Construction of the viability kernel for a non-linear controlled system with a target set

A non-linear controlled system in a finite time interval with phase constraints and a given target set is considered. The problem of constructing the viability kernel in the phase constraints is investigated. The viability kernel is the set of all initial positions from which at least one viable trajectory emerges, that is, the trajectory of the system constrained by the phase constraints and which enters the target set. A method of constructing time-discrete approximations of the viability kernel is proposed.