Asymptotic stability of switching systems
In this article, we study the uniform asymptotic stability of the switched system $u'=f_{ u(t)}(u)$, $uin mathbb{R}^n$, where $ u:mathbb{R}_{+}o {1,2,dots,m}$ is an arbitrary piecewise constant function. We find criteria for the asymptotic stability of nonlinear systems. In particular, fo...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2010-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2010/21/abstr.html |
| _version_ | 1818207043655303168 |
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| author | Driss Boularas David Cheban |
| author_facet | Driss Boularas David Cheban |
| author_sort | Driss Boularas |
| collection | DOAJ |
| description | In this article, we study the uniform asymptotic stability of the switched system $u'=f_{ u(t)}(u)$, $uin mathbb{R}^n$, where $ u:mathbb{R}_{+}o {1,2,dots,m}$ is an arbitrary piecewise constant function. We find criteria for the asymptotic stability of nonlinear systems. In particular, for slow and homogeneous systems, we prove that the asymptotic stability of each individual equation $u'=f_p(u)$ ($pin {1,2,dots,m}$) implies the uniform asymptotic stability of the system (with respect to switched signals). For linear switched systems (i.e., $f_p(u)=A_pu$, where $A_p$ is a linear mapping acting on $E^n$) we establish the following result: The linear switched system is uniformly asymptotically stable if it does not admit nontrivial bounded full trajectories and at least one of the equations $x'=A_px$ is asymptotically stable. We study this problem in the framework of linear non-autonomous dynamical systems (cocyles). |
| first_indexed | 2024-12-12T04:22:39Z |
| format | Article |
| id | doaj.art-acad2ab641a54f87b9e768006db1478b |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-12T04:22:39Z |
| publishDate | 2010-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-acad2ab641a54f87b9e768006db1478b2022-12-22T00:38:17ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912010-02-01201021,118Asymptotic stability of switching systemsDriss BoularasDavid ChebanIn this article, we study the uniform asymptotic stability of the switched system $u'=f_{ u(t)}(u)$, $uin mathbb{R}^n$, where $ u:mathbb{R}_{+}o {1,2,dots,m}$ is an arbitrary piecewise constant function. We find criteria for the asymptotic stability of nonlinear systems. In particular, for slow and homogeneous systems, we prove that the asymptotic stability of each individual equation $u'=f_p(u)$ ($pin {1,2,dots,m}$) implies the uniform asymptotic stability of the system (with respect to switched signals). For linear switched systems (i.e., $f_p(u)=A_pu$, where $A_p$ is a linear mapping acting on $E^n$) we establish the following result: The linear switched system is uniformly asymptotically stable if it does not admit nontrivial bounded full trajectories and at least one of the equations $x'=A_px$ is asymptotically stable. We study this problem in the framework of linear non-autonomous dynamical systems (cocyles).http://ejde.math.txstate.edu/Volumes/2010/21/abstr.htmlUniform asymptotic stabilitycocyclesglobalattractorsuniform exponential stabilityswitched systems |
| spellingShingle | Driss Boularas David Cheban Asymptotic stability of switching systems Electronic Journal of Differential Equations Uniform asymptotic stability cocycles globalattractors uniform exponential stability switched systems |
| title | Asymptotic stability of switching systems |
| title_full | Asymptotic stability of switching systems |
| title_fullStr | Asymptotic stability of switching systems |
| title_full_unstemmed | Asymptotic stability of switching systems |
| title_short | Asymptotic stability of switching systems |
| title_sort | asymptotic stability of switching systems |
| topic | Uniform asymptotic stability cocycles globalattractors uniform exponential stability switched systems |
| url | http://ejde.math.txstate.edu/Volumes/2010/21/abstr.html |
| work_keys_str_mv | AT drissboularas asymptoticstabilityofswitchingsystems AT davidcheban asymptoticstabilityofswitchingsystems |