Extension of Pontryagin's equation to nonlinear sampled-data control systems

Equations are derived that determine the probability of the describing point leaving the confines of a specified area in the phase space for a multi-dimensional random process whose components are considered at fixed equidistant time moments. For the case of a Markov process these equations are extensions of the Pontriagin equation to processes with specified components. The application of the resultant equations to determination of the probability of follow-up failure is considered for nonlinear sample-data systems.