Asymptotic methods for analyzing large deviations in this work are used to convert information about the state of a controlled diffusion process into probabilistic estimates of the normal or abnormal development of the process. Thus, over the reflex contour of local stabilization a system of global semantic control is implemented, a kind of second signal system. A functional analytical approach similar to the weak convergence of probabilistic measures is used as an analysis tool, which makes it possible to significantly expand the conditions for applying the method. Global control is reduced to solving the Lagrange problem in the form of Pontryagin for the system of ordinary differential equations (system of paths), the Ventzel-Freidlin action functional (or "rate function" in some English literature), which is presented here as an integral-quadratic criterion for control functions in the system of paths, and the boundary condition in the form of the critical state of the system. A bounded solution of the Lagrange — Pontryagin problem on the half-line, which gives a prototype of the quasipotential of the system of paths, is called the A-profile of the critical state. The A-profile makes it possible to significantly simplify the procedure for analyzing large deviations, up to its implementation in real time and the implementation of the global control loop (2nd signaling system). The resulting two-tier architecture is positioned as a baseline to achieve the functional stability of the control system. It is speculated that this role of the apparatus of large deviations takes place in biological evolving systems, including the formation of languages and other attributes of the evolution of higher human nervous activity.
The paper presents a new method of analyzing large deviations for nonlinear systems defined through matrices with state-dependent coefficients. Large deviations of the controlled process from some standard state is the basis to forecast any critical situation. The task of forecasting is limited to the optimal control problem of Lagrange-Pontryagin optimal control. Two effective methods – State-Dependent Riccati Equations (SDRE) and approximated sequence of Riccati equations (ASRE) – are used. The presented approach to the Lagrange-Pontryagin problem differs from the approach previously used for linear cases by the fact that it uses control in the form of a feedback instead of software open-loop control. This eliminates the need to calculate the end-time boundary value for a conjugate variable, which is the most time-consuming task in nonlinear cases.
A stochastic dynamic system is considered, for which a decision-making system is built over the stabilization loop, an analogue of the second signaling system. For this purpose, the principle of large deviations and the theory of optimal control are used.
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