Fabio Camilli

Mean Field Games: Numerical Methods for the Planning Problem

Mean field games describe the asymptotic behavior of differential games in which the number of players tends to

A Generalization of Zubov's Method to Perturbed Systems

+\infty

Control Lyapunov Functions and Zubov's Method

. Here we focus on the optimal planning problem, i.e., the problem in which the positions of a very large number of identical rational agents, with a common value function, evolve from a given initial spatial density to a desired target density at the final horizon time. We propose a finite difference semi-implicit scheme for the optimal planning problem, which has an optimal control formulation. The latter leads to existence and uniqueness of the discrete control problem. We also study a penalized version of the semi-implicit scheme. For solving the resulting system of equations, we propose a strategy based on Newton iterations. We describe some numerical experiments.

Viscosity solutions of Eikonal equations on topological networks

A generalization of Zubovs theorem on representing the domain of attraction via the solution of a suitable partial differential equation is presented for the case of perturbed systems with a singular fixed point. For the construction it is necessary to consider solutions in the viscosity sense. As a consequence, maximal robust Lyapunov functions can be characterized as viscosity solutions.

Mean Field Games: Convergence of a Finite Difference Method

For finite-dimensional nonlinear control systems we study the relation between asymptotic null-controllability and control Lyapunov functions. It is shown that control Lyapunov functions (CLFs) may be constructed on the domain of asymptotic null-controllability as viscosity solutions of a first order PDE that generalizes Zubovs equation. The solution is also given as the value function of an optimal control problem from which several regularity results may be obtained.

An approximation scheme for the maximal solution of the shape-from-shading model

In this paper we introduce a notion of viscosity solutions for Eikonal equations defined on topological networks. Existence of a solution for the Dirichlet problem is obtained via representation formulas involving a distance function associated to the Hamiltonian. A comparison theorem based on Ishii’s classical argument yields the uniqueness of the solution.

Homogenization of Hamilton-Jacobi equations: Numerical Methods

Mean field type models describing the limiting behavior of stochastic differential games as the number of players tends to

Time-Dependent Measurable Hamilton–Jacobi Equations

+\infty

Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs

have been recently introduced by Lasry and Lions. Numerical methods for the approximation of the stationary and evolutive versions of such models have been proposed by the authors in previous works. Here, convergence theorems for these methods are proved under various assumptions on the coupling operator.

Hamilton-Jacobi equations on networks

The shape-from-shading model leads to a first order Hamilton-Jacobi equation coupled with a boundary condition, i.e. of Dirichlet type. The analytical characterization of the solution presents some difficulties since this is an eikonal type equation which has several weak solutions (in the viscosity sense). The lack of uniqueness is also a big problem when we try to compute a solution. In order to avoid those difficulties the problem is usually solved by using some additional information such as the height at points where the brightness has a maximum, or the complete knowledge of the level curve. We use results obtained from viscosity theory to characterize the maximal solution without extra information and we construct an algorithm which converges to that solution. Some examples show the accuracy of the algorithm.