Backward reachability analysis computes the set of states that reach a target
set under the competing influence of control input and disturbances. Depending
on their interplay, the backward reachable set either represents all states
that can be steered into the target set or all states that cannot avoid
entering it -- the corresponding solutions can be used for controller synthesis
and safety verification, respectively. A popular technique for backward
reachable set computation solves Hamilton-Jacobi-Isaacs equations, which scales
exponentially with the state dimension due to gridding the state space. In this
work, we instead use set propagation techniques to design backward reachability
algorithms for linear time-invariant systems. Crucially, the proposed
algorithms scale only polynomially with the state dimension. Our numerical
examples demonstrate the tightness of the obtained backward reachable sets and
show an overwhelming improvement of our proposed algorithms over
state-of-the-art methods regarding scalability, as systems with well over a
hundred states can now be analyzed.Comment: 16 page