Viscosity Solutions on Lagrangian Manifolds and Connections with Tunnelling Operators

We consider a geometrical approach to constructing viscosity solutions to Hamilton-Jacobi-Bellman equations. This takes the Lagrangian manifold M on which the characteristic curves for the Cauchy problem lie and forms a viscosity solution V as the minimum of the generating functions of the branches of M lying over state space. This construction is well-known in the case of finite time variational problems. However, recent work in the literature has shown that it also works for infinite time (i.e. stationary) problems, provided the resulting function V is locally Lipschitz. This last condition is satisfied naturally in the finite time case, but is not obvious in the stationary case. We describe how, for Hamiltonians convex in the momentum variable, the local Lipschitz property follows from some natural assumptions on M, where M is of arbitrary dimension. We then discuss connections with idempotent analysis in particular that V is the log-limit of a canonical tunnelling operator constructed using a 1/h-Laplace type transform and with the concept of a graph selector from symplectic topology. Lastly we extend the construction of V to apply to non-convex Hamiltonians. 1. Constructing a viscosity solution from a Lagrangian manifold Define phase space to be the real 2n-dimensional vector space with coordinates (x, p) where x ∈ R and p ∈ R. Let M be a Lagrangian submanifold of phase space. This means that M is an n-dimensional submanifold of phase space on which the canonical two-form dp∧ dx vanishes. It follows that the one form pdx is locally exact on M . For later use, we note some further consequences of the Lagrangian property, all of which can be found in standard references on symplectic geometry such as [11, 13]. Let I denote a subset of the set {1, . . . , n} and Ī denote its compliment. Let x denote the set of coordinates {xi : i ∈ I} and pĪ denote the set {pk : k ∈ Ī}. Then it follows from the Lagrangian property that, at any point on M , there exists a collection of indices I such that (x , pĪ) form a local system of coordinates on M . 2000 Mathematics Subject Classification. Primary 49L25, 53D12, 35B40; Secondary 81Q20.