Averaging the controlled motion equations in a metric space
type: Item
instance of: scholarly article
title: Averaging the controlled motion equations in a metric space
zbMATH Open document ID: 0940.49006
DOI: 10.1007/BF02733119
author: V. A. Plotnikov
author: Liliya I. Plotnikova
published in: Cybernetics and Systems Analysis
publication date: 1999-04-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 49J24
Mathematics Subject Classification ID: 93B03
Mathematics Subject Classification ID: 49J27
zbMATH DE Number: 1262985
zbMATH Keywords: nonlinear control systems
zbMATH Keywords: differential inclusions
zbMATH Keywords: asymptotic optimisation
reviewed by: Władysław Hejmo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1262985