustainable heritage sites is still missing from the literature. Our paper fills the gap by discussing the importance of dynamic pricing solutions for heritage sustainability. Dynamic pricing concerns the price policy of a product over its lifecycle. The product price, which may change over the lifecycle, accounts for its effects on both current and future profits. Our research points to the important role of time awareness and the future effects of expensive policies aimed at heritage conservation. The use of a dynamic control setting seems appropriate for this study. It allows the simultaneous application of an adapted pricing policy control for the number of visitors and a preservation or protection policy funded in part by the revenues generated by visits. Both policies should be rationally generated using a single intertemporal objective function of the heritage operator, integrating the influence of expenses devoted to maintaining the heritage site with the quality of this heritage and the visitors' propensity to pay.

We propose a model in which heritage degrades with visits and may recover by appropriate expenditure. A private operator is the concessionaire of a World Heritage Site, seen as a profit center. Optimization of the operator's profit determines dynamically both the entrance fee, that is the price, paid by visitors and the amount of spending required to maintain the quality of the heritage. We consider a general form of demand for visits, but also introduce the usual additive and multiplicative specifications. We limit the study to the monopolistic case, which corresponds to many situations where a private operator is the sole manager of a unique heritage site. This case is relevant to the most outstanding World Heritage Sites such as the Effeil Tower in Paris, the Forbidden City in Beijing, and the Vatican Palace in Rome. For simplicity, we omit other ways for the operator to extract revenues, like selling souvenirs or filling hotels and restaurants.

We contribute to the literature on dynamic pricing and tourism management by examining dynamics policies for World Heritage Sites. The following findings emerge from our analysis.

First, the price proposed by an operator aware of the need to preserve the future value of the heritage is higher than the price of a myopic operator (Proposition 1) and depends directly on demand elasticities (Proposition 2). The evolution of optimal heritage conservation expenses determines the evolution of the shadow price applied to the heritage over time (Proposition 3). If the operator is aware of the depreciation caused to the heritage by visitors, the price dynamics will depend on both the heritage dynamics and the shadow price of the heritage (Proposition 3). Over time, for more specific classes of the demand function, stronger pricing policies will emerge (Propositions 4-5).

Our article tries to provide a better understanding of the tension between short-term profit maximization and long-term conservation of World Heritage Sites. More broadly, it proposes dynamic pricing and heritage conservation rules for an operator managing a site, taking into account both profit and heritage preservation issues. These rules address some concerns of residents and administrative authorities whose interests are in economic development and site conservation. Thus, our results suggest more sustainable practices regarding tourism management.

The rest of the paper is as follows. Section 2 reviews the related literature. Section 3 formulates the model and section 4 describes the analysis. The results of the analysis appear in section 5. Section 7 is the discussion and section 8 concludes the paper.

Related Literature

This section surveys the existing literature on sustainable tourism and, more specifically, on heritage sites and on optimal control applications to dynamic pricing and tourism management. The section ends by stressing the essential ingredients of our approach, as compared to the literature.

Sustainable Tourism and Heritage

We first focus on sustainable tourism. The most useful definition of cultural heritage site comes from Chen and Lee (2017, p. 259):

The term "Cultural Heritage Site" encompasses cultural ruins, archeological artifacts, monuments, buildings, and other heritage sites (. . . ) placed under protection.

Cultural heritage sites contribute to a society through aesthetic quality, spiritual meanings, historical and symbolic value; they represent for the community a shared experience and sense of continuity [START_REF] Wright | Drivers of heritage value: A meta-analysis of monetary valuation studies of cultural heritage[END_REF]. Thus, cultural heritage is at the heart of sustainable development due to its contribution to the richness of the living environment for present and future generations and is included in the United Nations Sustainable Development Goals (United Nations and [START_REF] Affairs | Transforming our world: The 2030 agenda for sustainable development[END_REF] and the United Nations Educational, Scientific and Cultural Organization's (UNESCO) Sustainable Development Agenda [START_REF] Boccardi | Introducing cultural heritage into the sustainable development agenda[END_REF].

Since the 1990s, environmental protection and sustainability have become major issues. In this line tourists, and generally, citizens are more aware of environmental issues and conscious of damages caused by mass tourism and modify their perceptions and preferences, as surveyed by [START_REF] Meleddu | Tourism, residentswelfare and economic choice: A literature review[END_REF]. As a result, demand for sustainable products has emerged in this sector, and major tourist destinations are seeking to preserve high quality environmental, cultural, and heritage attributes, and are exploring sustainable development strategies and techniques in order to be still attractive in the future [START_REF] Hassan | Determinants of market competitiveness in an environmentally sustainable tourism industry[END_REF]. Following Wright and Eppink (2016, p. 278), maintenance of heritage include preservation, conservation, and renovation.

The fact that cultural heritage sites present the characteristics of a public good and contribute to social welfare justifies public subsidies to cover repair, maintenance, and enhancement costs (Choi et al., 2010, pp. 213-214). However, in most cases, the existing public subsidies are not sufficient to cover all of these costs [START_REF] Voltaire | Pricing future nature reserves through contingent valuation data[END_REF]. In this context, it has become especially important to identify additional sources of funding and to develop strategies to help to maintain the sites.

One solution to maintain heritage would be an adequate entrance price. This strategy is in line with the "user-pays principle" [START_REF] Throsby | The economics of uniqueness. Investing in Historic City Cores and Cultural Heritage Assets for Sustainable Development?[END_REF]. Site operators did not subscribe to this idea for many years, preferring to charge a "token" low entrance fee [START_REF] Garrod | Managing heritage tourism[END_REF]. Yet, a higher admission price helps site managers to control the number of visitors, matching site capacity constraints. A consequent entrance price represents a good solution to overcrowding issues and capacity management [START_REF] Wu | Determinants of tourism ticket pricing for ancient villages and towns: Case studies from Jiangsu, Zhejiang, Shanghai and Anhui provinces[END_REF].

Price discrimination also represent a policy tool enabling underprivileged to visit heritage.

People accept price discrimination, as it is based on an assessment of visitors' willingness to pay or residents' willingness to accept the costs of maintaining the site [START_REF] Tuan | Capturing the benefits of preserving cultural heritage[END_REF]. Plus, research show that consumers' willingness to pay is higher for sustainable and better preserved destinations [START_REF] Dodds | Does the tourist care? a comparison of tourists in koh phi phi, thailand and gili trawangan, indonesia[END_REF][START_REF] Hedlund | The impact of values, environmental concern, and willingness to accept economic sacrifices to protect the environment on tourists intentions to buy ecologically sustainable tourism alternatives[END_REF]. By taking into account the preferences of the environmentally conscious population, the operator modernizes the heritage site, making it more sustainable [START_REF] Brau | Demand-driven sustainable tourism? a choice modelling analysis[END_REF][START_REF] Minciu | Commercialization of holidays in the protected natural areas-form of the sustainable development in tourism[END_REF].

A dynamic pricing perspective is missing from the above literature specialized on sustain-able heritage. We contribute to this literature by adding the temporal element.

Dynamic Pricing and Tourism

We review now prior literature on optimal control applications to dynamic pricing and tourism.

We start with the literature on dynamic pricing. In a seminal article, [START_REF] Kalish | Monopolist pricing with dynamic demand and production cost[END_REF] examines the dynamic pricing of a new product. Price and quality are studied together by [START_REF] Vörös | The dynamics of price, quality and productivity improvement decisions[END_REF] whereas, [START_REF] Chenavaz | Dynamic pricing, product and process innovation[END_REF] and [START_REF] Vörös | Multi-period models for analyzing the dynamics of process improvement activities[END_REF] examine both pricing and productivity policies. The impact of product quality on selling price is discussed by [START_REF] Chenavaz | Better product quality may lead to lower product price[END_REF] and is generalized by [START_REF] Ni | When better quality or higher goodwill can result in lower product price: A dynamic analysis[END_REF] with goodwill and by [START_REF] Vörös | An analysis of the dynamic price-quality relationship[END_REF] with a salvage value.

The above research uses structural (as opposed to parametric) demand and cost functions and provides general and robust results. [START_REF] Dye | Optimal dynamic pricing and preservation technology investment for deteriorating products with reference price effects[END_REF] and [START_REF] Xue | Optimal dynamic pricing for deteriorating items with reference-price effects[END_REF] characterize the dynamic optimal pricing of products that deteriorate over time. The literature above is essential to understand the price-quality relationship. Therefore, it represents a sound basis to investigate our price-heritage relationship. Now, we turn our attention to the literature on dynamic tourism. The link between environment and tourism is investigated in [START_REF] Greiner | Optimal periodic development of a pollution generating tourism industry[END_REF] and [START_REF] Sirilersuang | Dynamic optimal joint policies for tourism promotion and environmental restoration[END_REF]. Heritage sites must be maintained over time [START_REF] Hutter | Economic Perspectives on Cultural Heritage[END_REF] and defended against threats related to natural degradation and also man-made actions such as wars, political and other instabilities, including terrorism, outright neglect, uncontrolled economic exploitation, and excessive tourism pressure [START_REF] Feichtinger | Terrorism control in the tourism industry[END_REF]). Destinations's attributes attract tourists, which in turn damage those attributes. Tourism damage is modeled by [START_REF] Kort | Environmental effects of tourism industry investments: An inter-temporal tradeoff[END_REF] in the case of a dynamic trade-off between environment and tourism.

More optimal control applications have considered tourism. [START_REF] Piga | Pigouvian taxation in tourism[END_REF] studies Pigouvian taxation in tourism. [START_REF] Candela | Investment in tourism market: a dynamic model of differentiated oligopoly[END_REF] consider the tourism market with an oligopoly setting. Claude and [START_REF] Zaccour | Investment in tourism market and reputation[END_REF] look at reputation in the tourism market. Hernández andLeón (2007, 2013) examine tourism with a neoclassical demand function. [START_REF] Piazza | Deforestation and optimal management[END_REF] consider deforestation, which is similar to site degradation.

Dynamic pricing and tourism literature considered neither the link between price and heritage quality nor the sustainability of heritage. Our work contributes to this literature by analyzing the price-heritage relationship and the sustainable heritage.

