  We consider the problem of approximating the reachable set of a discrete-time
polynomial system from a semialgebraic set of initial conditions under general
semialgebraic set constraints. Assuming inclusion in a given simple set like a
box or an ellipsoid, we provide a method to compute certified outer
approximations of the reachable set.
  The proposed method consists of building a hierarchy of relaxations for an
infinite-dimensional moment problem. Under certain assumptions, the optimal
value of this problem is the volume of the reachable set and the optimum
solution is the restriction of the Lebesgue measure on this set. Then, one can
outer approximate the reachable set as closely as desired with a hierarchy of
super level sets of increasing degree polynomials. For each fixed degree,
finding the coefficients of the polynomial boils down to computing the optimal
solution of a convex semidefinite program. When the degree of the polynomial
approximation tends to infinity, we provide strong convergence guarantees of
the super level sets to the reachable set. We also present some application
examples together with numerical results.
