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The Aronsson equation, Lyapunov functions, and local Lipschitz regularity of the minimum time function. (English) Zbl 1456.49022

Summary: 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 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 \(C^1\)-solutions (even classical) explicitly.

MSC:

49K40 Sensitivity, stability, well-posedness
49K15 Optimality conditions for problems involving ordinary differential equations
93B27 Geometric methods
37B25 Stability of topological dynamical systems
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