  In the Maslov idempotent probability calculus, expectations of random
variables are defined so as to be linear with respect to max-plus addition and
scalar multiplication. This paper considers control problems in which the
objective is to minimize the max-plus expectation of some max-plus additive
running cost. Such problems arise naturally as limits of some types of risk
sensitive stochastic control problems. The value function is a viscosity
solution to a quasivariational inequality (QVI) of dynamic programming.
Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous
Hamiltonian is used to prove a comparison theorem for viscosity sub- and
super-solutions. An example from math finance is given, and an application in
nonlinear H-infinity control is sketched.
