α-Systems of differential inclusions and their unification

Abstract

This paper introduces α-systems of differential inclusions on a bounded time interval [t₀, ϑ] and defines α-weakly invariant sets in [t₀, 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 ⊂ [t₀, ϑ] × ℝⁿ 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 ⊂ [t₀, ϑ] × ℝⁿ 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.