DCDS
A semi-Lagrangian scheme for a degenerate second order mean field game system
Elisabetta Carlini Francisco J. Silva
Discrete & Continuous Dynamical Systems - A 2015, 35(9): 4269-4292 doi: 10.3934/dcds.2015.35.4269
In this paper we study a fully discrete Semi-Lagrangian approximation of a second order Mean Field Game system, which can be degenerate. We prove that the resulting scheme is well posed and, if the state dimension is equals to one, we prove a convergence result. Some numerical simulations are provided, evidencing the convergence of the approximation and also the difference between the numerical results for the degenerate and non-degenerate cases.
keywords: degenerate second order system Mean field games numerical methods. semi-Lagrangian schemes convergence analysis
DCDS
A model problem for Mean Field Games on networks
Fabio Camilli Elisabetta Carlini Claudio Marchi
Discrete & Continuous Dynamical Systems - A 2015, 35(9): 4173-4192 doi: 10.3934/dcds.2015.35.4173
In [14], Guéant, Lasry and Lions considered the model problem ``What time does meeting start?'' as a prototype for a general class of optimization problems with a continuum of players, called Mean Field Games problems. In this paper we consider a similar model, but with the dynamics of the agents defined on a network. We discuss appropriate transition conditions at the vertices which give a well posed problem and we present some numerical results.
keywords: Mean Field Games numerical methods. stochastic optimal control Networks
DCDS-S
A flame propagation model on a network with application to a blocking problem
Fabio Camilli Elisabetta Carlini Claudio Marchi
Discrete & Continuous Dynamical Systems - S 2018, 11(5): 825-843 doi: 10.3934/dcdss.2018051
We consider the Cauchy problem
$\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+H(x,Du) = 0&(x,t)\in \Gamma \times (0,T) \\ u(x,0) = {{u}_{0}}(x)&x\in \Gamma \\\end{array} \right.$
where
$\Gamma$
is a network and
$H$
is a positive homogeneous Hamiltonian which may change from edge to edge. In the first part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a flame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.
keywords: Evolutive Hamilton-Jacobi equation viscosity solution network Hopf-Lax formula approximation

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