From constraint satisfactions to periodic positive invariance for discrete-time systems

This paper deals with the analysis of the trajectories of autonomous dynamical systems with respect to static constraints. Two notions of constraints satisfaction and set-invariance are introduced in order to relax the classical definitions and offer a new perspective with potentially increased flexibility in the topology of the candidate sets. The relaxation comes from the possibility to violate the constraints for intervals of finite length along the evolution of the trajectory. In this line of developments, two alternative definitions emerge: a weak one which allows the validation of each constraint taken independently with an upper bound on the violation interval and the other one which imposes the length of the interval with a guarantee of constraint satisfaction. The main interest of those concepts is to handle simpler sets as positively invariant candidates with clear advantages in constrained control design.