On Generalized Differentials, Viability and Invariance of Differential Inclusions

Forward viability and invariance of time-dependent differential inclusions are studied with the aid of generalized differentials. Contingent derivative is compared with a newer concept of generalized differential quotient. It is shown that the latter is more suitable for expressing criteria of viability and invariance, as it better describes the directions tangent to invariant trajectories of differential inclusions. The concept of generalized differential quotient is related to Cellina continuously approximable set-valued functions whose properties are used.