The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function

Pierpaolo Soravia

doi:10.1155/2019/6417074

Abstract

We define and study C1-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 C1-solutions are absolutely minimizing functions. We discuss how C1-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 showing that it should only be Hölder continuous unless appropriate conditions hold. We provide two examples for Hörmander and Grushin families of vector fields where we construct C1-solutions (even classical) explicitly.

Highlights

In this note we want to describe a possible new, nonstandard way of using the Aronsson equation, a second order partial di erential equation, to obtain controllability properties of deterministic control systems
We can nd counterexamples to the fact that the Hamiltonian is constant along trajectories of the Hamiltonian dynamics, as we show later
If satis es appropriate decay in a neighborhood of the origin only at points where the Hamiltonian stays away from zero, we show that the corresponding minimum time function is locally Lipschitz continuous outside the singular set, despite the fact that even if the origin is small time locally attainable, the minimum time function can only be proven to be Hölder continuous in its domain, in general, under appropriate conditions. us the loss of regularity of the minimum time function is only concentrated at points in the singular set

Summary

Introduction

In this note we want to describe a possible new, nonstandard way of using the Aronsson equation, a second order partial di erential equation, to obtain controllability properties of deterministic control systems. As a side result, we show that our 1-solutions are absolutely minimizing functions, i.e. local minimizers of the functional that computes the ∞ norm of the Hamiltonian It is a well known equivalent property to being a viscosity solution of (AE) at least when is coercive or possibly in some Carnot Caratheodory spaces, but this fact is not completely understood in general. Our regularity results rather go in the direction of those contained in two recent papers by Albano et al [10, 11], where they show, by completely di erent methods, that if a family of smooth vector elds satis es the Hörmander condition, the set where the local Lipschitz continuity of the minimum time function fails is the union of singular trajectories, and that it is analytic except on a subset of null measure. Is paper appeared as preprint on ArXiv with number 1907.07436

Control Theory and the Aronsson Equation

Some Smooth Explicit Solutions of the Aronsson Equation

Conclusions