In this work we consider an extension of the viability concept introduced by Aubin in [1] to the regional case for Distributed Parameters Systems (DPS). This consists in coupling the recent works developed on regional analysis by El Jai and his team and the previous one developed on viability by Aubin. The aim of this paper is to introduce the regional viability and to study its connection with spreadability [6]. To illustrate the concepts some examples are presented.Then the state z 0 ∈ L 2 (Ω) is σ-regionally viable in K or K σ -viable if: χ σ z 0 ∈ K σ and χ σ z(x, t; z 0 ) ∈ K σ ∀t ∈]0, T [ In the definition 1, we have considered the regional viability for a state z 0 ∈ L 2 (Ω) (z 0 is defined on the whole domain Ω) with the regional constraints (on σ ⊂ Ω) i.e. χ σ z 0 ∈ K σ and χ σ z 0 (., t) ∈ K σ ∀t ∈]0, T [ because in regional analysis even if we are interesting by a regional aspect/aim (σ)

Abstract. In this work we consider the problem of stability, for distributed parameter systems, through the space variable. We give an extension of the stability radius, introduced by A. J. Pritchard and S. Townley [7,10], to the regional case. This consists to determine the "smallest disturbance" which destabilizes regionally an exponentially stable system. We prove in particular that for a certain given class of distributed parameter systems, it is possible to destabilize regionally an exponential stable system without destabilizing it totally.Résumé. Nous considérons dans ce travail le problème de la stabilité pour les systèmesà paramètres distribués avec une variable espace. Nous donnons une extension au cas régional du rayon de stabilité, introduit par A. J. Pritchard et S. Townley [7,10]. Il s'agit de déterminer la "plus petite perturbation" qui déstabilise régionalement un système exponentiellement stable. Nous montrons en particulier que pour certaines classes de systèmes distribués données, il est possible de déstabiliser régionalement un système initialement exponentiellement stable sans pour autant le déstabiliser totalement.