Our Approach

The above research on tourism management did not consider the price as a lever to control the visitor number, and maintain or improve heritage. Instead, the price only raises revenues, but without an explicit goal and with no systematic account for side effects. Despite its managerial relevance, no research examined the entrance fee over time of a World Heritage Site. The gap in the literature applying optimal control to tourism management is that of providing a policy that jointly maximizes profits and maintain heritage through a smart entrance fee mechanism. In other words, there is a need to examine sustainable joint pricing and heritage intertemporal policies. Such entrance free policy would be profitable in the long run only if the desirable condition of heritage preservation is met. Our research fills the remaining gap.

Our paper supplements the above-cited studies by proposing an innovative approach based on optimal control theory which, to our knowledge, was unused to analyze the joint issue of pricing and cultural heritage site conservation. We analyze the explicit over time trade-off related to the number of visitors to a site since more visitors mean increased current revenues, but reduced future heritage quality. In our model, the site operator decides the level of the entrance fee and the investment in heritage quality. The entrance fee has a direct effect on the number of visitors, and in turn an indirect effect on the degradation of the site.

At the same time, heritage conservation attracts visitors, allowing the extraction of higher rents and thus higher revenues. For simplicity, we don't model other ways of generating profits, such as souvenir shops or hospitality facilities. Following a structural approach, we use general formulations for the demand and site maintenance functions to provide general results regarding the dynamic pricing and maintenance policies of a heritage site operator.

Model Formulation

In this section, we formulate our model. We propose an optimal control model of a monopolistic operator, that is a a concession managed as a firm, in charge of a heritage site.

The operator sets the price, that is the entrance fee, p(t), and conservation expenditure, also called preservation investment, u(t), over time, t. The conservation expenditure goes to maintaining or improving the attractiveness of the heritage site, h(t). Visitors to the site, who prefer a lower price and better-preserved heritage, are aggregated in the demand D(p(t), h(t)). On the one hand, increasing the heritage attractiveness represents a current cost since it implies conservation expenditure; on the other hand, it generates future revenue by stimulating future demand. A higher number of visitors both increases the operator's current revenue and degrades future heritage quality. So, if no preservation action is undertaken, the site declines over time, implying the heritage becomes less attractive. The operator's profit results from the revenues minus the cost (investment in site conservation), that is π(p(t), u(t), h(t)) = p(t)D(p(t), h(t))-u(t). The operator trades off current profit, tied to higher demand, and future profit, linked to the preservation of heritage. This intertemporal focus requires a dynamic analysis. In the following sub-sections, we propose an optimal control framework to analyze the model. The operator solves an infinite horizon problem where the time variable t in [0, +∞) is continuous. For ease of reading, Table 1 defines the main notations used in the model.

dh dt = -αD(p, h) -δh + K(u) = heritage attractiveness dynamics, λ(t) = shadow price of heritage at time t, co-state variable, π(p, u, h) = pD(p, h) -u = current profit, e x = ∂D ∂x x D
= elasticity of demand D with respect to x.

Demand

Demand aggregates the number of visitors to the site. Using a general functional form, demand (or visitor affluence) function

D : (R + ) 2 → R + is twice continuously differentiable.
The demand for the site depends on the entrance fee and the site's attractiveness. Demand reads as, p, h 0,

D = D(p(t), h(t)). (1)
Hereafter for simplicity, unless there is a risk of confusion, we omit the dependence from certain variables, in particular the time variable. Demand decreases with price and increases with heritage quality [START_REF] Kort | Environmental effects of tourism industry investments: An inter-temporal tradeoff[END_REF][START_REF] Zaccour | Investment in tourism market and reputation[END_REF][START_REF] Sirilersuang | Dynamic optimal joint policies for tourism promotion and environmental restoration[END_REF]. There is a diminishing impact of heritage quality on demand. Moreover, it is more difficult to increase demand by decreasing the price if heritage attractiveness is high rather than if it is low. In other words, visitors are less price-sensitive for a higher level of heritage attractiveness. (A similar assumption of submodularity of the demand function concerning the arguments is made in [START_REF] Jørgensen | Differential Games in Marketing[END_REF][START_REF] Chenavaz | Better product quality may lead to lower product price[END_REF][START_REF] Ni | When better quality or higher goodwill can result in lower product price: A dynamic analysis[END_REF] These four assumptions about the demand function translate formally into, p, h 0,

∂D ∂p < 0, ∂D ∂h > 0, ∂ 2 D ∂h 2 < 0, ∂ 2 D ∂p∂h 0. (2) p D(p, •) D(p, h) D(p, h + )
Figure 1: In this plot, we give a graphical interpretation of condition ∂ 2 D ∂p∂h 0. We plot the function D, just as a function of p and by fixing different values of heritage attractiveness, h, h + . Since, we assume that visitors are less price sensitive if heritage attractiveness is high, we have that D(p, h) tend to be close to zero, faster then D(p, h + ). Indeed it is clear form the plot that ∂D(p,h) ∂p > ∂D(p,h+ ) ∂p . Then, writing the different quotient with respect to the variable h, the previous implies that ∂ 2 D(p,h) ∂p∂h 0.

Assumptions (2) are confirmed using either a linear demand function D = a 0 -a 1 p + a 2 h or a Cobb-Douglas demand function D = a 0 p -a 1 h a 2 with a 0 , a 1 , a 2 > 0. Additionally, the demand function is assumed to be not "too" convex in price, p, h 0,

2 -D ∂ 2 D ∂p 2 ∂D ∂p 2 > 0, (3)
which is a condition required for profit optimization to ensure a unique profit maximizing point. This is a standard assumption in the dynamic pricing literature using structural demand functions. (See for instance [START_REF] Dockner | Differential Games in Economics and Management Science[END_REF][START_REF] Vörös | The dynamics of price, quality and productivity improvement decisions[END_REF][START_REF] Vörös | An analysis of the dynamic price-quality relationship[END_REF][START_REF] Jørgensen | Differential Games in Marketing[END_REF][START_REF] Chenavaz | Dynamic pricing, product and process innovation[END_REF][START_REF] Chenavaz | Better product quality may lead to lower product price[END_REF][START_REF] Ni | When better quality or higher goodwill can result in lower product price: A dynamic analysis[END_REF] So that the sufficiency condition of Mangasarian holds1 , we also need to assume that

   -D ∂ 2 D ∂h 2 •    2 -D ∂ 2 D ∂p 2 ∂D ∂p 2    - ∂D ∂h -D ∂ 2 D ∂p∂h ∂D ∂p 2    > 0 (4)
The Mangasarian condition (4) implies ∂ 2 D ∂h 2 < 0 in (2). Although (4) appears difficult to check, it is satisfied in our cases of interest. Eventually, this condition (4) allow us to prove that the controls furnished by the maximum principle, are actually optimal point for the optimization problem.

Heritage

Next, we characterize heritage site attractiveness. The operator spends u(t) ∈ R + to preserve the site heritage h(t). Expenses for heritage conservation u(t) and heritage attractiveness h(t) are control and state variables respectively. The heritage conservation function K : R + → R is twice continuously differentiable. Heritage preservation depends on the associated expenditure Expenditure u increases heritage attractiveness h, but with diminishing returns (see Mc-Donagh and Nahkies 2010;[START_REF] Rama | Investing in the sense of place: The economics of urban upgrading projects with a cultural dimension[END_REF]. It can be quite easy, and therefore cheap, to improve the quality of a low-quality heritage site but it can be difficult, and therefore expensive, to improve an already high-quality heritage site. Thus, mathematically, we have, u 0, K > 0, K < 0.

(5)

The dynamic of heritage attractiveness h(t) can be written, for all t > 0,

dh(t) dt = -αD(p(t), h(t)) -δh(t) + K(u(t)), with h(0) = h 0 . (6)
Formulation ( 6) accounts for several features of heritage dynamics. First, heritage decreases with the number of visitors, with α > 0 the sensitivity of heritage dynamics to the visits [START_REF] Kort | Environmental effects of tourism industry investments: An inter-temporal tradeoff[END_REF][START_REF] Zaccour | Investment in tourism market and reputation[END_REF][START_REF] Sirilersuang | Dynamic optimal joint policies for tourism promotion and environmental restoration[END_REF]).2 Second, heritage attractiveness declines at a constant proportional rate δ > 0.3 Heritage preservation depends on the associated expenditure. Indeed, if no investment maintains a site, then it deteriorates naturally over time (see for instance [START_REF] Patin | The economy of cultural heritage, tourism and conservation[END_REF][START_REF] Tan | Sense of place and sustainability of intangible cultural heritage-The case of George Town and Melaka[END_REF]).4 Third, heritage can be improved thanks to the heritage conservation policy K as discussed above [START_REF] Kort | Environmental effects of tourism industry investments: An inter-temporal tradeoff[END_REF][START_REF] Sirilersuang | Dynamic optimal joint policies for tourism promotion and environmental restoration[END_REF]. Integrating (6) measures the stock (or cumulative level) of heritage attractiveness to the flow of demand, autonomous depreciation, and

expenditure on heritage conservation h(t) = h 0 + t 0 [-αD(p(s), h(s)) -δh(s) + K(u(s))]ds.

Profit

The current profit function π : (R + ) 3 → R is twice continuously differentiable. Profit corresponds to current revenue minus conservation expenses. The heritage expense u, is decided independently of demand. In this economic sense it represents a fixed cost. The cost of a visit c, that is the unit production cost, is constant. Without loss of generality, for simplicity, c is normalized to zero. Based on the revenues and costs, the profit function can be written,

p, u, h, D 0, π(p, u, h) = pD(p, h) -u.
Here, conservation expenses maintain or increase heritage, modeled as the capital stock.

In turn, increased heritage attractiveness stimulates visitor interest, allowing the operator to extract higher rents. Formally, ∂π ∂h > 0 follows from ∂D ∂h > 0 assumed in (3).

