Optimal scenario for road evacuation in an urban environment (Q6597372)

The authors consider a road network as a directed graph of \(N_{r}\in \mathbb{ N}^{\ast }\) roads, denoted as real intervals \([a_{i},b_{i}]\), and the time evolution of the density \(\rho _{i}\) on the \(i\)-th road according to the system: \(\partial _{t}\rho _{i}(t,x)+\partial _{t}f_{i}(\rho _{i}(t,x))=0\), \( (t,x)\in (0,T)\times \lbrack a_{i},b_{i}]\), with the initial and boundary conditions: \(\rho _{i}(0,x)=\rho _{i}^{0}(x)\), \(x\in \lbrack a_{i},b_{i}]\), \( f_{i}(\rho _{i}(t,a_{i}))=\gamma L_{i}(t)\), \(f_{i}(\rho _{i}(t,b_{i}))=\gamma R_{i}(t)\), \(t\in (0,T)\), where the flux \(f_{i}(\rho )\) is given by \(f_{i}(\rho )=\rho v_{i}^{\max}(1-\frac{\rho }{\rho _{i}^{\max}})\), \(v_{i}^{\max}\) and \(\rho_{i}^{\max}\) respectively denoting the maximal velocity and density allowed on the \(i\)-th road. The initial density \(\rho _{i}^{0}(x)\) and \(\gamma L_{i}\), \(\gamma R_{i}\) are functions allowing prescribing the flux at the left and right boundaries of the interval \([a_{i},b_{i}]\). At a junction \(J\) between \(J_{in}\) in-going and \(J_{\mathrm{out}}\) outgoing roads, a statistical traffic distribution matrix is introduced: \( A_{J}=(\alpha _{ji})_{(j,i)\in J_{\mathrm{out}}\times J_{in}}\), where \(\alpha _{ji}\) is the proportion of vehicles going to the \(j\)-th outgoing road among those coming from the \(i\)-th incoming road that satisfies \(0<\alpha _{ji}<1\), \( \sum_{j\in J_{\mathrm{out}}}\alpha _{ji}=1\). It allows to describe conditions imposed on the flux: \((\gamma L_{j})_{j\in J_{\mathrm{out}}}=A_{J}(\gamma R_{i})_{i\in J_{in}}\). The authors write the solution as: \(\gamma (t)=\phi ^{LP}(\rho (t))\), and they give the explicit expression of \(\phi ^{LP}\) for four types of junctions. They also introduce a problem with a vector of controls at the junctions and they write the flux as \(\gamma =\phi ^{LP}(\rho ,u)\), assuming that \(\phi ^{LP}\) is Lipschitz, and \(C^{1}\) with respect to its second variable \(u\). To define the sensitivity of the different data of the problem with respect to the control, they discretize the model with a first-order finite volume scheme. They introduce the optimal control problem: \( \inf_{u\in \mathcal{U}_{ad}}\mathcal{J}_{\theta }(u)\), where the admissible set \(\mathcal{U}_{ad}\) of controls is defined by \(\mathcal{U}_{ad}=L^{\infty }([0,T],[0,1]^{N_{r}})\), and \(\mathcal{J}_{\theta }\) is the regularized cost functional defined as: \(\mathcal{J}_{\theta }(u)=C_{T}(u)+\frac{\theta _{S}}{ 2}S(u)+\theta _{B}B(u)\), with \(C_{T}(u)=\sum_{i\in \chi _{\mathrm{path}}}\sum_{j=1}^{N_{c}}\rho _{i,j}(T;u)\), \(S(u)=\int_{0}^{T}( \sum_{i=1}^{N_{r}}u_{i}(t)-N_{\max})_{+}^{2}dt\), and \(B(u)= \sum_{i=1}^{N_{r}}TV(u_{i})\). Here \(\rho (T;u)\) is the solution to the semi-discrete control problem, \(N_{\max}\) the maximal number of active controls, and \(TV(u)=sup\{\int_{0}^{T}u(t)\phi ^{\prime }(t)dt\), \(\phi \in C_{c}^{1}([0,T])\), \(\left\Vert \phi \right\Vert _{L^{\infty }(0,T)}\leq 1\}\) . Assuming hypotheses on the data and \(\theta _{B}>0\), the authors prove the existence of a solution to this optimal control problem, introducing a minimizing sequence and proving that the sequence of associated densities is uniformly bounded in \(W^{1,\infty }(0,T)\). They determine the optimality conditions, first defining an admissible perturbation of \(u\) in \(\mathcal{U} _{ad}\). They define a perturbed optimal control problem as: \(\inf_{u\in \mathcal{U}_{ad}}\mathcal{J}_{\theta ,\nu }(u)\), where \(\mathcal{J}_{\theta \nu }(u)=C_{T}(u)+\frac{\theta _{S}}{2}S(u)+\theta _{B}B_{\nu }(u)\), \(\nu >0\) standing for a regularization parameter and \(B_{\nu }(u)\) being the sum of differentiable approximations in \(L^{2}(0,T)\) \(TV_{\nu }(u_{i})\) of \( TV(u_{i})\). They write the first-order optimality conditions for this approximated problem in a very concise and workable way. The rest of the paper is devoted to the presentation of a numerical algorithm, based on a finite volume scheme for the primal problem and an Euler scheme for the adjoint problem. For the optimization problem, they use a projected gradient descent method or a fixed point method, or even hybrid methods between the two preceding ones. The authors present the numerical simulations in the cases of simple junctions, of a traffic circle, and of a three routes network. They analyze the impact of the parameters on the results.