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.