Vladimir S. Lerner

Vladimir S. Lerner

The paper introduces the recent results related to an entropy functional on trajectories of a controlled diffusion process, and the information path functional (IPF), analyzing their connections to the Kolmogorov's entropy, complexity and the Lyapunov's characteristics. Considering the IPF's essentials and specifics, the paper studies the singularities of the IPF extremal equations and the created invariant relations, which both are useful for the solution of important mathematical and applied problems. Keywords: Additive functional; Entropy; Singularities, Natural Border Problem; Invariant

Vladimir S. Lerner

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.