Recent trends in optimal control and partial differential equations

On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians

speaker: Andrea Davini (Universita' di Roma La Sapienza)

abstract: We study the asymptotic behavior of the viscosity solutions u λ G of the Hamilton-Jacobi (HJ)

equation

λu(x) + G(x, u0

) = c(G) in R

as the positive discount factor λ tends to 0, where G(x, p) := H(x, p) − V (x) is the perturbation of a Hamiltonian H ∈ C(R×R), Z–periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential V ∈ Cc(R). The constant c(G) appearing above is defined as the infimum of values a ∈ R for which the HJ equation G(x, u0

) = a in R admits

bounded viscosity subsolutions. We prove that the functions u λ G locally uniformly converge, for

λ → 0 +, to a specific solution u 0 G of the critical equation G(x, u0 ) = c(G) in R.

We identify u 0 G in terms of projected Mather measures for G and of the limit u 0

H to the unper- turbed periodic problem. This is joint work with I. Capuzzo-Dolcetta.

timetable:
Mon 8 May, 17:50 - 18:20, Aula Dini
documents:

ABSTRACT

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