  For any class of channel conditional distributions, with finite memory
dependence on channel input RVs $A^n {\stackrel{\triangle}{=}} \{A_i: i=0,
\ldots, n\}$ or channel output RVs $B^n {\stackrel{\triangle}{=}} \{B_i: i=0,
\ldots, n\}$ or both, we characterize the sets of channel input distributions,
which maximize directed information defined by $ I(A^n \rightarrow B^n)
{\stackrel{\triangle}{=}} \sum_{i=0}^n I(A^i;B_i|B^{i-1}) $ and we derive the
corresponding expressions, called "characterizations of Finite Transmission
Feedback Information (FTFI) capacity". The main theorems state that optimal
channel input distributions occur in subsets ${\cal P}_{[0,n]}^{CI}\subseteq
{\cal P}_{[0,n]} {\stackrel{\triangle}{=}}\big\{ {\bf P}_{A_i|A^{i-1},
B^{i-1}}: i=0, \ldots, n\big\}$, which satisfy conditional independence on past
information. We derive similar characterizations, when general transmission
cost constraints are imposed. Moreover, we also show that the structural
properties apply to general nonlinear and linear autoregressive channel models
defined by discrete-time recursions on general alphabet spaces, and driven by
arbitrary distributed noise processes. We derive these structural properties by
invoking stochastic optimal control theory and variational equalities of
directed information, to identify tight upper bounds on $I(A^n \rightarrow
B^n)$, which are achievable over subsets of conditional distributions ${\cal
P}_{[0,n]}^{CI} \subseteq {\cal P}_{[0,n]}$, which satisfy conditional
independence and they are specified by the dependence of channel distributions
and transmission cost functions on inputs and output symbols. We apply the
characterizations to recursive Multiple Input Multiple Output Gaussian Linear
Channel Models with limited memory and we show a separation principle between
the computation of the elements of the optimal strategies.
