Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem

This paper is devoted to establish continuous dependence estimates for the ergodic problem for Bellman operators (namely, estimates of (v_1-v_2) where v_1 and v_2 solve two equations with different coefficients). We shall obtain an estimate of ||v_1-v_2||_\infty with an explicit dependence on the L^\infty-distance between the coefficients and an explicit characterization of the constants and also, under some regularity conditions, an estimate of ||v_1-v_2||_{C^2(\R^n)}. Afterwards, the former result will be crucial in the estimate of the rate of convergence for the homogenization of Bellman equations. In some regular cases, we shall obtain the same rate of convergence established in the monographs [11,26] for regular linear problems

Stationary Mean Field Games systems defined on networks

We consider a stationary Mean Field Games system defined on a network. In this framework, the transition conditions at the vertices play a crucial role: the ones here considered are based on the optimal control interpretation of the problem. We prove separately the well-posedness for each of the two equations composing the system. Finally, we prove existence and uniqueness of the solution of the Mean Field Games system

First report outside Eastern Europe of West Nile virus lineage 2 related to the Volgograd 2007 strain, northeastern Italy, 2014

open11noWest Nile virus (WNV) is a Flavivirus transmitted to vertebrate hosts by mosquitoes, maintained in nature through an enzootic bird-mosquito cycle. In Europe the virus became of major public health and veterinary concern in the 1990s. In Italy, WNV re-emerged in 2008, ten years after the previous outbreak and is currently endemic in many areas of the country. In particular, the northeastern part of Italy experience continuous viral circulation, with human outbreaks caused by different genovariants of WNV lineage 1, Western-European and Mediterranean subcluster, and WNV lineage 2, Hungarian clade. Alongside the WNV National Surveillance Program that has been in place since 2002, regional surveillance plans were implemented after 2008 targeting mosquitoes, animals and humans.openRavagnan, Silvia; Montarsi, Fabrizio; Cazzin, Stefania; Porcellato, Elena; Russo, Francesca; Palei, Manlio; Monne, Isabella; Savini, Giovanni; Marangon, Stefano; Barzon, Luisa; Capelli, GioiaRavagnan, Silvia; Montarsi, Fabrizio; Cazzin, Stefania; Porcellato, Elena; Russo, Francesca; Palei, Manlio; Monne, Isabella; Savini, Giovanni; Marangon, Stefano; Barzon, Luisa; Capelli, Gioi

Liouville properties and critical value of fully nonlinear elliptic operators

We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.Comment: 18 pp, to appear in J. Differential Equation

A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks

Three definitions of viscosity solutions for Hamilton-Jacobi equations on networks recently appeared in literature ([1,4,6]). Being motivated by various applications, they appear to be considerably different. Aim of this note is to establish their equivalence

Optimal control of multiscale systems using reduced-order models

We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"--rather than "first optimize, then reduce"--is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls

Homogenization of a mean field game system in the small noise limit

This paper concerns the simultaneous effect of homogenization and of the small noise limit for a second order mean field game (MFG) system with local coupling and quadratic Hamiltonian. We show under some additional assumptions that the solutions of our system converge to a solution of an effective first order system whose effective operators are defined through a cell problem which is a second order system of ergodic MFG type. We provide several properties of the effective operators, and we show that in general the effective system loses the MFG structure

Ergodic Mean Field Games with H\"ormander diffusions

We prove existence of solutions for a class of systems of subelliptic PDEs arising from Mean Field Game systems with H\"ormander diffusion. These results are motivated by the feedback synthesis Mean Field Game solutions and the Nash equilibria of a large class of $N$-player differential games