\(\alpha\)-systems of differential inclusions and their unification (Q2834407)

This paper introduces \(\alpha\)-systems of differential inclusions (\(\alpha\)-systems) on a time interval \([t_0,\vartheta],\) as well as defines \(\alpha\)-weakly invariant sets on \([t_0,\vartheta]\times \mathbb{R}^n,\) where \(\mathbb{R}^n\) is a phase space.NEWLINENEWLINETwo main problems related to the \(\alpha\)-systems are considered, namely: 1. In \([t_0,\vartheta]\times \mathbb{R}^n\) extract the set \(W\) of all positions \((t_*,x_*),\) from which the motions of the differential inclusions of an \(\alpha\)-system can be brought to a given compact set \(M\subset \mathbb{R}^n\) at the moment \(\vartheta\).NEWLINENEWLINE2. In \([t_0,\vartheta]\times \mathbb{R}^n\) extract the maximal \(\alpha\)-weakly invariant set \(W^c, W^c(\vartheta) \subset M,\) where \(W^c(\vartheta) = \{x \in \mathbb{R}^n: (\vartheta, x) \in W\}\).NEWLINENEWLINEThe issues connected with the calculation of \(W\) and \(W^c\) are also discussed. Exact calculation or analytical description of \(W\) and \(W^c\) can be performed in some cases only. Therefore, approximate calculation of \(W\) and \(W^c\) naturally represents a topical field of research. The paper gives an approximate calculation scheme for the sets \(W\) and \(W^c\). To this end, the authors introduce an \(\alpha\)-approximating system of sets that corresponds to a finite partition of the time interval \([t_0,\vartheta]\). The convergence of the approximating system to \(W^c\) is established as the partition step vanishes.NEWLINENEWLINETo extract the sets \(W, W^c\) and calculate them approximately, it is possible to substitute \(\alpha\)-systems by some other systems satisfying similar conditions and facilitating theoretical study and approximate calculations. One of such convenient systems can be obtained by unification of an \(\alpha\)-system.NEWLINENEWLINEThe paper also describes a unification of an \(\alpha\)-system that is similar to the unification of differential games. The authors define the quasi-Hamiltonian of an \(\alpha\)-system (\(\alpha\)-Hamiltonian), a core element in the unification structure.NEWLINENEWLINEThis paper continues the investigations of N.N. Krasovskii and A.I. Subbotin in the field of positional differential games and the generalized (minimax) solutions of the Hamilton-Jacobi equations. In what follows, the authors extend the idea of unification to the approach problems that involve the systems of differential inclusions not interconnected initially by a Hamilton-type function.