Overview of viability results ∗

Let K be a multivalued map from the interval [0, 1] to R. On its graph, denoted by Gr(K), another multivalued map F is defined. Its values are closed subsets of R. The map F , called an orientor field, gives rise to a differential inclusion ẋ(t) ∈ F (t, x(t)). The inclusion is called viable, if for every (t0, x0) ∈ Gr(K) there is a global absolutely continuous forward trajectory x : [t0, 1] → R of this inclusion, satisfying the initial condition x(t0) = x0. As the differential inclusion may come from a control system ẋ(t) = f(t, x(t), u(t)), the viability of the differential inclusion may be interpreted as a kind of controllability of the control system with time dependent constraints. The first result on viability, by N. Nagumo, was formulated for a constant multifunction K and a single-valued time-independent F . In its full complexity, viability was studied by J.-P. Aubin, D. Bothe, K. Deimling, H. Frankowska, S. Hu, N. Papageorgiu, S. Plaskacz, T. Rzeżuchowski, and many others. The obtained results give conditions on K and F that guarantee the viability of the inclusion. Besides various measurability and continuity assumptions, the essential one states that the intersection of F (t, x) with some generalized derivative of K at (t, x) is nonempty for almost all t ∈ [0, 1] and all x ∈ K(t). This requirement is a direct extension of the Nagumo condition. We recall all these conditions and present our own viability criteria. As a generalized derivative we choose Generalized Differential Quotient (GDQ), introduced recently by H. Sussmann. Since GDQ of a multivalued function is not unique, we use one, denoted by SGDQ, that seems to fit best. SGDQ is the closure of the union of all minimal GDQs. Finally, we compare conditions introduced by different authors, setting a road map of the viability problem. ∗This work was supported by Bialystok Technical University grant W/IMF/1/04.