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Optimal Control of the Mean Field Equilibrium for a Pedestrian Tourists’ Flow Model

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Abstract

Art heritage cities are popular tourist destinations but for many of them overcrowding is becoming an issue. In this paper, we address the problem of modeling and analytically studying the flow of tourists along the narrow alleys of the historic center of a heritage city. We initially present a mean field game model, where both continuous and switching decisional variables are introduced to respectively describe the position of a tourist and the point of interest that he/she may visit. We prove the existence of a mean field equilibrium. A mean field equilibrium is Nash-type equilibrium in the case of infinitely many players. Then, we study an optimization problem for an external controller who aims to induce a suitable mean field equilibrium.

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Notes

  1. Note that Eq. 16 are integral equalities, and that every flow function gi,j is built by means of the incoming flow g, the optimal controls and the split functions, as explained in Appendix A. Hence the use weak star convergence of the split functions is appropriate.

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Acknowledgements

This research was partially funded by a INdAM-GNAMPA project 2017

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Correspondence to Fabio Bagagiolo.

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Appendices

Appendix A: On Functions g and g i,j

In this appendix, we discuss the relationship between the flow functions g and gi,j and the components of \(\mathcal {M}\), and hence of ρ both from a spatial and a temporal perspective. We consider only the branch B1,1 as analogous arguments apply to the other branches of our model.

The agents’ flow, represented by the time depending g, enters the branch B1,1 at 𝜃S. Then g can be interpreted as number of agents per unit of time and represents an incoming density with respect to time. Differently, the total mass ρ1,1 on branch B1,1 using Eq. 17, is obtained integrating over the branch the density with respect to spacem1,1, representing the number of agents per space unit. The mass spread along the branch reaches in time the switching points 𝜃1,𝜃2, where it is again converted in time dependent flows gi,j entering the subsequent branches, with gi,j again densities with respect to time. The process repeats at every switching point.

We then describe the mathematical relationship between these two different kinds of densities. Our argument is connected to what is called disintegration of a measure (Ambrosio et al. 2008; Camilli et al. 2017).

Let us consider an agent at the position 𝜃 at time s and assume that it is going towards 𝜃1. It arrived in 𝜃S at time \(\underline {\sigma }(\theta ,s) \leq s\) and it will reach 𝜃1 at a time \(\overline {\tau }(\theta ,s) \geq s\). By Remark 1, in s and in all of the interval \([\underline {\sigma }, \overline {\tau })\) the agent has a constant velocity \(u^{1,1}(s) = u^{1,1}(\underline {\sigma })\) such that \(|u^{1,1}(\underline {\sigma })|= |\theta _{1}-\theta _{S}|/(\overline {\tau }-\underline {\sigma })\), until leaving the branch. When the total mass ρ is given, then \(\overline {\tau }\) realizes the minimum in \(\inf _{\tau \in (\underline {\sigma },T]}\left \{\frac {1}{2}\frac {(\theta _{1}-\theta _{S})^{2}}{\tau -\underline {\sigma }} +{\int }_{t}^{\tau }\mathcal {F}^{1,1}{\kern 1.7pt}ds +V(\theta _{1},\tau ,0,1)\right \}\), i.e., the second argument of the minimum in Eq. 13; the value of \(\underline {\sigma }\) is evaluable using conditions Eqs. 713. In addition, \(\underline {\sigma }\) enjoys the properties linking g and gi,j which are described below.

Let us denote \(q(\underline {\sigma })=1/u^{1,1}(\underline {\sigma })\) and, for the sake of simplicity, let us suppose that \(\underline {\sigma }\) and q are differentiable (actually, they are Lipschitz and at least a.e. differentiable).