Operator's Dynamic Optimization Problem

The operator maximizes its intertemporal profit (or present value of the profit stream) over the planning horizon by choosing conservation and pricing policies simultaneously, accounting for the dynamics of heritage. For simplicity, we let the salvage value of heritage be null. Let the discount rate be r in R + and the objective function of the firm be max

u, p 0 ∞ 0 exp (-rt) π(p(t), u(t), h(t))dt, (7) subject to dh(t) dt = -αD(p(t), h(t)) -δh(t) + K(u(t))
, for all t > 0, with h(0) = h 0 . along with the demand function. Profit may be maximized either by a higher markup and lower sales, as with p * 1 , or by a lower markup and higher sales, as with p * 3 . In the situation p * 2 , the markup and sales effects cancel each other out.

Model Intuitions

Section 4 quantifies the effects on the price-heritage policies which are depicted qualitatively in Figure 2. It is necessary to derive precise analytical results to ensure that the operator maximizes profit by applying the appropriate pricing and conservation policies.

Model Analysis

Here we analyze the model. The shadow price (or current-value adjoint variable) λ(t) represents the marginal value of the heritage at time t. The current-value Hamiltonian H with the current-value adjoint variable (or shadow price) λ for heritage dynamics reads, u, p, h 0, λ in R,

H(u, p, h, λ) = pD(p, h) -u + λ [-αD(p, h) -δh + K(u)] . (8)
The Hamiltonian H measures the intertemporal profit from summing the current profit pD -u and the future profit λ[-αD -δh + K]. The maximum principle for our optimization problem writes, for all t ≥ 0. Recall that the * notation refers to the optimal solution.

                               dh * dt = -αD(p * (t), h * (t)) -δh * (t) + K(u * (t)), dλ * dt = -p * ∂D(p * (t),u * (t)) ∂h + r + α ∂D(p * (t),u * (t))) ∂h + δ λ * (t), h * (0) = h 0 , lim t→+∞ e -rt λ * (t) = 0 H(u * (t), p * (t), h * (t), λ * (t)) = max u,p≥0 H(u, p, h * (t), λ * (t)), for each t ≥ 0. ( 9
)
The dynamic of the shadow price is combined with the transversality condition for a free terminal state and infinite terminal time. See [START_REF] Aseev | The pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons[END_REF] for a theoretical explanation of the transversality condition and Chenavaz et al. ( 2020) for a recent application on dynamic pricing.

Lemma 1. The quadruplets (u * (t), p * (t), h * (t), λ * (t)), solution of the system (9) return a local maximum for the optimization problem (7). If (9) admits an inner solution, then, it is given by the resolution of the following system,

         λ * • K (u * ) = 1, p * = - D(p * , h * ) ∂D(p * ,h * ) ∂p + αλ * .
Moreover, λ * (t) ≥ 0 and u * (t), p * (t) ≥ 0 for all t ≥ 0.

Proof. The proof is in Appendix A.2.

In the remainder of the article, we eliminate the * notation; all equations refer to the optimal solution, unless explicitly written. The shadow price of heritage, λ, represents the impact on future profits of a one-unit increase in current heritage attractiveness. As such, λ exerts an influence on the intertemporal trade-off between current and future profits and is worthy of further investigation. Note the heritage elasticity of demand is e h = ∂D ∂h h D . Substituting e h and ( 19) in ( 9) implies for all t > 0

dλ dt -r + δ + α ∂D ∂h λ = -e h pD h
, with lim t→+∞ e -rt λ(t) = 0.

Lemma 2. The value of λ over time can be written for all t ≥ 0

λ(t) = ∞ t e h pD h exp - t r + δ + α ∂D ∂h ds d .
Proof. The proof is in Appendix A.1.

The shadow price of heritage, λ, increases with the heritage elasticity of demand, e h , and the heritage deflated revenue, pD h . Also, λ increases with the discount rate, r, and increases with autonomous heritage decay, δ. Eventually, λ decreases with the degrading impact of visitors, α, because ∂D ∂h < 0.

Results

We present our analysis results below. The main objective of the model is to determine the optimal dynamic pricing and conservation policies of a heritage site operator. Remember that for simplicity, when presenting the results we may omit some function variables, especially the time variable. We determine the pricing rules through Propositions 1-2. We consider variations in conservation and pricing levels in Propositions 3 for a general demand function formulation. We specify the demand function and obtain stronger results for the price variations in Propositions 4-5. All proofs are in Appendix A.

Pricing Rules

The optimal price p, comes from the first-order optimality condition,

p = - D ∂D ∂p + αλ.
If the firm ignore the long run effect of the shadow price of heritage (λ = 0), then the static (myopic) price is considered,

p static = - D ∂D ∂p
.

See [START_REF] Fruchter | Dynamic brand-image-based production location decisions[END_REF]; [START_REF] Fruchter | Signaling quality: Dynamic price-advertising model[END_REF] for a similar distinction. Here, we derive the two main pricing rules, namely Propositions 1 and 2.

Proposition 1. For all t ≥ 0, the difference between the optimal dynamic (foresighted) and static (myopic) price reads

p -p static = α K > 0.
Proof. The proof is in Appendix A.3

Note then that the difference between optimal dynamic and static prices does not depend on the discount factor, r, or on heritage deterioration, δ, while the shadow price of heritage depends on both of these parameters. The rationale for this result is that while the infinite horizon of the forecast tends to decrease with the shadow price of the heritage, the current optimal decision continues to be to set rather high fees if the objective is intertemporal. The dynamic pattern of λ is only technical. When, in each period, the operator of the heritage site also considers future profits, it always sets a higher price dynamic (foresighted) than the static (myopic) price. The difference between the optimal dynamic and the static price is larger with greater visitor damage α and larger with the difficulty involved in repairing the site K .

Proposition 1 compares with the reformulation of the price optimality condition of Kalish (1983, Equation (2c), p. 138)

p -p static = -λ < 0,
In our setting, the substitution of ( 19) in ( 10) allows rewriting Proposition 1 as

p -p static = αλ > 0.
The difference is that in [START_REF] Kalish | Monopolist pricing with dynamic demand and production cost[END_REF], which considers the sales of a new product, the dynamic price is lower than the static price to stimulate current sales. Greater current sales, in turn, boosts future goodwill and profits. In contrast to our model of tourism management, the dynamic price is larger than the static price to reduce current visits. Eventually, fewer current visits maintain or improve future heritage and profits.

Consider now the variation in the optimal price related to variation in demand. Thus, note the price elasticity of demand, e p = -∂D ∂p p D .

Proposition 2. For t ≥ 0, the optimal relationship between price and heritage expenses implies

p = e p e p -1 α K . ( 10
)
Proof. The proof is in Appendix A.4.

Proposition 2 originates from the joint optimal price and conservation policies if the heritage site operator is maximizing profit. Proposition 2 builds on the classic static part e p e p -1 and on our model's specific dynamic part α K which integrates future heritage value. When visitors' willingness to pay the fee has a strong negative effect on the heritage, the operator will charge a higher price to reduce the number of visitors, and preserve the heritage; p increases with λ. When spending on conservation allows easy recovery of the heritage, then the operator charges less to increase demand and accepts more visitors; p decreases with K .

Proposition 2 is best understood compared with the classical dynamic pricing rule of Kalish (1983, Equation (3), p. 138)

p = e p e p -1 (c -λ),
in which c is the marginal cost.

In our framework, the substitution of ( 19) in (10) rewrites Proposition 2 as a similar result to [START_REF] Kalish | Monopolist pricing with dynamic demand and production cost[END_REF] with p = e p e p -1 αλ.

The difference is that we don't consider a marginal cost, which is low for a visitor and approximated by zero, but we integrate the impact of visitors on heritage degradation, α.

Proposition 2 is the consequence of the equalization of marginal revenues and marginal costs over the planning period. For maximal profit, and given the optimal control variable values, changes to marginal revenue must balance changes to marginal costs. In turn, changes to the heritage drive changes to pricing. The causality in the changes results in a relationship between price and heritage.

Proposition 2 highlights the optimal relationship between price and preservation expenses.

However, it does not specify the conditions under which the optimal price increases or decreases after a heritage improvement. Proposition 2 equates marginal revenues to marginal costs at all times. It must hold throughout the planning period. To achieve the maximal profit, changes in marginal revenue must offset changes in marginal costs. In turn, changes to the heritage drive price changes. The causality among these changes is linked to the relationship between price and heritage.

To the best of our knowledge, former optimal control models investigating tourism management do not provide any dynamic pricing rule [START_REF] Feichtinger | Terrorism control in the tourism industry[END_REF][START_REF] Greiner | Optimal periodic development of a pollution generating tourism industry[END_REF][START_REF] Kort | Environmental effects of tourism industry investments: An inter-temporal tradeoff[END_REF][START_REF] Piga | Pigouvian taxation in tourism[END_REF][START_REF] Candela | Investment in tourism market: a dynamic model of differentiated oligopoly[END_REF][START_REF] Zaccour | Investment in tourism market and reputation[END_REF]Hernández andLeón, 2007, 2013;[START_REF] Sirilersuang | Dynamic optimal joint policies for tourism promotion and environmental restoration[END_REF]. Consequently, Propositions 1 and 2 represent the first dynamic pricing rules applied to tourism management.

Variations of p(t) and u(t)

Next, we study the evolution of heritage preservation expenses and of the price over time, t.

Proposition 3. For all t > 0, the dynamics of price p reads

dp dt      2 -D ∂ 2 D ∂p 2 ( ∂D ∂p ) 2 +      = dh dt      e h e p p h + + D ∂ 2 D ∂p∂h ( ∂D ∂p ) 2 -      + dλ dt α + . ( 11
)
For all t > 0, the dynamics of u imitate the dynamics of λ. Formally,

sgn du dt = sgn dλ dt . (12)
Proof. The proof is in Appendix A.5.

The operator preserves the heritage in accordance with the shadow price of the heritage.

In other words, the operator spends more for a higher shadow price and less for a lower shadow price. This simple rule of thumb is easy to apply.

Corollary 1. The optimal conservation expense u is dependent on the sole dynamics of λ.

Proof. The proof is in Appendix A.6.

