  We define and study $C^1-$solutions of the Aronsson equation (AE), a second
order quasi linear equation. We show that such super/subsolutions make the
Hamiltonian monotone on the trajectories of the closed loop Hamiltonian
dynamics. We give a short, general proof that $C^1-$solutions are absolutely
minimizing functions. We discuss how $C^1-$supersolutions of (AE) become
special Lyapunov functions of symmetric control systems, and allow to find
continuous feedbacks driving the system to a target in finite time, except on a
singular manifold. A consequence is a simple proof that the corresponding
minimum time function is locally Lipschitz continuous away from the singular
manifold, despite classical results show that it should only be H\"older
continuous unless appropriate conditions hold. We provide two examples for
H\"ormander and Grushin families of vector fields where we construct
$C^1-$solutions (even classical) explicitly.
