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The paper introduces the recent results related to an entropy functional ontrajectories of a controlled diffusion process, and the information pathfunctional (IPF), analyzing their connections to the Kolmogorov's entropy,complexity and the Lyapunov's characteristics. Considering the IPF's essentialsand specifics, the paper studies the singularities of the IPF extremalequations and the created invariant relations, which both are useful for thesolution of important mathematical and applied problems.  Keywords: Additive functional; Entropy; Singularities, Natural BorderProblem; Invariant

The entropy functional, the information path functional's essentials and their connections to Kolmogorov's entropy, complexity and physics