Abstract
We study the representation formulae for the fundamental solutions and viscosity solutions of the Hamilton-Jacobi equations of contact type. We also obtain a vanishing contact structure result for relevant Cauchy problems which can be regarded as an extension to the vanishing discount problem.
Highlights
In the previous work ([19] and [20]), the authors developed an analogy to weak KAM theory for contact systems on compact manifolds
H(x, u(x), Du(x)) = c, x ∈ M, (HJs) using implicit variational principle, where M is a C2 connected closed manifold and c is contained in the set of critical values
The main purpose of this paper is to understand the limit of the viscosity solutions of (HJe) in the case M = Rn when Hu is uniformly bounded and tends to 0
Summary
In the previous work ([19] and [20]), the authors developed an analogy to weak KAM theory for contact systems on compact manifolds. This leads to a representation formula of the viscosity solutions of the Hamilton-Jacobi equation. A classical problem in ergodic control consists of studying the limit behavior of the optimal value uλ of a discounted cost functional with infinite horizon as the discount factor λ tends to zero. In the literature, this problem has been addressed under various conditions ensuring that the rescaled value function λuλ converges uniformly to a constant limit.
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