Corollary 1 simplifies the marketing-mix of the operator. Of course, p and u are tied through λ as attested in the first order conditions ( 19) and ( 20). However, when determining the conservation policy, the operator can focus on λ only, and disregard p.

Proposition 3, more specifically (11), relates the price dynamics dp dt to the heritage dynamics dh dt and the shadow price dλ dt . On the left-hand side of ( 11), the second factor 2 -D

∂ 2 D ∂p 2
( ∂D ∂p ) 2 is strictly positive because of the second-order condition (3). On the right-hand side, for a given λ, the effect of heritage on price originates from two opposing effects: the markup effect e h e p p h and the sales effect

D ∂ 2 D ∂p∂h ( ∂D ∂p ) 2 :
• The markup effect e h e p p h measures the rise in consumers' willingness to pay for a better heritage. The markup effect is positive. It is more pronounced for a superior product with a low level of competition (a unique monument) such as the Eiffel Tower in Paris or the Alhambra in Granada; it is less pronounced for an inferior product with strong competition such as the many historical villages and city centers in Europe.

• The sales effect D

∂ 2 D ∂p∂h ∂D ∂p
2 quantifies the increase in sales after a decrease in price combined with a heritage improvement. The sales effect is negative. It is larger for a mass consumption product such as the Colosseum in Rome or the Chateau de Versailles; it is lower for a niche product such as the military heritage in Northern France, which includes the Omaha Beach Memorial in Normandy.

Proposition 3 states also that with a fixed heritage ( dh dt = 0), the dynamics of p follows the dynamics of λ. In other words, the price increases, if the future heritage is more valuable; a higher price reduces the number of visitors, which preserves the heritage. Conversely, the price decreases, if the future heritage is less valuable; a lower price increases revenues by attracting a higher number of visitors, with the consequence that heritage might decline.

In brief, Proposition 3 shows how the price dynamics account for the current markup and sales effects on the demand side, and the future heritage effect on the supply side.

Special Cases

The general demand function provides an information rule regarding the price-heritage relationship in Proposition 3. The caveat is that it can sometimes be difficult to apply. Specifying the demand function can be a way of providing clearer price-heritage relationship rules. The gain in applicability can be set off against the loss of generality.

Additive Separable Demand Function A demand function additively separable in price and heritage is the simplest and most natural modeling approach. With the additive separable case, demand (1) can be written, p, h 0,

D = m(p) + n(h), (13)
where m < 0, n > 0 and

2 -(m(p) + n(h)) m (p) (m (p)) 2 > 0. -n (h) 2 -(m(p) + n(h)) m (p) (m (p)) 2 (m(p) + n(h)) > (n (h)) 2 ,
According to this assumptions, conditions (2)-( 3)-( 4) are automatically satisfied.

Proposition 4. For t > 0, with an additively separable demand function, the dynamics of price p is written

dp dt 2 -(m(p) + n(h)) m (p) (m (p)) 2 + = dh dt - n (h) m (p) + + dλ dt α + . ( 14
)
Proof. The proof is in Appendix A.7.

With an additive separable demand function, the markup effect (positive) holds, whereas the sales effect (negative) disappears. Therefore, price is more likely to increase with greater heritage because of the sole role of the markup effect. Yet, because of the dynamics of the shadow price, a positive price-heritage relationship is not sure.

Example 1. Linear price demand function.

Assume the demand function D = a 0 -a 1 p + a 2 h α , with a 0 , a 1 , a 2 > 0 and 0 < α < 1.

The derivatives are written

∂D ∂p = -a 1 , ∂ 2 D ∂p 2 = ∂ 2 D ∂p∂h = 0.
Substituting the derivatives in ( 14) yields

2 dp dt = dh dt a 2 a 1 αh α-1 + + dλ dt α + ,
confirming that price increases with heritage, provided that λ is not "overly" decreasing.

The property of additive separability is observed for instance when demand aggregates two different types of individual populations, the first mainly motivated by the quality of the heritage and the second by the price of the visits. This is a case when the heritage attracts both an expert and an intrinsically motivated population, and families or average revenue standard visitors. Those two populations react differently to the dynamics of price and heritage. In the myopic case, as the level of preservation of the heritage is given in the short term, the demand of the first population is inelastic and the price depends on the price elasticity of the second population. During the time the same price is then maintained, but given the heritage depletion, the profit decreases due to the collapse of the experts' demand, until the destruction of the heritage. When the price is determined dynamically, it is higher during all periods, with the effect of restoring heritage but also the demand of the first population and the profit of the operator. In definitive, the dynamic approach is in line with experts' interest while the static method would be, in the short run, in line with the wishes of low revenue tourists.

Multiplicative Separable Demand Function A demand function multiplicatively separable in price and heritage remains relatively general with few constraints. Multiplicative separability is both simple and natural and ensures analytical tractability and fits well with the data [START_REF] Chenavaz | Dynamic pricing, product and process innovation[END_REF][START_REF] Helmes | Dynamic advertising and pricing with constant demand elasticities[END_REF]. With the multiplicative separable case, demand (1) is written as, p, h 0,

D = m(p)n(h), (15)
with m > 0, n < 0, n < 0 and 2 -m m (m ) 2 > 0.

Then ∂D ∂p = nm < 0, ∂D ∂h = mn > 0, ∂ 2 D ∂p∂h = m n < 0, ∂ 2 D ∂p 2 = m n.
According to this assumptions, conditions (2)-( 3)-( 4) are automatically satisfied.

Proposition 5. For t > 0, the dynamics of price p is written

dp dt     2 -m m (m ) 2 +     = dλ dt α + . ( 16
)
Proof. The proof is in Appendix A.8.

With a multiplicative separable demand function, the markup effect and the sales effect outweigh each other exactly. Therefore, the dynamics of price are independent of the dynamics of heritage. In other words, the price-heritage relationship is null.

Corollary 2. For a multiplicatively separable demand function, the dynamics of both p and u imitate the dynamics of λ. Formally,

sgn dp dt = sgn du dt = sgn dλ dt .
Proof. The proof is in Appendix A.9.

The marketing-mix implications of Proposition 5 and Corollary 2 are straightforward. The dynamics of price and heritage preservation expenses emulate the dynamics of the shadow price. To decide on price and preservation expenses, the operator only needs to know the value of the shadow price.

Example 2. Cobb-Douglas price-heritage demand function.

Suppose the demand function D = a 0 p -a 1 h a 2 , with a 0 , a 1 > 1, and 0 < a 2 < 1. The

derivatives writte ∂D ∂p = -a 1 D p , ∂D ∂h = a 2 D h , ∂ 2 D ∂p 2 = a 1 (1 + a 1 ) D p 2 , ∂ 2 D ∂h 2 = a 2 (a 2 -1) D h 2 , ∂ 2 D ∂p∂h = -a 1 a 2 D ph . Substituting the derivatives in (16) yields dp dt 1 - 1 a 1 + = dλ dt α + , (17)
verifying that the dynamics of p simulates the dynamics of λ.

The case of multiplicative separability is a sort of "textbook case." It expresses the imperfect substitutability between price and heritage in the arguments of indirect utility and consumer demand. This situation may correspond to a demand expressed by visitors both motivated by the price of the visit and the quality of heritage, and able to confirm or not their demand according to the nature of the trade-off between the two determinants of their choice. The dynamic profile of prices exhibited by Proposition 5 could then be considered as a normative result presenting a sufficient degree of generality when the heritage belongs to standard leisure consumption which corresponds to these characteristics.

Discussion

In this section, we discuss the theoretical and managerial implications of our results.

Theoretical Implications

For illustrative reasons, we focus on the case where lim t→∞ λ(t) = 0. Figure 3 summarizes the main insights of our research regarding the conservation and pricing policies over time.

The first subfigure depicts the conservation policy, u, which mimics the evolution of the shadow price, λ (Proposition 3 and Corollary 1). The implication of the pricing condition (19) guarantees that λ is always positive as a greater heritage invariably increases future profits. Moreover, since lim t→∞ λ(t) = 0, the operator has an incentive to invest in heritage preservation. Further, conservation expenses may increase in the first phase but always de-crease after some time. As heritage does not generate any additional rent at the end of the horizon, the operator does not maintain the heritage anymore, that is lim t→∞ u(t) = 0.

The following subfigures depict the pricing policy under different classes of demand functions. The price at the end of the horizon is the same for each class, with p ∞ = lim t→∞ p(t) = lim t→∞ -D(p(t),h(t))

∂D ∂p (p(t),h(t)) . In effect, the vanishing shadow price of heritage at the end of the horizon, lim t e -ρt λ(t) = 0, dictates a single first-order condition on price (20) at infinite. The class of demand functions generates different temporal paths for the pricing policies. Indeed with multiplicative separability, D = m(p)n(h), the pricing policy exactly mimics the conservation policy (Proposition 5 and Corollary 2). With additive separability, D = m(p) + n(h), the pricing policy is similar, but does not exactly match the conservation policy (Proposition 4). With a general demand function, D = D(p, h), the shape of the pricing policy may not be inferred from the conservation policy. Indeed Proposition 3 points out different elements exerting influence in opposing directions for the dynamics of price. As such, the pricing policy is undetermined, and distinct pricing schemes may appear.

We considered above a dynamic operator in the sense that it accounts for the impact of its pricing and conservation policies on future heritage attractiveness and thus on future profits.

In such a situation the shadow price, λ(t), is given at time t by Lemma 2. Alternatively, in the final subfigure, we consider a myopic (or static) operator, which disregards future heritage.

In this situation the discount factor is infinite, r = ∞. Consequently the shadow price, given by Lemma 2, is null over the whole planning horizon, that is λ(t) = 0 for all t ≥ 0. The implication is that the operator considers the current profit alone. Proposition 1 reveals that the myopic price is always lower than the dynamic price. The reason is that the myopic operator charges less to increase current sales, as it disregards future site degradation and the consequences on future profits. The final subfigure depicts that, even if the operator is myopic and behaves statically by maximizing current profit at each time t, price is not constant over time. Indeed the optimal price is tied to demand and in turn to heritage attractiveness.

