Evolution equations on locally closed graphs and applications Original Research Article

Let XX be a Banach space, let A:D(A)⊆X→XA:D(A)⊆X→X be the infinitesimal generator of a C0C0-semigroup, let II be a nonempty, bounded interval and let K:I↝XK:I↝X be a given multi-valued function. By using the concept of AA-quasi-tangent set introduced by Cârjă, Necula, Vrabie [O. Cârjă, M. Necula, I.I. Vrabie, Necessary and sufficient conditions for viability for a semilinear differential inclusions, Trans. Amer. Math. Soc., 361 (2009) 343–390; O. Cârjă, M. Necula, I.I. Vrabie, Viability, Invariance and Applications, North-Holland a Mathematics Studies, vol. 207, Elsevier, 2007] and using a tangency condition expressed in the terms of this concept, we establish a necessary and sufficient condition for mild viability referring to evolution inclusions of the form u′(t)∈Au(t)+F(t,u(t))u′(t)∈Au(t)+F(t,u(t)), where FF is a multi-function defined on the graph of KK. As applications, we deduce a sufficient condition for null-controllability and a comparison result for semilinear evolution inclusions.