Probability criterion: its use in stabilization problems (Q1881848)

The paper deals with a (simple) control system of the form \[ y_{n+1} = y_{n} +\xi_{n} u_{n}. \] \(u_{n}\) denotes a control vector, \(y_{n} \) is the output at the \(n\)th instant, \(\xi_{n}\) is a random variable (corresponding to a noise) that takes values in the interval \((0; \, +\infty). \) The ``underlying'' probability measure is assumed to be unknown, consequently it is necessary to base a control on the empirical frequency. Applications corresponding to this model are introduced at the beginning of the paper. The aim of the paper is to suggest a stabilization algorithm (based on empirical data) such that the probability of a fulfilling of the prescribed admissible stabilization error converges to \(1.\) The introduced algorithm is illustrated on the problem of a motion of an artificial stationary earth satellite. The paper is written in a very understandable way. The presented assertions are formulated as mathematical theorems which are also proven in the paper.