  In this paper we examine how Lagrangian techniques can be used to compute
underapproximations and overapproximation of the finite-time horizon,
stochastic reach-avoid level sets for discrete-time, nonlinear systems. This
approach is applicable for a generic nonlinear system without any convexity
assumptions on the safe and target sets. We examine and apply our methods on
the reachability of a target tube problem, a more generalized version of the
finite-time horizon reach-avoid problem. Because these methods utilize a
Lagrangian (set theoretic) approach, we eliminate the necessity to grid the
state, input, and disturbance spaces allowing for increased scalability and
faster computation. The methods scalability are currently limited by the
computational requirements for performing the necessary set operations by
current computational geometry tools. The primary trade-off for this improved
extensibility is conservative approximations of actual stochastic reach set. We
demonstrate these methods on several examples including the standard
double-integrator, a chain of integrators, and a 4-dimensional space vehicle
rendezvous docking problem.
