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arXiv · 2311.07000

Topological and control theoretic properties of Hamilton-Jacobi equations via Lax-Oleinik commutators

Piermarco Cannarsa·Wei Cheng·Jiahui Hong
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Abstract

In the context of weak KAM theory, we discuss the commutators $\{T^-_t\circ T^+_t\}_{t\geqslant0}$ and $\{T^+_t\circ T^-_t\}_{t\geqslant0}$ of Lax-Oleinik operators. We characterize the relation $T^-_t\circ T^+_t=Id$ for both small time and arbitrary time $t$. We show this relation characterizes controllability for evolutionary Hamilton-Jacobi equation. Based on our previous work on the cut locus of viscosity solution, we refine our analysis of the cut time function $\tau$ in terms of commutators $T^+_t\circ T^-_t-T^+_t\circ T^-_t$ and clarify the structure of the super/sub-level set of the cut time function $\tau$.

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Piermarco Cannarsa, Wei Cheng, Jiahui Hong. 2023-11-13. Topological and control theoretic properties of Hamilton-Jacobi equations via Lax-Oleinik commutators. https://arxiv.org/abs/2311.07000

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