Generalized viscosity solutions for Hamilton-Jacobi equations with time-measurable hamiltonians

Abstract

The objective of this paper is to extend the idea of viscosity solution for nonlinear first order Hamilton-Jacobi equations, U, + H(t, x, D.,u) = 0, with time-continuous H to time-measurable hamiltonians. We are primarily motivated by the fact that control problems, for which the value function is the viscosity solution of a Hamilton-Jacobi equation, should not be restric- ted to continuity in time. In Barron and Jensen [ 1 ] we were confronted with a linear Hamilton-Jacobi equation arising in the proof of the Pon- tryagin which had time-measurable coeflicients. The existing theory of viscosity solutions did not apply to even this case. Moreover, important models of controlled first order HamiltonJacobi equations require time- measurable controls. For example, models of controlled traffic flow or queueing processes might be of this type. We are about then to extend the definition of viscosity solution to “generalized viscosity solution” applying to time-measurable hamiltonians. In this paper we give the definition and derive the corresponding uni- queness result with an implicit domain of dependence consequent. The fun- damental technique used in the proof is the so-called “blow up” method which has also been used elsewhere [S], although not in the generality discussed here. This method effectively freezes the point (to, x0) under consideration and essentially removes it from the problem. Of course t, is