Because attractiveness decreases over time, if not maintained by conservation investment, then demand and price change over the planning horizon.

Managerial Implications

The main motivation of this paper is that World Heritage Sites depreciate rapidly when widely accessible to visitors. Consequently, after the first period of intense exploitation, private or public organizations operating the asset may severely restrict or even prohibit any access and visits. Some sites or parts of them may be closed temporarily to the public, like for instance Tutankhamun's Tomb, which fully reopened in 2019 after 10 years of restoration. Others, because of irreversible damage caused by visiting, like Lascaux cave, will never be accessible to the public again.5

We questioned this policy with a dynamic optimization setting. It proved unsustainable because the first period accelerates the depreciation of assets and the second deprives managers of financial resources necessary for their restoration. This sequence rapidly destroys the heritage and makes its restoration difficult when the exploitation of heritage becomes unprofitable. Our model supports the thesis that visiting, considered as the main cause of the problem, could also be its remedy if controlled by a suitable pricing policy. The number of visitors could be reduced using monitored access prices.

The preservation of World Heritage Sites requires a sound pricing policy. Operators aim at resource preservation and at the same time making the site accessible to visitors in the long run. Therefore, they must charge higher access prices than those which maximize short-term profits (Proposition 1). The revenues from the access prices finance heritage preservation, with a balance between access price and preservation expense (Proposition 2). Higher prices trade off a lower heritage degradation with higher exploitation revenues (Maximization problem (7)).

Proposition 3 examines the link between the dynamics of price and heritage attractiveness in the general case (general demand function). Counterintuitively, it shows that price does not necessarily increase with heritage. In other words, it may be in the interest of the operator to charge less a better heritage site. Propositions 4 and 5 consider specific cases of demand behavior, obtaining stronger results, at the cost of a generality loss.

Premium access prices are necessary to reduce the pressure on the resource, especially when numerous potential visitors are highly price-sensitive and ask for discounts or when some shareholders promote other policies. The same people, depending on their current situation, may have contradictory and incoherent claims. As a citizen, intellectual, or activist one may promote heritage preservation and argue for higher entrance prices, but as a tourist one would prefer no or low entrance fees to cultural heritage.

Propositions 1 and 2 guide the calibration of the spread between dynamic and static prices, preserving a sustainable heritage. This spread increases with the damage generated by tourists and with the difficulty to repair the site. It is important to disentangle these two effects: some sites could be both resilient and easy to repair, like castles, fortifications, or military installations. Others integrate fragile and not reproducible components, such as cave painting or old stained-glass windows. In the first case, the dynamic component of prices could be neglected, while they should impose a very high access price in the second case. In many cases, the situation is between those two extreme cases: cathedrals, pagodas, palaces, can be restored quite in their original form and the damage resulting from visits is generally neither negligible nor dramatic.

Conclusion

To conclude, we summarize our approach, list our contributions, and show future research areas.

Summary

This article has examined the joint dynamic policies of heritage conservation and pricing for World Heritage Sites. Working on the structural properties of the general (non-parametric) class of demand functions, we have highlighted different elements influencing the interplay between price and heritage. We extend the literature on heritage sites by proposing analytic (as opposed to quantitative) pricing and heritage conservation rules (Propositions 1, 2, and 3). Such optimality rules balance the total effect on profit of current demand-side effects linked to markup and sales, and future supply-side effects linked to heritage Propositions 3).

More precisely, Proposition 1 indicates that World Heritage Sites' operators, who ignore the long-term consequences of their pricing strategies (in terms of heritage degradation) will consistently price too low, thereby systematically increasing current profit at the cost of future profit and heritage damage. Our approach offers a new lens on the demand and supply sides of World Heritage Sites managed by profit-maximizing operators. Proposition 2 provides a deeper understanding of the linkage between pricing and investment strategies related to World Heritage Sites. Proposition 3 informs the operator about the heritage investment over time. Eventually, Proposition 3 offers clear-cut theoretical implications regarding the critical challenge of achieving a sustainable balance between the conflicting goals of World Heritage Site economic exploitation and conservation.

Contributions

The challenge of sustainable policies for World Heritage Sites resides in two opposing forces.

On the one hand, the heritage deteriorates with the number of visitors; on the other, the heritage quality may increase thanks to the revenue generated by entrance fees. Therefore, the question is no longer whether to allow or to prohibit visits to the site, but how to regulate this access. We contribute to extant literature by focusing on pricing policies rather than quantitative restrictions to optimize the trade-off between current profit and preservation of the heritage, i.e. the source of future profits.

Further, our contributions show that dynamic prices targeting intertemporal profit maximization are always higher than static prices targeting short-term objectives. More fundamentally, we were able, with general demand functions, to distinguish the components of the spread between both dynamic and static prices. Stronger is the depreciation of the heritage, and the more efficient are the investments able to reconstitute it, the greater is the difference between dynamic and myopic prices. When the time pattern of prices is considered, the elasticities of demand on prices and quality are fundamental determinants of these dynamics.

Eventually, our contributions emphasize that there is not one single recommendation for all cases, independently of the type of visitors, and of the existence of more or less close substitutes to the site or monument under consideration. Managerial policies should be adapted to the nature of the heritage, to the more or less heterogeneity of demand, and the competitive environment. But there are no cases such that dynamic optimization would impose lower access fees than a static one.

Research Limitations and Future Research

Our model may be extended in several ways to address several policy issues that we discuss hereafter. A public policy concern may be the access to all people, that is, access should be possible not only to the 'rich,' but also to the 'underpriviledged.' Such concern may be achieved by a discriminating pricing policy, with a low or free entrance fee for underpriviledged, as for children, seniors, large families, unemployed, or local people of poor countries. [START_REF] Apollo | Dual pricing-two points of view (citizen and non-citizen). Case of entrance fees in tourist facilities in Nepal[END_REF] reviews such policies in poor countries: to access the TajMahal, foreign visitors pay 3,500% higher fees than local visitors. The model generalization may be done in future work.

Another policy issue is that of the founding of a high conservation cost. Our model models an entrance fee; it could be extended to restrict also visitor access by a quota. For instance, Barcelona's Park Güell manages jointly its access pricing policy with the limitation of the number of visitors admitted per hour [START_REF] Goodwin | Managing tourism in Barcelona[END_REF][START_REF] Crespi-Vallbona | Managing sociocultural sustainability in public heritage spaces[END_REF]. Plus, all conservation costs are paid by the entrance fee. In the case of quota, a high conservation cost may also be born by government subsidies. The adequate private/public partnership remains an open question, calling for other model extensions.

Our model framework could also be useful to protect natural heritage sites. Indeed, some natural sites set an entrance fee to both raise revenues and limit visits. That's the case of the neolithic cave of Galdar in Gran Canaria Isle. However, for natural heritage sites, the model could be adapted or extended to allow flow regulation by restricting access, with or without an entrance fee. This model would correspond to the practice of restricting the access of Lascaut's cave (with entrance fee) and of the Callanques of Marseille (without entrance fee). Such model extension to natural heritage conservation would complement our model framework.

A Proofs

A.1 Proof of Lemma 2

We found the explicit solution for λ, by using integrating factor method, (for further details on the method see [START_REF] Boyce | Elementary Differential Equations and Boundary Value Problems[END_REF],

dλ dt -r + δ + α ∂D ∂h λ = -e h pD h , with lim t→+∞ e -rt λ(t) = 0.
We look for a function u(t) such that

d dt (uλ) = u dλ dt -u r + δ + α ∂D ∂h λ. ( 18
) d dt (uλ) = u dλ dt -u r + δ + α ∂D ∂h λ ⇒ du dt λ + dλ dt u = u dλ dt -u r + δ + α ∂D ∂h λ ⇒ du dt λ = -u r + δ + α ∂D ∂h λ ⇒ du dt = -u r + δ + α ∂D ∂h ⇒ u(t) = exp - t 0 r + δ + α ∂D ∂h ds + C .
Combining equation ( 18) and the equation satisfied by λ, we get

d dt (uλ) = -e h pD h u ⇒ λ(t) = 1 u(t) - t 0 e h pD h u( )d + C 1 ⇒ λ(t) = exp t 0 r + δ + α ∂D ∂h ds - t 0 e h pD h exp - 0 r + δ + α ∂D ∂h ds d + C 2 .
In order to have the transversality condition to be verified, we must impose that

A.2 Results about Optimization

Lemma 3. The shadow price is non negative, λ * ≥ 0. In particular, two possible scenarios can occur, 1. λ * (t) > 0 for t ≥ 0;

2. λ * (t) > 0 for t ∈ [0, T * ) and λ * (t) = 0 for t ∈ [T * , ∞).

Proof. From Lemma 2, we have that λ * (t) can be written as e t 0 g(s)ds ∞ t f ( )d with some f ( ) ≥ 0 for each ∈ [0, T ]. If there exists T > 0 such that f ( ) > 0 for a.e. ∈ [T, ∞], then for each t ≥ T , λ(t) ≥ 0 and moreover for t < T λ(t)

∞ T f ( )d > 0.
Now, we assume that the condition above is not verified. This means that for each T > 0, there exists a non negligible set A (T ) ⊆ [T, +∞) such that f ( ) = 0, for each ∈ A (T ). If for each T > 0, A (T ) ⊂ [T, ∞), then λ(t) > 0 for each t ≥ 0. Otherwise, there exists T * such that A (T ) = [T, ∞). This means that for t < T * λ

(t) = e t 0 g(s)ds [t,∞)∩A (t) f ( )d > 0,
and for t ≥ T * , λ(t) = 0.

Proof of Lemma 1. The solution of the Pontryagin system is a candidate optimal solution.

Its shape depends strongly on λ * (t). Indeed, in order to solve the Pontryagin system, we need to find

(u * (t), p * (t)) ∈ argmax (u,p)∈(R + ) 2 H(u, p, h * (t), λ * (t)), for each t ≥ 0.
In the next lines, we will find out that if λ * = 0 the there exists a solution in the inner, namely u * , p * > 0. Otherwise, we can only find a solution on the border. Since, a priori we do not know the sign of λ * (t), we study the different cases presented in the Lemma 3.

