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2019 The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function
Pierpaolo Soravia
Abstr. Appl. Anal. 2019: 1-9 (2019). 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.

Citation

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Pierpaolo Soravia. "The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function." Abstr. Appl. Anal. 2019 1 - 9, 2019. https://doi.org/10.1155/2019/6417074

Information

Received: 17 July 2019; Accepted: 8 October 2019; Published: 2019
First available in Project Euclid: 13 May 2020

zbMATH: 07176256
MathSciNet: MR4047596
Digital Object Identifier: 10.1155/2019/6417074

Rights: Copyright © 2019 Hindawi

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