  In this paper, we consider the (upper) semigroup envelope, i.e. the least
upper bound, of a given family of linear Feller semigroups. We explicitly
construct the semigroup envelope and show that, under suitable assumptions, it
yields viscosity solutions to abstract Hamilton-Jacobi-Bellman-type partial
differential equations related to stochastic optimal control problems arising
in the field of Robust Finance. We further derive conditions for the existence
of a Markov process under a nonlinear expectation related to the semigroup
envelope for the case where the state space is locally compact. The procedure
is then applied to numerous examples, in particular, nonlinear PDEs that arise
from control problems for infinite dimensional Ornstein-Uhlenbeck and L\'evy
processes.
