α-Systems of differential inclusions and their unification

This paper introduces α -systems of differential inclusions on a bounded time interval [ t 0 , ϑ ] and defines α-weakly invariant sets in [ t 0 , ϑ ] × ℝ n , where ℝ n is a phase space of the differential inclusions. We study the problems connected with bringing the motions (trajectories) of the differential inclusions from an α -system to a given compact set M ⊂ ℝ n at the moment ϑ (the approach problems). The issues of extracting the solvability set W ⊂ [ t 0 , ϑ ] × ℝ n in the problem of bringing the motions of an α -system to M and the issues of calculating the maximal α -weakly invariant set W c ⊂ [ t 0 , ϑ ] × ℝ n are also discussed. The notion of the quasi-Hamiltonian of an α-system (α-Hamiltonian) is proposed, which seems important for the problems of bringing the motions of the α -system to M .