  We study a stochastic control system, described by Ito controllable equation,
and evaluate the solutions by an entropy functional (EF), defined by the
equation functions of controllable drift and diffusion. Considering a control
problem for this functional, we solve the EF control variation problem (VP),
which leads to both a dynamic approximation of the process entropy functional
by an information path functional (IPF) and information dynamic model (IDM) of
the stochastic process. The IPF variation equations allow finding the optimal
control functions, applied to both stochastic system and the IDM for joint
solution of the identification and optimal control problems, combined with
state consolidation. In this optimal dual strategy, the IPF optimum predicts
each current control action not only in terms of total functional path goal,
but also by setting for each following control action the renovated values of
this functional controllable drift and diffusion, identified during the optimal
movement, which concurrently correct this goal. The VP information invariants
allow optimal encoding of the identified dynamic model operator and control.
The introduced method of cutting off the process by applying an impulse control
estimates the cutoff information, accumulated by the process inner connections
between its states. It has shown that such a functional information measure
contains more information than the sum of Shannon entropies counted for all
process separated states, and provides information measure of Feller kernel.
Examples illustrate the procedure of solving these problems, which has been
implemented in practice. Key words: Entropy and information path functionals,
variation equations, information invariants, controllable dynamics, impulse
controls, cutting off the diffusion process, identification, cooperation,
encoding.
