Analysis of piecewise linear stochastic systems in quarter-spaces by means of the Pugachev-Sveshnikov equation

An analytic approach is presented to obtain a probability distribution function of the state-vector of piecewise linear systems which have four domains (quarter spaces) of linearity. The approach is based on the use of the Pugachev-Sveshnikov equation for the characteristic function and its reduction to the parametric Riemann Boundary Value Problem for bi-half planes. The Crandall's problem for the controlled dry friction, which is switched off when body's velocity is over a critical level, is solved as an instance of application of the derived theory. The asymptotic behavior of the displacement of a body, placed on a randomly oscillating foundation, and occupation time, while velocity is under the critical level, are explored.