Absolute minimizers, Aronsson equation and eikonal equations with Lipschitz continuous vector fields

We define viscosity solutions of the Aronsson equation arising from Hamilton-Jacobi eikonal equations for systems with Lipschitz-continuous vector fields and prove that value functions of the corresponding deterministic optimal control problems are solutions if they are bilateral viscosity solutions of the HJ equation. We then introduce and characterize absolute minimizers in two ways. We show that value functions are absolute minimizers if and only if they are bilateral viscosity solution of the HJ equation and determine that bilateral solutions of HJ equations are unique among absolute minimizers with prescribed Dirichlet boundary conditions. Finally we characterize absolute minimizers by comparison with {\it generalized cone functions}. This is a key step to prove existence of absolute minimizers by Perron's method, hence of solutions of the Aronsson equation, and the fact that they satisfy the Harnak inequality.