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Necessary and sufficient conditions for viability for semilinear differential inclusions

Given a set A" in a Banach space X, we define: the tangent set, and the quasi-tangent set to K atC G K, concepts more general than the one of tangent vector introduced by Bouligand (1930) and Severi (1931). Both notions prove very suitable in the study of viability problems referring to differential inclusions. Namely, we establish several new necessary, and even necessary and sufficient conditions for viability referring to both differential inclusions and semilinear evolution inclusions, conditions expressed in terms of the tangency concepts introduced.