Singularly Perturbed Control Systems with Noncompact Fast Variable

We deal with a singularly perturbed optimal control problem with slow and fast variable depending on a parameter {\epsilon}. We study the asymptotic, as {\epsilon} goes to 0, of the corresponding value functions, and show convergence, in the sense of weak semilimits, to sub and supersolution of a suitable limit equation containing the effective Hamiltonian. The novelty of our contribution is that no compactness condition are assumed on the fast variable. This generalization requires, in order to perform the asymptotic proce- dure, an accurate qualitative analysis of some auxiliary equations posed on the space of fast variable. The task is accomplished using some tools of Weak KAM theory, and in particular the notion of Aubry set

Existence and regularity of strict critical subsolutions in the stationary ergodic setting

We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict subsolutions of class C^1,1 in R^N. The proofs are based on the use of Lax–Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set

Existence and regularity of strict critical subsolutions in the stationary ergodic setting

We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict subsolutions of class \CC^{1,1} in $\R^N$. The proofs are based on the use of Lax--Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set

Discounted Hamilton-Jacobi Equations on Networks and Asymptotic Analysis

We study discounted Hamilton Jacobi equations on networks, without putting any restriction on their geometry. Assuming the Hamiltonians continuous and coercive, we establish a comparison principle and provide representation formulae for solutions. We follow the approach introduced in 11, namely we associate to the differential problem on the network, a discrete functional equation on an abstract underlying graph. We perform some qualitative analysis and single out a distinguished subset of vertices, called lambda Aubry set, which shares some properties of the Aubry set for Eikonal equations on compact manifolds. We finally study the asymptotic behavior of solutions and lambda Aubry sets as the discount factor lambda becomes infinitesimal.Comment: Corrected typos in the titl

Efficient provision of public goods with endogenous redistribution

We study a continuous and balanced mechanism that is capable of implementing in Nash equilibrium all the Pareto-efficient individually rational allocations for an economy with public goods. The Government chooses a set of weights directly related to the Lindahl prices corresponding to the Pareto-efficient allocation it wants to implement. The mechanism then guarantees that initial endowments are re-allocated so that the chosen vector of Lindahl prices is indeed a Lindahl equilibrium, and implements the corresponding Lindahl allocation. Previously known mechanisms that implement the Lindahl correspondence do not allow the Government to choose which point on the Pareto frontier should be implemented, unless it can also redistribute initial endowments in the appropriate way. By contrast, in our case the Government directly controls the distribution of welfare in the economy. Finally, besides being balanced and continuous, our mechanism is `simple'. Each agent has to declare a desired increase in the amount of public good, and a vector of redistributive transfers of initial endowments (across other agents).

Homogenization on arbitrary manifolds

We describe a setting for homogenization of convex hamiltonians on abelian covers of any compact manifold. In this context we also provide a simple variational proof of standard homogenization results.Comment: 17 pages, 1 figur