Let us also assume 𝜃S = 0, 𝜃 > 0. We can implicitly define the value of \(\underline {\sigma }(\theta ,s)\) through the following equation:

$$ s = \underline{\sigma}(\theta,s)+q(\underline{\sigma}(\theta,s))\theta. $$
(25)

Differentiating Eq. 25 with respect to s and 𝜃 we derive

$$ \begin{array}{@{}rcl@{}} \underline{\sigma}_{s}(\theta,s)&=&\frac{1}{1+q^{\prime }(\underline{\sigma}(\theta,s))\theta}, \quad \underline{\sigma}_{\theta}(\theta,s)=\frac{q(\underline{\sigma}(\theta,s))}{1+q^{\prime }(\underline{\sigma}(\theta,s))\theta} \quad \Rightarrow \quad \underline{\sigma}_{s}(\theta,s)\\ &=&-\frac{\underline{\sigma}_{\theta}(\theta,s)}{q(\underline{\sigma}(\theta,s))} \end{array} $$

Now, suppose that all agents entering (𝜃s, 1, 1) at any time move towards 𝜃1. Then the flow g of agents arriving in 𝜃S and moving towards 𝜃1 is spread over B1,1, according to the law \(m^{1,1}(\theta ,s)=-g(\underline {\sigma }(\theta ,s))\underline {\sigma }_{\theta }(\theta ,s)\). In addition, the flow of agents crossing 𝜃 in B1,1 at time s is given by \(m^{1,1}(\theta ,s)u^{1,1}(s) = g(\underline {\sigma }(\theta ,s))\underline {\sigma }_{s}(\theta ,s)\). Both relationships may be verified by standard mass balance/conservation arguments. They obviously hold only if at time s agents have already arrived at 𝜃, that is when \(\underline {\sigma }(\theta ,s) \geq 0\), otherwise the density m1,1(𝜃,s) is equal to zero. In particular, at the switching point 𝜃1 the arriving flow, coinciding with the flow g0,1(s) entering the new branch B0,1 in time, is \(s\mapsto g_{0,1}(s) = g(\underline {\sigma }(\theta _{1},s))\underline {\sigma }_{s}(\theta _{1},s)\).

If differently, agents entering through (𝜃s, 1, 1) split among different choices, and the corresponding split fraction that moves towards 𝜃1 is \(\lambda _{2}^{(\theta _{S},1,1)}\) (see Eq. 20 and Definition 3), then the entering flow in B0,1 through 𝜃1 is \( s\mapsto g_{0,1}(\underline {\sigma }(\theta _{1},s))=\lambda _{2}^{(\theta _{S},1,1)}g(\underline {\sigma }(\theta _{1},s))\underline {\sigma }_{s}(\theta _{1},s)\). This is also the flow g01 to be considered in Eq. 16. Similar considerations (with different function \(\underline {\sigma }\) and q) hold in the case of agents moving towards 𝜃2 in the branch B1,1 and for all other cases in the other branches (with the corresponding flows gi,j).

Let us finally argue on the uniqueness of \(\underline {\sigma }\) and \(\overline {\tau }\). Specifically, consider the agents entering at time \(\underline {\sigma }\) at (𝜃S, 1, 1) and moving towards 𝜃1, and that reach such state at time \(\overline {\tau }\). We claim: (1) any arrival time \(\overline {\tau }\) originates from a unique entering time \(\underline {\sigma }\); (2) any entering time \(\underline {\sigma }\) generates a unique arrival time \(\overline {\tau }\), for any nonzero flow of agents at 𝜃1. A sketch of the proofs follows:

  1. (1)

    Let us suppose that the agents are optimally moving from 𝜃S to 𝜃1 in the branch B1,1,, and that the agents respectively starting at \(\underline {\sigma }_{1}\) and at \(\underline {\sigma }_{2}>\underline {\sigma }_{1}\) reach 𝜃1 at the same time \(\overline {\tau }<T\). This means that \(\overline {\tau }\) optimizes the second term inside the minimum in Eq. 13 for both \(t=\underline {\sigma }_{1}\) and \(t=\underline {\sigma }_{2}\). Suppose that the function τV (𝜃S,τ, 1, 1) is differentiable at \(\overline {\tau }\). Since \(\overline {\tau }\) is interior to \(]\underline {\sigma }_{2},T[\subset ]\underline {\sigma }_{1},T[\), first order conditions read as

