The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function
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 minimi...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2019/6417074 |
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| author | Pierpaolo Soravia |
| author_facet | Pierpaolo Soravia |
| author_sort | Pierpaolo Soravia |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-16092feffa1c48b0baf6977de5788cfa |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-16092feffa1c48b0baf6977de5788cfa2025-02-03T06:12:35ZengWileyAbstract and Applied Analysis1085-33751687-04092019-01-01201910.1155/2019/64170746417074The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time FunctionPierpaolo Soravia0Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, ItalyWe 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.http://dx.doi.org/10.1155/2019/6417074 |
| spellingShingle | Pierpaolo Soravia The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function Abstract and Applied Analysis |
| title | The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function |
| title_full | The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function |
| title_fullStr | The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function |
| title_full_unstemmed | The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function |
| title_short | The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function |
| title_sort | aronsson equation lyapunov functions and local lipschitz regularity of the minimum time function |
| url | http://dx.doi.org/10.1155/2019/6417074 |
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