Envelopes and nonconvex Hamilton–Jacobi equations

Abstract

This paper introduces a new representation formula for viscosity solutions of nonconvex Hamilton-Jacobi PDE using "generalized envelopes" of affine solutions. We study as well envelope and singular characteristic constructions of equivocal surfaces and discuss also differential game theoretic interpretations.In memory of Arik A. Melikyan.

A plethora of open problems in particle flow research for nonlinear filters, Bayesian decisions, Bayesian learning, and transport

“…The fourth open problem is to generalize the Hopf-Lax formula to solve our transport PDE, analogous to the recent work reported in [56] & [57]. Osher reports beating the curse of dimensionality for certain optimal control problems using the Hopf-Lax formula for convex Hamiltonians.…”

“…Moreover, the HJB equation is nonlinear, whereas our transport equation is linear, which makes it even easier to solve. But the second feature of the HJ equation solved by Evans [57] is that the Hamiltonian is a function only of the gradient of the solution, and this is obviously the same as in our problem, because div is the trace of the gradient. The first two pages of Evans [57] explains how to use the envelope method to solve the HJ equation explicitly.…”

“…But the second feature of the HJ equation solved by Evans [57] is that the Hamiltonian is a function only of the gradient of the solution, and this is obviously the same as in our problem, because div is the trace of the gradient. The first two pages of Evans [57] explains how to use the envelope method to solve the HJ equation explicitly. Furthermore, the simple problem solved by Evans in the first two pages of [57] has nothing to do with optimization, which is the same as our application to particle flow.…”

“…Another approach is to follow Gromov, and deal with the underdetermined PDE directedly, as explained in [9]. Both [56] and [57] are focused on solving the Hamilton-JacobiBellman PDE for optimal control or the Hamilton-Jacobi PDE for physics. There are several reasons that we believe the Hamilton-Jacobi-Bellman PDE is relevant to our transport problem.…”

“…The first two pages of Evans [57] explains how to use the envelope method to solve the HJ equation explicitly. Furthermore, the simple problem solved by Evans in the first two pages of [57] has nothing to do with optimization, which is the same as our application to particle flow. This is a very simple but crucial insight into the HJ PDE, which I was taught by Professor David Mayne many years ago.…”