  We present a method for synthesizing controllers to steer trajectories from
an initial set to a target set on a finite time horizon. The proposed control
synthesis problem is decomposed into two steps. The first step
under-approximates the backward reachable set (BRS) from the target set, using
level sets of storage functions. The storage function is constructed with an
iterative algorithm to maximize the volume of the under-approximated BRS. The
second step obtains a control law by solving a pointwise min-norm optimization
problem using the pre-computed storage function. A closed-form solution of this
min-norm optimization can be computed through the KKT conditions. This control
synthesis framework is then extended to uncertain nonlinear systems with
parametric uncertainties and L_2 disturbances. The computation algorithm for
all cases is derived using sum-of-squares (SOS) programming and the
S-procedure. The proposed method is applied to several robotics and aircraft
examples.