    $$ \begin{array}{@{}rcl@{}} \displaystyle 0&=&-\frac{1}{2}\left( \frac{\theta_{S}-\theta_{1}}{\overline{\tau}-\underline{\sigma}_{1}}\right)^{2}+\mathcal{F}^{1,1}(\mathcal{M}(\overline{\tau}))+V^{\prime}(\theta_{1},\overline{\tau},1,1)\\ \displaystyle &&-\frac{1}{2}\left( \frac{\theta_{S}-\theta_{1}}{\overline{\tau}-\underline{\sigma}_{2}}\right)^{2}+\mathcal{F}^{1,1}(\mathcal{M}(\overline{\tau}))+V^{\prime}(\theta_{1},\overline{\tau},1,1) \end{array} $$

    contradicting \(\underline {\sigma }_{1}\neq \underline {\sigma }_{2}\). Note that V is not necessarily differentiable in time, however it has a super-differential at any instant. Indeed it is easy to see that the value function for (w1,w2) = (0, 0) has such a property, and then obtain the super-differentiability of the others arguing backward in Eqs. 7911 and 13. The super-differentiability at \(\overline {\tau }\) implies that, at least locally in time around \(\overline {\tau } <T\), one V (𝜃1,τ, 1, 1) ≤ h(τ) where h is a suitable differentiable function, with equality holding at \(\overline {\tau }\). Hence the argument above would hold with V replaced by h.

  2. (2)

    Let us suppose that agents arriving at time \(\underline {\sigma }\) in (𝜃S, 1, 1) and moving towards 𝜃1 may reach this latter significant state in more than one optimal time, say \(\overline {\tau }_{1},\overline {\tau }_{2}\) with \(\overline {\tau }_{1}< \overline {\tau }_{2}\). This means that \(\underline {\sigma }(\theta _{1},\overline {\tau }_{1})=\underline {\sigma }(\theta _{1},\overline {\tau }_{2})=\underline {\sigma }\). We now observe that only agents entering 𝜃S at time \(\underline {\sigma }\) may reach 𝜃1 between τ1 and τ2, as agents cannot overtake each other (easy to prove). As a consequence \(\underline {\sigma }(\theta _{1},s)\) is constant in the interval [τ1,τ2] and hence its time derivative \(\underline {\sigma }_{s}(\theta _{1},s)\) is null. This last fact in turn implies that the value of the entrance flow at \(\underline {\sigma }\), i.e., \(g(\underline {\sigma })\), would uniformly spread over the interval [τ1,τ2] and hence would become equal to 0.

Remark that the above arguments imply that no Dirac masses can arise at any point and at any time, apart from the case of a significant point at the final time T, or, just after a switching, when the new choice of the optimal control is u = 0, i.e. to not move. Both situations do not affect the flow functions gi,j.

Appendix B: On HBJ and Transport Equations

For the reader convenience, we here recall the Hamilton-Jacob-Bellman (3)–(6) and the transport equations (8a)–(8d) contained in Bagagiolo and Pesenti (2017). We point out once again that the final cost in the present paper is different from that in Bagagiolo and Pesenti (2017): here \(c_{S}\xi _{\theta =\theta _{S}}(T)\) replaces \(c_{3}(\theta -\theta _{S})^{2}\).

In Bagagiolo and Pesenti (2017, Theorem 1) was proved that the value function V (𝜃,t,w1,w2) is a continuous and bounded viscosity solution of the following HJB equations (subscript indicate derivatives).

In B1,1:

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \displaystyle-V_{t}(\theta,t,1,1)+\frac{1}{2}|V_{\theta}(\theta,t,1,1)|^{2}=\mathcal{F}^{(1,1)}(\mathcal{M}(t))&\text{in } {]\theta_{1},\theta_{2}[\times]0,T[}\\ \displaystyle V(\theta_{1},t,1,1)=V(\theta_{1},t,0,1)&\text{in } {]0,T]}\\ \displaystyle V(\theta_{2},t,1,1)=V(\theta_{2},t,1,0)&\text{in } {]0,T]}\\ \displaystyle V(\theta,T,1,1)=c_{1}+c_{2}+c_{3}(\theta-\theta_{S})^{2}&\text{in } {]\theta_{1},\theta_{2}[} \end{array}\right. \end{array} $$
(26a)

In B0,1:

