On the stability property in positional differential games
In the space \(\mathbb R^m\), a pursuit game problem for a conflict control system, the behavior of which is described by the differential equation \[ \dot{x}=f(t,x,u,v),\;t\in [t_0,\vartheta],\;u\in P,\;v\in Q, \] \[ x(t_0)=x_0, \] is considered. Here, \(u\) and \(v\) are the controls of the first and second players, \(P\subset \mathbb R^p\) and \(Q\subset \mathbb R^q\) are compact sets. The goal of the first player is to ensure that the state vector \(x[\vartheta]\) of the system reaches a given compact set \(M\subset \mathbb R^m\). In this paper, some basic constructions and properties are presented concerning the stability. The sets (stable bridges) from which the game can be successfully completed for the first (pursuing) player are distinguished. Moreover, ``it is shown how the stability defect can be used to estimate the neighborhood of the target set that the first player guarantees the conflict control system reaches''.
scientific article; zbMATH DE number 6106439