Graph selectors and viscosity solutions on lagrangian manifolds

McCaffrey, David

doi:10.1051/cocv:2006023

McCaffrey, David

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 795-815.

Résumé

Let $Λ$ be a lagrangian submanifold of $T^{*} X$ for some closed manifold $X .$ Let $S (x, ξ)$ be a generating function for $Λ$ which is quadratic at infinity, and let $W (x)$ be the corresponding graph selector for $Λ,$ in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset $X_{0} \subset X$ of measure zero such that $W$ is Lipschitz continuous on $X,$ smooth on $X ∖ X_{0}$ and $(x, \partial W / \partial x (x)) \in Λ$ for $X ∖ X_{0} .$ Let $H (x, p) = 0$ for $(x, p) \in Λ$ . Then $W$ is a classical solution to $H (x, \partial W / \partial x (x)) = 0$ on $X ∖ X_{0}$ and extends to a Lipschitz function on the whole of $X .$ Viterbo refers to $W$ as a variational solution. We prove that $W$ is also a viscosity solution under some simple and natural conditions. We also prove that these conditions are satisfied in many cases, including certain commonly occuring cases where $H (x, p)$ is not convex in $p$ .

MR Zbl

DOI : 10.1051/cocv:2006023

Classification : 49L25, 53D12
Mots-clés : viscosity solution, lagrangian manifold, graph selector

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McCaffrey, David. Graph selectors and viscosity solutions on lagrangian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 795-815. doi : 10.1051/cocv:2006023. https://www.numdam.org/articles/10.1051/cocv:2006023/

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Bernard, Patrick Semi-concave singularities and the Hamilton-Jacobi equation, Regular and Chaotic Dynamics, Volume 18 (2013) no. 6, p. 674 | DOI:10.1134/s1560354713060075
McCaffrey, David A representational formula for variational solutions to Hamilton-Jacobi equations, Communications on Pure Applied Analysis, Volume 11 (2012) no. 3, p. 1205 | DOI:10.3934/cpaa.2012.11.1205
McCaffrey, D. Geometric existence theory for the control-affine H∞ problem, Journal of Mathematical Analysis and Applications, Volume 324 (2006) no. 1, p. 682 | DOI:10.1016/j.jmaa.2005.12.034

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