  In this paper we propose a convex programming based method to address a
long-standing problem of inner-approximating backward reachable sets of
state-constrained polynomial systems subject to time-varying uncertainties. The
backward reachable set is a set of states, from which all trajectories starting
will surely enter a target region at the end of a given time horizon without
violating a set of state constraints in spite of the actions of uncertainties.
It is equal to the zero sub-level set of the unique Lipschitz viscosity
solution to a Hamilton-Jacobi partial differential equation (HJE). We show that
inner-approximations of the backward reachable set can be formed by zero
sub-level sets of its viscosity super-solutions. Consequently, we reduce the
inner-approximation problem to a problem of synthesizing polynomial viscosity
super-solutions to this HJE. Such a polynomial solution in our method is
synthesized by solving a single semi-definite program. We also prove that
polynomial solutions to the formulated semi-definite program exist and can
produce a convergent sequence of inner-approximations to the interior of the
backward reachable set in measure under appropriate assumptions. This is the
main contribution of this work. Several illustrative examples demonstrate the
merits of our approach.
