Use of Quasipotentials for Monitoring of Large Deviations in the Control Processes

Abstract

The Lagrange problem arising in the analysis of large deviations in the states of dynamical systems was analyzed with the use of Wentzell-Freidlin method. For the linear case and Hurwitz state matrix of an unperturbed system the author obtained relations for quasipotential extremals, providing estimates of the probabilities of events for the initial conditions close to zero. On this basis, the author proposes an algorithm for prediction of the critical states of the dynamical systems, with the perturbed vector, "white noise", multiplied by a small parameter. Examples of application of the method to the task of controlling the angle of heel for a marine vessel in rough seas, and the angle of attack of an aircraft are presented. The case of lack of Hurwitz is analyzed on the example of the financial mathematics - instruments known as the Black-Scholes model The results show that the quasipotential extremals can be effectively used for the tasks of stabilization as a means of additional quality assurance management: passage of the stochastic system state through a small neighborhood of a quasipotential extremal signalizes about an abnormal movement of the controlled process. More accurate conclusions about the danger rate and the need to switch to the crisis management can be made with the estimated probability of a crisis and action functional, calculated on the basis of the quasipotential (equations included). In case of instability the question can be solved directly on the action functional, as an example of a financial mathematics model shows.