  A method is presented to obtain an inner-approximation of the backward
reachable set (BRS) of a given target tube, along with an admissible controller
that maintains trajectories inside this tube. The proposed optimization
algorithms are formulated as nonlinear optimization problems, which are
decoupled into tractable subproblems and solved by an iterative algorithm using
the polynomial S-procedure and sum-of-squares techniques. This framework is
also extended to uncertain nonlinear systems with L_2 disturbances and
L_{\infty} parametric uncertainties. The effectiveness of the method is
demonstrated on several nonlinear robotics and aircraft systems with control
saturation.
