From Relaxed Constraint Satisfaction to $p-$Invariance of Sets

The present paper proposes a general framework for the analysis of constraints satisfaction with respect to the trajectories of a dynamical system. The results are presented in a discrete-time framework and pertain to the class of set-theoretic methods. The main objective is to go beyond the state-of-the-art by characterizing the intermittent constraint satisfaction along the evolution of the trajectories of a dynamical system. Two relaxed notions are introduced in this sense, one characterizing the validation of constraints within a given finite window and the other imposing the validation after a fixed number of time-steps following a violation. The constraint satisfaction with respect to a controlled trajectory will then be extended to a set of constraints and then to tubes of trajectories. It is shown that all these notions can be accordingly anchored to the well-known controlled positive set invariance, thus offering a generalized framework for the analysis of dynamical systems in a set-theoretic framework. The technical note is completed with illustrations of the constructions on both linear and nonlinear case.