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Characterization of minimizable Lagrangian action functionals and a dual Mather theorem. (arXiv:1810.03433v1 [math.OC])
来源于:arXiv
We show that a necessary and sufficient condition for a smooth function on
the tangent bundle of a manifold to be a Lagrangian density whose action can be
minimized is, roughly speaking, that it be the sum of a constant, a nonnegative
function vanishing on the support of the minimizers, and an exact form.
We show that this exact form corresponds to the differential of a Lipschitz
function on the manifold that is differentiable on the projection of the
support of the minimizers, and its derivative there is Lipschitz. This function
generalizes the notion of subsolution of the Hamilton-Jacobi equation that
appears in weak KAM theory, and the Lipschitzity result allows for the recovery
of Mather's celebrated 1991 result as a special case. We also show that our
result is sharp with several examples.
Finally, we apply the same type of reasoning to an example of a finite
horizon Legendre problem in optimal control, and together with the Lipschitzity
result we obtain the Hamilton-Jacobi-Bellma 查看全文>>