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \displaystyle -V_{t}(\theta,t,0,1)+\frac{1}{2}|V_{\theta}(\theta,t,0,1)|^{2}=\mathcal{F}^{(0,1)}(\mathcal{M}(t))&\text{in } ]\theta_{2}-2\pi,\theta_{2}[\times]0,T[\\ \displaystyle V(\theta_{2}-2\pi,t,0,1)=V(\theta_{2},t,0,0)&\text{in } ]0,T]\\ \displaystyle V(\theta_{2},t,0,1)=V(\theta_{2},t,0,0)&\text{in } ]0,T]\\ \displaystyle V(\theta,T,0,1)=c_{2}+c_{3}(\theta-\theta_{S})^{2}&\text{in } ]\theta_{2}-2\pi,\theta_{2}[ \end{array}\right. \end{array} $$
(26b)

In B1,0:

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \displaystyle -V_{t}(\theta,t,1,0)+\frac{1}{2}|V_{\theta}(\theta,t,1,0)|^{2}=\mathcal{F}^{(1,0)}(\mathcal{M}(t))&\text{in } ]\theta_{1},\theta_{1}+2\pi[\times]0,T[\\ \displaystyle V(\theta_{1},t,1,0)=V(\theta_{1},t,0,0)&\text{in } ]0,T]\\ \displaystyle V(\theta_{1}+2\pi,t,1,0)=V(\theta_{1},t,0,0)&\text{in } ]0,T]\\ \displaystyle V(\theta,T,1,0)=c_{1}+c_{3}(\theta-\theta_{S})^{2}&\text{in } ]\theta_{1},\theta_{1}+2\pi[ \end{array}\right. \end{array} $$
(26c)

and in B0,0:

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \displaystyle -V_{t}(\theta,t,0,0)+\frac{1}{2}|V_{\theta}(\theta,t,0,0)|^{2}=\mathcal{F}^{(0,0)}(\mathcal{M}(t))&\text{in } \mathbb{R}\times]0,T[\\ \displaystyle V(\theta,T,0,0)=c_{3}(\theta-\theta_{S})^{2}&\text{in } [0,2\pi]. \end{array}\right. \end{array} $$
(26d)

Moreover, the four transport equations for the density m, one per every branch, are

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \displaystyle m^{1,1}_{t}(\theta,t)-[V_{\theta}(\theta,t,1,1)m^{1,1}(\theta,t)]_{\theta}=0\ \text{in } B_{1,1}\times[0,T]\\ \displaystyle m^{1,1}(\theta_{S},t)=g(t) \end{array}\right. \end{array} $$
(27a)
$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \displaystyle m^{1,0}_{t}(\theta,t)-[V_{\theta}(\theta,t,1,0)m^{1,0}(\theta,t)]_{\theta}=0\ \text{in } B_{1,0}\times[0,T]\\ \displaystyle m^{1,0}(\theta_{2},t)=m^{1,1}(\theta_{2},t) \end{array} \right. \end{array} $$
(27b)
$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \displaystyle m^{0,1}_{t}(\theta,t)-[V_{\theta}(\theta,t,0,1)m^{0,1}(\theta,t)]_{\theta}=0\ \text{in } B_{0,1}\times[0,T]\\ \displaystyle m^{0,1}(\theta_{1},t)=m^{1,1}(\theta_{1},t) \end{array} \right. \end{array} $$
(27c)
$$ \begin{array}{@{}rcl@{}} \left\{ \begin{array}{ll} \displaystyle m^{0,0}_{t}(\theta,t)-[V_{\theta}(\theta,t,0,0)m^{0,0}(\theta,t)]_{\theta}=0\ \text{in } B_{0,0}\times[0,T]\\ \displaystyle m^{0,0}(\theta_{1},t)=m^{1,0}(\theta_{1},t)+m^{1,0}(\theta_{1}+2\pi,t)\\ \displaystyle m^{0,0}(\theta_{2},t)=m^{0,1}(\theta_{2},t)+m^{0,1}(\theta_{2}-2\pi,t). \end{array}\right. \end{array} $$
(27d)

We recall that, consistently with Eq. 5, one has u = −V𝜃.

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Bagagiolo, F., Faggian, S., Maggistro, R. et al. Optimal Control of the Mean Field Equilibrium for a Pedestrian Tourists’ Flow Model. Netw Spat Econ 22, 243–266 (2022). https://doi.org/10.1007/s11067-019-09475-4

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