If the first case occurs, we study

(u * (t), p * (t)) ∈ argmax (u,p)∈(R + ) 2 H(u, p, h * (t), λ * (t) =0
), for each t ≥ 0.

In this case the critical point satisfies the following equations,

∂H(u * , p * , h * , λ * ) ∂u = 0 =⇒ λ * • K (u * ) = 1, ( 19
) ∂H(u * , p * , h * , λ * ) ∂p = 0 =⇒ p * = - D(p * , h * ) ∂D(p * ,h * ) ∂p + αλ * . ( 20
)
Since u * is obtained by reversing the function K , then u * > 0. Assume that the function

F (p) = - D(p, h * ) ∂D(p,h * ) ∂p
+ αλ * admits a fixed point different from zero, p * . Then the positivity of λ * , together with the ∂D ∂p < 0 implies that p * > 0. Moreover, we can say that the critical point furnishes a candidate maximum for (7). Indeed, since λ * > 0 and K < 0 Given ( 21) and ( 22),

∂ 2 H(u * , p * , h * , λ * ) ∂u 2 = λK (u * ) =⇒ ∂ 2 H(u * , p * , h * , λ) ∂u 2 < 0. ( 21
∂ 2 H ∂u 2 ∂ 2 H ∂p 2 - ∂ 2 H ∂u∂p 2 = ∂ 2 H ∂u 2 ∂ 2 H ∂p 2 =⇒ ∂ 2 H ∂u 2 ∂ 2 H ∂p 2 - ∂ 2 H ∂u∂p 2 > 0,
Now, we observe that the Hamiltonian H, evaluated in the critical point, is negative definite.

Indeed, let us consider the Hessian matrix associated to the hamiltonian H, H H , evaluated in the critical point obtained in ( 19)-( 20) , Notice that H(u, p, h * (t), 0) = pD(p, h * (t)) -u. It can be easily proven that the point (0, p * ) with p * = -D ∂D∂p is a local maximum for H(u, p, h * (t), 0). Moreover, the hessian of the function pD(p, h) is negative definite in the point (p * , h * ), which implies that the point (p * , h * ) is a local maximum for pD(p, h). Since u > 0 we conclude that (0, p * , h * ) is a local maximum for H(u, p, h, 0), H(u, p, h, 0) = pD(p, h) -u ≤ pD(p, h) ≤ p * D(p * , h * ) = H(0, p * , h * , 0).

To conclude, we prove that for all h, u that stay close enough to h * , u Then, we conclude that the last line is negative, by repeating the same argument proposed in the proof of Theorem 2.5 in [START_REF] Yong | Stochastic controls: Hamiltonian systems and HJB equations[END_REF].

A.3 Proof of Proposition 1

Following (20), the dynamic price is p = -D(p(t), h(t))

∂D(p(t),h(t)) ∂p

+ αλ, with λ given by Lemma 2. For the static price, the shadow price of heritage is null over the whole planning period, that is λ = 0 for all t ≥ 0, and p static = -D ∂D ∂p

. Note with (19) that λ = 1 K and substituting it in (20) offers p = -D(p(t), h(t))

∂D(p(t),h(t)) ∂p + α K . The difference between the two prices is straightforward.

A.4 Proof of Proposition 2

Isolating λ from ( 19), and substituting λ in (20) yields p = -D ∂D ∂p + α K . Then we rearrange this result with the price elasticity notation.

A.5 Proof of Proposition 3

In order to study the evolution of the price variable, we differentiate the first-order condition on price D + p ∂D ∂p -αλ ∂D ∂p = 0, rearranged from (20), with respect to time t. Mathematically, for all t > 0, we have:

∂D

(

  see for instance Throsby 1999; Creigh-Tyte 2000; Rizzo and Throsby 2006; Artal-Tur et al. 2018; Sirilersuang and Pongkijvorasin 2018) and reads K = K(u).

  Figure 2: The Effect of Heritage on Price and Demand

Figure 3 :

 3 Figure 3: Conservation and Pricing Policies over Time for Different Classes of Demand Functions.

H.

  H (u * , p * , h * , λ * ) = * ,h * ) ∂p 2 -D(p * , h * ) ∂ 2 D(p * ,h * ) ∂p 2 ∂D(p * ,h * ) ∂p 2 ∂D(p * ,h * ) ∂h -D(p * , h * ) ∂ 2 D(p * ,h * ) ∂p∂h ∂D(p * ,h * ) ∂p 0 ∂D(p * ,h * ) ∂h -D(p * , h * ) ∂ 2 D(p * ,h * ) ∂p∂h ∂D(p * ,h * ) ∂p -D(p * , h * ) ∂ 2 D(p * ,h * )We call D k the leading minor of order k. Thus, from the hypothesis on K and the sign onλ * D 1 = λK (u * ) < 0, D 2 = λK (u * ) * , h * ) ∂h -D(p * , h * ) ∂ 2 D(p * ,h * ) ∂p∂h ∂D(p * ,h * )study the case when λ * satisfies the second case of Lemma 3. Then, we need to find (u * (t), p * (t)) ∈ argmax (u,p)∈(R + ) 2 H(u, p, h * (t), 0), t ∈ [T * , ∞).

  rt) (H(u(t), p(t), h(t), 0) -H(u * (t), p * (t), h * (t), 0rt) (H(u(t), p(t), h(t), λ(t))-H(u * (t), p * (t), h * (t), λ(t)))dt+ T * 0 exp (-rt) λ(t)(b * -b)dt.

Table 1 :

 1 Main Notations

	t	= continuous time,
	r	= discount factor, parameter,
	p(t)	= price (or entrance fee) at time t, control variable,
	h(t)	= heritage attractiveness at time t, state variable,
	D(p, h)	= demand (or affluence) of visitors,
	u(t)	= expenditure on heritage conservation at time t, control variable,
	α	= impact of visitors on heritage degradation, parameter,
	δ	= autonomous decay of heritage, parameter,

  exp (-rt) (π(t, h(t), u(t)) -π(p * (t), h * (t), u * (t)))dt ≤ 0 (23)where we indicate with p * , h * , u * the solution of the maximum principle. If case 1. of Lemma 3 is verified, then since H is concave in p * (t), h * (t), u * (t) for t ∈ [0, T ), to prove (23) we just adapt the proof of Theorem 2.5 in[START_REF] Yong | Stochastic controls: Hamiltonian systems and HJB equations[END_REF]. Now consider we focus on the case 2. of Lemma 3. We callb = -αD(p(t), h(t)) -δh(t) + K(u(t)), b * = -αD(p * (t), h * (t)) -δh * (t) + K(u * (t)),and since (u * , p * , h * ) is a local maximum for H, we get

* ∞ 0 ∞ 0 exp (-rt) (π(p(t), h(t), u(t)) -π(p * (t), h * (t), u * (t)))dt = = ∞ 0 exp (-rt) (H(u(t), p(t), h(t), λ(t))-H(u * (t), p * (t), h * (t), λ(t)))dt+ ∞ 0 exp (-rt) λ(t)(b * -b)dt = T * 0 exp (-rt) (H(u(t), p(t), h(t), λ(t))-H(u * (t), p * (t), h * (t), λ(t)))dt+

  Now we focus on the evolution of u. Note that the first order condition on conservation expenses (19) must hold for all optimal values and at any time. We differentiate both sides of (19) with respect to time, noting that both sides must remain equal for optimal values. Thus

	∂p	dp dt	+	∂D ∂h	dh dt	+	∂ 2 D ∂p 2	dp dt	+	∂ 2 D ∂p∂h	dh dt		p -	α K	+	∂D ∂p	dp dt	+	αK (K ) 2	du dt	= 0.	(24)
	Substituting (19) in (20) yields
										p -	α K	= -	∂p D ∂D	> 0.	(25)
	Note that
												-	∂D ∂h ∂D ∂p	=	e h e p	p h	.		(26)
	Substituting (27), (25), and (26) in (24) and rearranging provides the result (11).
	we obtain for all t > 0:
										dλ dt	K + λK		du dt	= 0.

The Hamiltonian (8) must be jointly concave in the control variable p and u and in the state variable h. See[START_REF] Mangasarian | Sufficient conditions for the optimal control of nonlinear systems[END_REF] andKamien and Schwartz (2012, Section 15).

Note that α cannot be "too" large, otherwise the number of visitors falls to zero, D = 0, which means that the site is closed to the public to be preserved. This corner solution has been implemented in some sites, such as Lascaux Cave, which was closed to the public in 1963 (see[START_REF] Bastian | Lights and shadows on the conservation of a rock art cave: the case of Lascaux Cave[END_REF][START_REF] Martin-Sanchez | Lascaux cave: an example of fragile ecological balance in subterranean environments[END_REF]. Since then, only a precise replica of the cave can be visited. Interior solutions, D > 0, are the more common cases of sites open to the public. Because we are interested in accessible sites, we rule out D = 0 by imposing an upper bound on α.

The case δ < 0 studied inHernández and León (2007, 2013);[START_REF] Sirilersuang | Dynamic optimal joint policies for tourism promotion and environmental restoration[END_REF] corresponds to a natural recovery process. This case makes sense in their context of the natural site, but not in our context of World Heritage Site.

The surveys by[START_REF] Jørgensen | Dynamic games in the economics and management of pollution[END_REF] and by[START_REF] Benchekroun | Collaborative environmental management: A review of the literature[END_REF] provide more information on the proportional depreciation of the state variable in sustainable economies.

It is interesting to notice that in the text of the decisions adopted during the 43 rd session of World Heritage Committee (UNESCO, Baku, 2019) the word "access" is used 44 times, the words "restrict," "restriction," "restrictive," 10 times, while the words "fees" and "price," are used respectively 1 and 0 times (UNESCO, 2019).

Substituting in the previous result λ = 1 K from (19), dividing by K , and rearranging yields

Recalling K < 0 from assumption (5) gives for all t > 0,

which proves the result.

A.6 Proof of Corollary 1

Immediate with Proposition 3.

