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We define and study $C^1-$solutions of the Aronsson equation (AE), a secondorder quasi linear equation. We show that such super/subsolutions make theHamiltonian monotone on the trajectories of the closed loop Hamiltoniandynamics. We give a short, general proof that $C^1-$solutions are absolutelyminimizing functions. We discuss how $C^1-$supersolutions of (AE) becomespecial Lyapunov functions of symmetric control systems, and allow to findcontinuous feedbacks driving the system to a target in finite time, except on asingular manifold. A consequence is a simple proof that the correspondingminimum time function is locally Lipschitz continuous away from the singularmanifold, despite classical results show that it should only be H\"oldercontinuous unless appropriate conditions hold. We provide two examples forH\"ormander and Grushin families of vector fields where we construct$C^1-$solutions (even classical) explicitly.

The Aronsson equation, Lyapunov functions and local Lipschitz regularity of the minimum time function