  In this paper we study the robust invariant sets generation problem for
discrete-time switched polynomial systems subject to disturbance inputs within
the optimal control framework. A robust invariant set of interest is a set of
states such that every possible trajectory starting from it never leaves a
specified safe set, regardless of actual disturbances. The maximal robust
invariant set is shown to be the zero level set of the unique bounded solution
to a Bellman type equation, which is a functional equation being widely used in
discrete-time optimal control. This is the main contribution of this work. The
uniqueness of bounded solutions enables us to solve the derived Bellman type
equation using numerical methods such as the value iteration, which provides an
approximation of the maximal robust invariant set. In order to increase the
scalability of the Bellman equation based method, a semi-definite program,
which is constructed based on the derived Bellman type equation, is also
implemented to synthesize robust invariant sets. Finally, three examples
demonstrate the performance of our methods.
