Admissibility of trajectories for control systems related by smooth mappings
The author seeks conditions guaranteeing that a given differential space transformation maps every trajectory of one control system into an admissible trajectory of another control system. Let two control systems \[ \dot{x}= f(x,u),\quad \dot{y} = g(y,v) \] be defined on two differential manifolds \(M\) and \(N\) and let \(\Phi: M\to N\) be a \(C^1\) mapping. The systems are called \(\Phi\)-related if for every \(x\in M\) \[ \{ d\Phi_x f(x,\omega),\;\omega\in\Omega\}\subseteq \{g(\Phi(x),\theta),\;\theta\in\Theta\} \] Here \(\Omega\) and \(\Theta\) are separable metric spaces where the respective control functions take on their values. A mapping \(\xi:\,Y\to Z\), where \(Y\) and \(Z\) are topological spaces, compactly covers compact subsets of its image if for every compact set \(K\subseteq\xi(Y)\) there exists a compact set \(C\subseteq Y\) such that \(K\subseteq \xi(C)\). It is obtained that if the mapping \[ \xi(x,\theta) = (x,g(\Phi(x),\theta)): M\times \Theta\to M\times TN \] compactly covers compact subsets of its image and systems \(f\) and \(g\) are \(\Phi\)-related, then the mapping \(\Phi\) maps trajectories of system \(f\) into trajectories of system \(g\). In this result, admissible controls for both systems are essentially compact-valued on compact intervals functions such that the systems satisfy \(C^1\)-Carathéodory condition. With an additional assumption of local compactness of the spaces \(\Omega\) and \(\Theta\), the above-mentioned result is strengthened. Namely, it is obtained that for a given transformation \(\Phi\) there exists a Borel-measurable transformation \(\nu: M\times\Omega\to\Theta\) such that \(\Phi\) maps trajectories of system \(f\) to trajectories of system \(g\), whose controls can be obtained using transformation \(\nu\).
scientific article; zbMATH DE number 1998220