A.7 Proof of Proposition 4

Substitute (13) and its derivatives in Proposition 3. With an additively separable demand function, the sales effect is null. Only the markup effect remains.

A.8 Proof of Proposition 5

Substitute (15) and its derivatives in Proposition 3. With a multiplicatively separable demand function, the markup and sales effects are of same magnitude, but have opposite signs. They cancel each other out.

A.9 Proof of Corollary 2

Immediate with Propositions 3 and 5.


Z Ning

M Rizk

A Baghdadi

J-Ph Diguet

Enhancing embedded AI-based object detection using multi-view approach

Keywords: Multi-view, Object detection, Embedded edge device

Object detection based on convolutional neural network (CNN) is widely used in multitude emergent applications. Yet, the deployment of CNNs on embedded devices at the edge with reduced resources and power budget poses a real challenge. In this paper, we address this issue by enhancing the detection performance without impacting the inference speed. We investigate the use of multi-view for the same scene to achieve better detection performance. A novel system of distributed smart cameras is proposed where each camera integrates a CNN for detection. Implementation results show that using light networks on the distributed cameras can lead to better detection performance and a reduction in the overall consumed power.

I. INTRODUCTION

Object detection is widely used in several modern application fields such as autonomous vehicles, medical imaging, crowd detection, exploration, etc. Currently, artificial intelligence (AI) is the key trend to detect objects in images and videos. CNN architectures allow the detection of multiple objects in a scene, yet at a cost of high computational resources and power consumption. This fact creates a challenge when deploying CNNs on embedded devices with limited resources and power budget especially for applications that require realtime inference. Other parameters such as the size and weight of the computational platform as well as its power consumption are critical for trendy applications such as wearable devices, drones and autonomous robots.

Mainly, compared to light neural networks, dense networks lead to higher detection performance in terms of accuracy and precision but at the cost of inference speed. Often, systems integrating CNN-based detection methods have a trade-off between inference rate and detection performance. Several techniques have been recently proposed to optimize the trained networks towards embedded devices. These techniques are highly dependent on the target hardware, and usually produce degraded detection performance. In addition, the training cost in terms of power and utilized resources increases with the size of the target network.

In this work we address this issue with a totally new approach that makes use of multi-view of the same scene to increase the detection accuracy. We propose a system of distributed nodes, where each node integrates a camera and an embedded device that performs inference at the edge using light CNNs. The distributed nodes collaborate in order to achieve best predictions with reduced overall power consumption. Fig. 1 illustrates the proposed system for the application of detecting ships. The distributed nodes record the same object with different angles of view so that each one has its own prediction. Then, the predictions are all gathered and the final decision is taken. Note that in this case the nodes share only the detection information rather than the captured images. Hence, privacy, security and transmission bandwidth are conserved. In addition to enhancing the detection, using multi-view concept during inference helps in avoiding occlusion, which can deteriorate the predictions. For example, as shown in Fig. 2, the pedestrian is hidden by a car. This generally brings a loss in the prediction confidence score. Multi-view can enhance the training of the network. Training only on one representation of an object can bring a bias, which damages the generality of the classifier. For instance, if the classifier is trained only on images of the front side of an object, it will be difficult for it to recognize the object from images of the backside. Therefore, the addition of this multi-view feature allows the classifier to train on vast various representations of an object, which brings a true added value.

II. RELATED WORK

Few research works have addressed object detection and classification in the context of multi-view. Arsalan Mousavian et al. [START_REF] Mousavian | 3d bounding box estimation using deep learning and geometry[END_REF] have proposed a method to construct 3D bounding boxes based on a single image and geometric relationships of 979-8-3503-9851-9/22/$31.00 ©2022 IEEE Fig. 2. Pedestrian hidden by a car the object, which can perform 3D object detection and pose estimation from a single image. Their main contribution is limited to the prediction of a 3D bounding box from a 2D one. The method proposed in [START_REF] Qi | Frustum pointnets for 3d object detection from rgb-d data[END_REF] has been able to accurately estimate 3D bounding boxes for even small objects, using both mature 2D object detectors and advanced 3D deep learning for object localization. The authors in [START_REF] Wang | Detr3d: 3d object detection from multi-view images via 3d-to-2d queries[END_REF] have proposed a framework for multi-camera 3D object detection called DETR3D, which extracts 2D features from multiple camera images and then uses a transformation matrix to relate 3D positions to multi-view images. Recently, the work proposed in [START_REF] Katircioglu | Human detection and segmentation via multi-view consensus[END_REF] has used multi-view approach during training on a skier's pose segmentation system, to address the lack of large datasets.

To the best of our knowledge, no published work has used a multi-view approach in the detection phase, and no one has investigated the enhancement of multi-view approach on the performance of object detection system.

III. METHODS

A. Available multi-view datasets

In order to apply our proposed idea of using multiple views, we need a dataset that includes several images for the same scene, which are captured from different points of view. As a starting point, we have investigated the published datasets.

The well known COCO (Common Objects in Context) [START_REF] Lin | Microsoft coco: Common objects in context[END_REF] is a dataset of 80 common objects provided by Microsoft and has been widely used as a baseline for object detection problems. The amount of data in this dataset is sufficient for a general target detection system training task. However, each object in this dataset usually appears only once per scene and there is no information about multiple views.

Several research works have already provided multiperspective datasets. The multi-view car dataset provided by EPFL [START_REF] Ozuysal | Pose estimation for category specific multiview object localization[END_REF] includes 2000 images of 20 cars with different positions and angles of view. However, all the images have the same background and constant light intensity. In our context above, this amount of data is not big enough for the training. In addition, the stated images' identical properties tend to cause overfitting of the model.

Another dataset, ELKI Multi-View Clustering Data Sets [START_REF] Schubert | ELKI Multi-View Clustering Data Sets Based on the Amsterdam Library of Object Images (ALOI)[END_REF], from ALOI (Amsterdam Library of Object Images) contains 1000 objects and 100 photos for each object. The variables in this dataset contain different angles with different light as well as the RGB information. However, this dataset uses only black background which has the same default as the previous one. They have only one object for each class, so they may not be capable of making good decisions on other objects of the same class. The PASMVS dataset [START_REF] Broekman | Pasmvs: a dataset for multi-view stereopsis training and reconstruction applications[END_REF] is generated with scripts, which do not include images, but the code and materials used to reproduce 18,000 scenes and backgrounds. It has only four classes, including imaginary animals which are not usable in real-life detection.

There are also other multi-view datasets that we don't mention here as they are not convenient in our case. Therefore, we decided to create our own dataset.

B. Creation of multi-view dataset

To build a multi-view dataset, we firstly define 7 classes to be included: mug, shoe, keyboard, wallet, pencil case, book, and mouse. For each class, we define its coordinate system to express the object's orientation by yaw, pitch, and roll. The mugs have their centers as the origin, the handle as the Xaxis, the Z-axis facing outward, and the left-handed system to establish the corresponding Y-axis. The origin of the shoe is defined at the heel, with the X-axis along the body of the shoe. We define the short side of the keyboard as the Y-axis and the long side as the X-axis. When the opening of the wallet is on the right side, the coordinate origin of the wallet is defined as the vertex in the lower left corner. Along the bottom edge to the right is the X-axis and up is the Y-axis. For the pencil case, the origin is defined as the position of the zipper lock when it is closed, and the Y-axis is along the long axis. We define the long side of the book as the Y-axis and the short side as the X-axis. The definition of the mouse axis is somewhat special, with the scroll wheel as the origin, when placed forward, pointing to the left for the Y-axis, pointing forward for the X-axis.

We choose three different rooms and fix several cameras at different locations. Different scenes are generated by changing the types, angles, and positions of the placed objects and changing the lighting conditions of the rooms. We pre-define more than ten positions in each room where objects could be placed, and use a randomized method to generate a table containing the placement of objects corresponding to the scenes. Then we manually enter the room to place the items according to the generated table and then use a program to control all cameras to take pictures at the same time and store the images as png files. After the shooting has been completed, we manually annotate them using CVAT [START_REF] Sekachev | v1.1.0[END_REF]. At the same time, blurring is used to erase parts of the images that might affect the learning process, such as the shoes worn by the person taking the picture. In most cases the objects are placed at different positions on the ground, whereas sometimes we use objects outside our defined classes (e.g. cardboard boxes) to increase the height of the object positions. Fig. 6 shows some example images from the dataset.

C. Baseline experiment

YOLO [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF] is a state-of-the-art CNN-based model for object detection, with great advantages in accuracy and speed. It is the acronym of "You Only Look Once", which means that you only need to browse the image once to recognize the class and location of the objects in the picture. It also offers pre-trained models on the COCO dataset. YOLOv5 [START_REF] Jocher | Ultralytics/yolov5: v3.1 -Bug Fixes and Performance Improvements[END_REF] is an extension of the YOLO series, and we can also consider it as an improved work based on YOLOv3 [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] and YOLOv4 [START_REF] Bochkovskiy | Yolov4: Optimal speed and accuracy of object detection[END_REF]. Although there is no corresponding paper description for YOLOv5, the authors are actively working on this open source project and have gained a lot of experience and advantages in practice together with other users. The model uses the PyTorch architecture, which is very extensible and convenient. Five versions are given in the official YOLOv5 code, namely YOLOv5n(nano), YOLOv5s(small), YOLOv5m(medium), YOLOv5l(large), and YOLOv5x(extreme). These different variants represent a good trade-off between accuracy and speed, making YOLOv5 convenient for users to implement.

We select a number of scenes in the approximate ratio of 3:5:4 based on the amount of data in the three rooms, then choose the images recorded by all cameras in these scenes as the test set so that the number of images in the test set is about 20% of the total. Another 80 scenes containing 8 cameras are selected from the total dataset, and these images are taken out and kept as the test set for the Camera position experiment that will be presented later. The remaining scenes are divided into training and validation sets according to the ratio 3:1, and all images of the same scene always belong to the same set.

In order to improve the training efficiency, we have conducted research on hyperparameters by doing a sweep using Wandb [START_REF] Biewald | Experiment tracking with weights and biases[END_REF]. For each set of hyperparameters, 5 training cycles have been conducted to observe the training effect. Then we have discarded the unreasonable and undesirable hyperparameter combinations and selected the ones with good training effects to continue the training for 300 complete training cycles.

