  In this paper, we study the zero-error capacity for finite state channels
with feedback when channel state information is known to both the transmitter
and the receiver. We prove that the zero-error capacity in this case can be
obtained through the solution of a dynamic programming problem. Each iteration
of the dynamic programming provides lower and upper bounds on the zero-error
capacity, and in the limit, the lower bound coincides with the zero-error
feedback capacity. Furthermore, a sufficient condition for solving the dynamic
programming problem is provided through a fixed-point equation. Analytical
solutions for several examples are provided.
