75 research outputs found
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 . 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
AN APPLICATION OF AUBRY-MATHER THEORY IN LORENTZIAN GEOMETRY(Mathematical Models of Phenomena and Evolution Equations)
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
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