Controllability of differential inclusions
type: Item
instance of: scholarly article
title: Controllability of differential inclusions
zbMATH Open document ID: 0953.93011
DOI: 10.1007/BF02255897
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: An autonomous differential inclusion on a Rimannian compact manifold with a convex-valued Lipschitz-continuous right-hand side is studied. A control set is defined as a maximal viable (in some sources also called weakly invariant) subset of the state space. A control set \(C\) is called locally attractive if attainable sets emanating from a small neighborhood of \(C\) asymptotically converges to \(C\) in the Hausdorff metric. The objective of the study is the behavior of control sets under perturbations of the attainable set semiflow. A differential inclusion on a manifold is characterized by a set field, i.e. at each point \(x\) there is specified a subset \(F(x)\) of the tangential space. However, a small perturbation of a set field is defined in terms of the respective perturbation of the attainable set semiflow. Such a trick allows the author to avoid a requirement of uniform Lipschitz-continuity of the perturbed set fields. Two main results are obtained. The first theorem asserts that control sets with nonempty interiors are lower semi-continuous with respect to small perturbations. The second theorem claims that the lower continuity can be upgraded to continuity in the Hausdorff metric under the additional assumption that the control set is locally attractive. The continuity is understood in the following weak sense: given a locally attractive control set \(C\) and an arbitrary positive \(\varepsilon\) there exists a positive \(\delta_0\) such that for any perturbation parameter \(\delta\), \(0<\delta<\delta_0\) there exists a control set \(C_\delta\) of the perturbed system, for which \(d_H(C_\delta,C)\leq\varepsilon\). As an illustration, the continuity result is applied to a singularly perturbed nonlinear control system. A sequence of invariant control sets of the perturbed systems which approximate the entire state space as the parameter of perturbation tends to zero is constructed.
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 93C30
Mathematics Subject Classification ID: 93B03
Mathematics Subject Classification ID: 93C70
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 93C73
zbMATH DE Number: 1431802
zbMATH Keywords: differential inclusion
zbMATH Keywords: singular perturbation
zbMATH Keywords: robustness
zbMATH Keywords: control set
zbMATH Keywords: attainable set
zbMATH Keywords: perturbations
zbMATH Keywords: lower continuity
author: Götz Grammel
reviewed by: D. B. Silin
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02255897
OpenAlex ID: W1977245881
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description: scientific article; zbMATH DE number 1431802