  Accurately modeling and verifying the correct operation of systems
interacting in dynamic environments is challenging. By leveraging parametric
uncertainty within the model description, one can relax the requirement to
describe exactly the interactions with the environment; however, one must still
guarantee that the model, despite uncertainty, behaves acceptably. This paper
presents a convex optimization method to efficiently compute the set of
configurations of a polynomial dynamical system that are able to safely reach a
user defined target set despite parametric uncertainty in the model. Since
planning in the presence of uncertainty can lead to undesirable conservatives,
this paper computes those trajectories of the uncertain nonlinear systems which
are $\alpha$-probable of reaching the desired configuration. The presented
approach uses the notion of occupation measures to describe the evolution of
trajectories of a nonlinear system with parametric uncertainty as a linear
equation over measures whose supports coincide with the trajectories under
investigation. This linear equation is approximated with vanishing conservatism
using a hierarchy of semidefinite programs each of which is proven to compute
an approximation to the set of initial conditions that are $\alpha$-probable of
reaching the user defined target set safely in spite of uncertainty. The
efficacy of this method is illustrated on four systems with parametric
uncertainty.
