Viscosity solutions on Lagrangian manifolds and connections with tunnelling operators. (English) Zbl 1089.49033

McCaffrey, D.

Litvinov, G. L. (ed.) et al., Idempotent mathematics and mathematical physics. Proceedings of the international workshop, Vienna, Austria, February 3–10, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3538-6/pbk). Contemporary Mathematics 377, 221-238 (2005).

Summary: We consider a geometrical approach to construct 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 spaces. 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) poblems, 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 it to non-convex Hamiltonians.
For the entire collection see [Zbl 1069.00011].

MSC:

49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
53D12	Lagrangian submanifolds; Maslov index
35F20	Nonlinear first-order PDEs
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
37J05	Relations of dynamical systems with symplectic geometry and topology (MSC2010)

Keywords:

viscosity solution; Lagrangian manifold; stationary idempotent solution; canonical tunnelling operator; graph selector