Lagrangian manifolds, viscosity solutions and Maslov index. (English) Zbl 0998.49017

McCaffrey, D.; Banks, S. P.

J. Convex Anal. 9, No. 1, 185-224 (2002).

Summary: Let \(M\) be a Lagrangian manifold, let the 1-form \(pdx\) be globally exact on \(M\) and let \(S(x,p)\) be defined by \(dS = pdx\) on \(M\). Let \(H(x,p)\) be convex in \(p\) for all \(x\) and vanish on \(M\). Let \(V(x) = \inf \{S(x, p) : p\) such that \((x, p) \in M\}\). Recent work in the literature has shown that (i) \(V\) is a viscosity solution of \(H(x, \partial V / \partial x) = 0\) provided \(V\) is locally Lipschitz, and (ii) \(V\) is locally Lipschitz outside the set of caustic points for \(M\). It is well known that this construction gives a viscosity solution for finite time variational problems - the Lipschitz continuity of \(V\) follows from that of the initial condition for the variational problem. However, this construction also applies to infinite time variational problems and stationary Hamilton-Jacobi-Bellman equations where the regularity of \(V\) is not obvious. We show that for dim \(M\leq 5\), the local Lipschitz property follows from some geometrical assumptions on \(M\) – in particular that the Maslov index vanishes on closed curves on \(M\). We obtain a local Lipschitz constant for \(V\) which is some uniform power of a local bound on \(M\), the power being determined by dim \(M\). This analysis uses Arnold’s classification of Lagrangian singularities.

MSC:

49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49L20	Dynamic programming in optimal control and differential games
58E25	Applications of variational problems to control theory
53D12	Lagrangian submanifolds; Maslov index

Keywords:

viscosity solutions; Lagrangian manifold; caustic points; Hamilton-Jacobi-Bellman equations; Maslov index