Averaging the controlled motion equations in a metric space
The authors introduce their own notion \(R\)-\textit{solution} that may be interpreted as a property of the system reaching a non-empty target set. The paper deals with some problems of controlled object the motion of which is defined by quasi-differential equations in a metric space. As a mapping of the controlled system dynamics there is taken the following quasi-differential equation in a metric space: \[ \delta (x(t+\Delta), \varphi(\varepsilon, \Delta, t, x(t)))=o(\Delta), \quad x(t_{0})=x_{0}, \] where \(\varphi:(0,\varepsilon] \times [0, \sigma] \times [t_{0}, t_{0}+T) \times X \times U \rightarrow X\), \(\varepsilon > 0, u \in U \subset Z,\;X\) is a metric space with the metric \(\delta(.,.)\), \(\varepsilon\) is a small parameter, \(Z\) is the metric space with the metric \(\delta_{1}(.,.)\). As the \textit{R-solution} of the above equation is understood a continuous mapping \(y:[t_{0}, T) \rightarrow X\), satisfying the above equation for each \(t \in [t_{0}, T)\), where \(X\) is a metric space with the metric \(\delta (.,.)\). Some examples of the averaging scheme for quasi-differential equations defining the motion of controlled object have been exhibited. The paper contains a small supplement to works published by the authors previously.
scientific article; zbMATH DE number 1262985