Dynamical qualitative analysis of evolutionary systems

Kuipers\´ QSIM algorithm for tracking the monotonicity properties of solutions to differential equations has been revisited by Dordan by placing it in a rigorous mathematical framework. The Dordan QSIM algorithm provides the transition laws from one qualitative cell to the others. We take up this idea and revisit it at the light of recent advances in the field of "hybrid systems" and, more generally, "impulse differential equations and inclusions". Let us consider a family of "qualitative cells Q(α)" indexed by a parameter a ϵ A: We introduce a dynamical system on the discrete set of qualitative states prescribing an order of visit of the qualitative cells and an evolutionary system govening the "continuous" evolution of a system, such as a control system. The question arises to study and characterize the set of any pairs of qualitative and quantitative initial states from which start at least one order of visit of the qualitative cells and an continuous evolution visiting the qualitative cells in the prescribed order. This paper is devoted to the issues regarding this question using tools of set-valued analysis and viability theory.