$\alpha$-systems of differential inclusions and their unification

In this article, $\alpha$-systems of differential inclusions are introduced on a bounded time segment $[t_0,\vartheta]$ and $\alpha$-weakly invariant sets in $[t_0,\vartheta] \times \mathbb R^n$ are defined, where $\mathbb R^n$ is a phase space of the differential inclusions. Problems are studied connected with bringing the motions (trajectories) of differential inclusions in an $\alpha$-system to a given compact set $M \subset \mathbb R^n$ at the time $\vartheta$. Questions are discussed of finding the solvability set $W \subset [t_0, \vartheta] \times \mathbb R^n$ of problem of bringing the motions of $\alpha$-system to $M$ and calculating the maximal $\alpha$-weakly invariant set $W^c \subset [t_0, \vartheta] \times \mathbb R^n$. The notion is introduced of quasi-Hamiltonian of $\alpha$-system ($\alpha$-Hamiltonian), which we see as important for studying problems of bringing motions of $\alpha$-system to $M$.