Hamilton-Jacobi equations for control problems of parabolic equations

Viscosity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems

Ergodic Control of Semilinear Stochastic Equations and Hamilton-jacobi Equations

In this paper we consider optimal control of stochastic semilinear equations with linearly increasing drift and cylindrical noise. We show existence and uniqueness (up to an additive constant) of solutions to the stationary Hamilton-Jacobi equation associated with the cost functional given by the asymptotic average per unit time cost. As a consequence we nd the optimizing controls given in the ...

Hamilton-Jacobi-Bellman Equations

This work treats Hamilton-Jacobi-Bellman equations. Their relation to several problems in mathematics is presented and an introduction to viscosity solutions is given. The work of several research articles is reviewed, including the Barles-Souganidis convergence argument and the inaugural papers on mean-field games. Original research on numerical methods for Hamilton-Jacobi-Bellman equations is...

Hypercontractivity of Hamilton–jacobi Equations

– Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton–Jacobi equations. By the infimum-convolution description of the Hamilton–Jacobi solutions, this approach provides a clear view of the connection between logarithmic Sobo...

Multiscale problems and homogenization for second-order Hamilton–Jacobi equations

We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-sca...