  In this work, we introduce a notion of reachability entropy to characterize
the smallest data rate which is sufficient enough to enforce reach-while-stay
specification. We also define data rates of coder-controllers that can enforce
this specification in finite time. Then, we establish the data-rate theorem
which states that the reachability entropy is a tight lower bound of the data
rates that allow satisfaction of the reach-while-stay specification. For a
system which is related to an another system under feedback refinement
relation, we show that the entropy of the former will not be larger than that
of the latter. We also provide a procedure to numerically compute an upper
bound of the reachability entropy for discrete-time control systems by
leveraging their finite abstractions. Finally, we present some examples to
demonstrate the effectiveness of the proposed results.
