Tangent sets, viability for differential inclusions and applications Original Research Article

Let XX be a real Banach space, let A:D(A)⊆X↝XA:D(A)⊆X↝X be a given operator, KK a nonempty and possible non-open subset in View the MathML sourceD(A)¯, F:K↝XF:K↝X a given multi-function. In this lecture, we consider the differential inclusion, u′(t)∈Au(t)+F(u(t))u′(t)∈Au(t)+F(u(t)), with (a) A=0A=0, (b) AA linear and (c) AA nonlinear and, in each one of these cases, we give a short survey of the most important and very recent necessary and sufficient conditions for viability expressed in terms of tangent sets and AA-quasi-tangent sets to KK at a given point ξ∈Kξ∈K, concepts recently introduced by the authors. From a rather long list of applications, we confined ourselves only to: solutions in moving sets, a comparison result for a reaction–diffusion system, a comparison result for a nonlinear diffusion inclusion and a sufficient condition for null controllability.