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Home » Issues » Archive of Issues » 2022 - 3 issue » pp. 447-462

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Russia Yekaterinburg
Year
2022
Volume
32
Issue
3
Pages
447-462
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Section Mathematics
Title On the parametric dependence of the volume of integral funnels and their approximations
Author(-s) Ushakov V.N.ab, Ershov A.A.ab
Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb
Abstract We consider a nonlinear control system in a finite-dimensional Euclidean space and on a finite time interval, which depends on a parameter. Reachable sets and integral funnels of a differential inclusion corresponding to a control system containing a parameter are studied. When studying numerous problems of control theory and differential games, constructing their solutions and estimating errors, various theoretical approaches and associated computational methods are used. The problems mentioned above include, for example, various types of approach problems, the resolving constructions of which can be described quite simply in terms of reachable sets and integral funnels. In this paper, we study the dependence of reachable sets and integral funnels on a parameter: the degree of this dependence on a parameter is estimated under certain conditions on the control system. The degree of dependence of the integral funnels is investigated for the change in their volume with a change in the parameter. To estimate this dependence, systems of sets in the phase space are introduced that approximate the reachable sets and integral funnels on a given time interval corresponding to a finite partition of this interval. In this case, the degree of dependence of the approximating system of sets on the parameter is first estimated, and then this estimate is used in estimating the dependence of the volume of the integral funnel of the differential inclusion on the parameter. This approach is natural and especially useful in the study of specific applied control problems, in solving which, in the end, one has to deal not with ideal reachable sets and integral funnels, but with their approximations corresponding to a discrete representation of the time interval.
Keywords control nonlinear systems, differential inclusions, reachable sets, parametric dependence, volume of integral funnel, discrete approximation
UDC 517.977.58
MSC 49M25, 93C15
DOI 10.35634/vm220307
Received 25 July 2022
Language Russian
Citation Ushakov V.N., Ershov A.A. On the parametric dependence of the volume of integral funnels and their approximations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 3, pp. 447-462.
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