After the training is completed, we select the bestperforming weights for each network size during the training process and test the model performance on the test set.

All trainings are done with a computer equipped with NVIDIA GeForce GTX 1060 6GB. All subsequent experiments were performed on the same machine.

D. Multi-view experiment

In this experiment, we want to verify that multiple cameras from different views working together can improve prediction accuracy. Unlike the previous experiment, we now ask the system to give predictions for one scene at a time, instead of predicting each image individually. In our example, each scene contains four images seen by four cameras at different locations. Therefore, we run four programs simultaneously, each using the model trained in the previous experiment, with four images of different viewpoint of the same scene as input. Each program gives all the bounding boxes above the set threshold and their corresponding prediction categories and confidence, and then the system considers these predictions together to give the final prediction.

Since each camera observes from a different angles of view, we first transform all the bounding boxes to the same plane as shown in Fig. 7(a), using Homography Transformation [START_REF] Dubrofsky | Homography estimation[END_REF]. This is a matrix transformation that allows us to project images obtained from cameras into the same coordinate system. This approach has been widely used in the field of computer vision. For each object, each camera may provide a bounding box, so the object will have multiple bounding boxes. We want to merge the bounding boxes so that all predictions for one object can be treated together. Inspired by the NMS (nonmaximum suppression) algorithm that has been used in the YOLO [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF], we propose a similar method as described in Algorithm 1. Each subsystem outputs all bounding boxes above the threshold, projecting them onto the plane where the object is located. Then, the merge algorithm is applied to these projected bounding boxes together. If some of these boxes are merged, then their corresponding predictions are also merged and treated together in later process. Fig. 7(b) shows the bounding boxes before and after merge.

All the predictions from the nodes are pre-processed to obtain some bounding boxes, each of which also contains several predicted categories and corresponding confidence scores.

Now we can focus on verifying that multi-view can improve the accuracy of prediction. For that, we consider first a simple situation, in which there is only one object within each scene and each camera sees only this one object, i.e., all the predicted bounding boxes belong to this only object. For each scene, we first compare the prediction of each node with the true value to get the accuracy of the prediction for the images. Then we combine the predictions of all nodes to give the final prediction and compare this prediction with the true value to get the accuracy of the prediction for the scene.

We currently propose several methods to give the final prediction: the maximum value method, the average method, and the summation method. The maximum value method means that the prediction with the highest confidence is directly selected as the final prediction. The average method and the summation method average or sum the confidence of all predictions of the same class, respectively, and then select the largest one as the final prediction. In practice, these three methods may yield different final predictions, for example, in the 7 cameras system example shown in the Table I, using the three methods will yield 0, 1, 2 as the prediction respectively. The overall process 1 of this multi-view experiment is shown in Fig. 8.

E. Camera position experiment

This experiment is designed to investigate the effect of camera placement on accuracy in the multi-view approach. We want to demonstrate that the camera placement affects the correct prediction rate. In other words, it is the multiple views that improve the correct prediction rate, not simply the increase in the number of cameras that brings the improvement.

We use scenes with 8 cameras as the test set for this experiment, which have been selected when segmenting the dataset before training and have not been used as the training or validation set. The relative relationship between the 8 cameras and the zone of objects is shown in Fig. 9. Four of the cameras are placed on the same table as the objects, and the other four cameras are placed higher than the table. For each scene, we select images taken by four of the eight cameras as input, give the final output using the same method as in the multiview experiment, and then repeat the process by replacing the selected camera combination. Finally, we compare the prediction accuracy of the different camera combinations. 1 Code published at https://github.com/mm0806son/Enhance embedded object detection system using multi view

A. Baseline experiment

In the baseline experiment, we verify the training and inference of different size models of YOLOv5 and record their accuracy on the multi-view dataset we produced. The results are considered as a reference for subsequent experiments. At the same time, we confirm that using smaller size models is more advantageous in terms of energy consumption and memory requirements, and thus more suitable for embedded systems.

We have trained 300 epochs on the training set for 5 models of different sizes and then used the validation set to evaluate them. The size of the models and their performance on the validation set are shown in Table II 2 . In general, the larger is the model, the better is the performance, yet the training and inference time is longer. The evaluation metrics are shown in Fig. 10. The GPU usage during training is shown in Fig. 11. For the three smaller models (i.e., medium, small, nano) we have used 16 as the batch size. For the two larger models (i.e., extreme, large) we have used 4 as the batch size due to the GPU memory limitation. The training of the larger models consumes 1.5× to 2× the energy required to train the smaller models and takes up a large amount of memory, placing higher demands on the GPU. Smaller models are also faster to train and require less computing power from the hardware.

C. Camera position experiment

Regarding this third experiment, we choose the cameras 0,1,3,7 as the first group and the cameras 1,2,4,5 as the second group. The first group of cameras is able to observe around the object, while that of the second group gather on one side of the object. The accuracy of each model for detection alone on images and for detection using multi-view approach on scenes is shown in Table IV.

When using the smallest nano network, we find that camera placement has a significant impact on performance. Regardless of which of the three strategies is used, the accuracy of the two groups of cameras differs by about 0.13. When using the other larger networks, the performance of the two groups of cameras fluctuates, with no one group being able to significantly outperform the other, but both outperforming the predictions made for each image individually.

It can also be noted from the results that until the YOLOv5l model, using a more complex model helps significantly for the second group of cameras, as the large model has a 30% improvement in accuracy over the nano model. This is very likely due to the fact that the images observed by these cameras did not have enough details to enable a very small network to perform accurate decisions.

The results show that the improvement in performance by changing the position can be comparable to the improvement by using a larger network. If the cameras can be placed in the right position to capture more details at the time of observation, its prediction results will be better. This makes a lot of sense for saving energy and computational power.

V. CONCLUSION

In this paper, we present the idea of using multiple subsystems with different views to work together to improve the overall accuracy of an object detection system. To that end, we have designed and collected a multi-viewpoint dataset on our own, containing several common objects in an office environment. For each scene in this dataset, we have multiple images from different angles of view to validate multi-view related experiments.

Using this dataset, we have conducted three experiments: baseline, multi-view, and camera position. The performance, energy consumption, and computational power requirements of different models of YOLOv5 have been measured. Smaller YOLOv5 models have less demand and faster training and inference, yet perform relatively worse. In the multi-view experiments, we have presented the complete flow of how the multi-view object detection system works. The predictions of the nodes are first preprocessed, and then the final prediction of the scene is obtained using three simple strategies: max, average, and sum. We found that a distributed system using several small models has been able to greatly improve detection performance, outperforming large models. This approach requires less computational power and is much faster. In the camera placement experiments, we have found that the location of the cameras can have a significant impact on system performance, especially for small networks. A good placement of the cameras can achieve an enhancement effect similar to the use of large models.

Overall, our work demonstrates that using multi-view nodes to form a global object detection system can save computational power and increase speed. This has great interest for models deployed on edge and embedded systems. Sometimes people also want to build prototypes first with few hardware resources. This approach can help them get reliable predictions before committing more resources. In future work, we can also take the orientation of the object into consideration to pursue further performance enhancement, since the observed images from different views have a correspondence of orientation.

Fig. 1 .

 1 Fig. 1. Boat detected using a multi-view network

Fig. 3 .

 3 Fig. 3. Example of objects

Fig. 4 .Fig. 5 .

 45 Fig. 4. Statistic of the constructed dataset

Fig. 4

 4 shows the number of times each class appears in the dataset and the statistics of the number of objects in each image. Most of the images (1520) have only one object, others (640) have two objects, a few (265) have three objects, and a very few (55) have no objects and are used as background images for training.

  Fig 5 shows the heat map of the object positions. Most of the objects are placed in the center of the view (green area). The blue color corresponds to the area with less number of objects.

Fig. 6 . 9 : 12 :

 6912 Fig. 6. Examples from the constructed dataset

  Fig. 7. Pre-processing of node output results

Fig. 8 .

 8 Fig. 8. Process of multi-view experiment

Fig. 9 .

 9 Fig. 9. Cameras' position IV. RESULTS

Fig. 11 .

 11 Fig. 11. Results of baseline experiment

TABLE I PREDICTION

 I EXAMPLE OF A 7 CAMERAS SYSTEM

		cam0 cam1 cam2 cam3	cam4 cam5 cam6
	Prediction	0	0	1	1	2	2	2
	Confidence	0.8	0.2	0.7	0.5	0.5	0.4	0.4

Visualised results can be found at https://wandb.ai/smart cam/ Enhance embedded object detection system using multi view?workspace= user-mm0806son

https://www.kaggle.com/datasets/zijiening/smart-cam-multiview

ACKNOWLEDGMENT

This work was supported in part by the Regional Council of Bretagne through the ODESSA FEDER project. The multiview dataset 3 used in this paper was built by Aziz Sellami, Yuming Wang, Sebastian Heyer and Zijie Ning. They worked together to design the collection scheme and completed the tasks of item placement, photography, and labeling.

B. Multi-view experiment

In the multi-view experiment, we use each model separately for image prediction, and then different strategies for scene prediction. The results of the experiment are shown in Table III.

All three methods described in Section III-D to give the final multi-view prediction have achieved good results, greatly improving the correctness of inference and enabling small models to outperform large ones. In the baseline experiment, we investigate the energy consumption (Fig. 11(a)), GPU memory requirements (Fig. 11(b)) of different models. We used 4 nano (YOLOv5n) models for scene prediction, and their total number of parameters and computation is about the same as 1 small model, yet they perform better. These 4 nano models can even outperform 1 extreme (YOLOv5x) model, while having only one-tenth of its number of parameters and executes 20× faster. It is also noticed from the Fig. 11(a) that the larger models (i.e. x,l) consume about twice as much energy as the smaller models when their batch size is a quarter of that of the smaller models (i.e. s, n). Therefore, under our hardware conditions, using 4 small models saves about half the energy. Using the multi-view system approach allows to achieve better detection performance while requiring much less computational power.