Publication detail

On stabilization of unstable steady states of autonomous ordinary differential equations via delayed feedback controls

ČERMÁK, J. NECHVÁTAL, L.

Original Title

On stabilization of unstable steady states of autonomous ordinary differential equations via delayed feedback controls

English Title

On stabilization of unstable steady states of autonomous ordinary differential equations via delayed feedback controls

Type

journal article in Web of Science

Language

en

Original Abstract

The paper discusses stabilizing effects of some time-delayed feedback controls applied to unstable steady states of an autonomous system of ordinary differential equations. First, we derive explicit delay-dependent stability conditions that are applicable to a family of time-delayed systems with simultaneously triangularizable system matrices. Then, using this criterion and other argumentation, we employ diagonal delayed feedback controls of conventional and Pyragas type to stabilize unstable steady states of the studied autonomous system. More precisely, we formulate explicit, non-improvable and immediately applicable conditions on time delay and feedback strength that enable such a stabilization. As an illustration, we stabilize the unstable steady states of the Rössler dynamical system considered under the standard choice of entry parameters when the uncontrolled system displays a chaotic behavior. Also, we consider a non-diagonal feedback control (whose rotational gain matrix, involving a feedback strength and phase, commutes with the Jacobi matrix of the uncontrolled system) and show its larger stabilization potential with respect to the appropriate diagonal control. The obtained results are tested by numerical experiments and confronted with the existing results. As a supplement, we provide MATLAB codes supporting theoretical conclusions.

English abstract

The paper discusses stabilizing effects of some time-delayed feedback controls applied to unstable steady states of an autonomous system of ordinary differential equations. First, we derive explicit delay-dependent stability conditions that are applicable to a family of time-delayed systems with simultaneously triangularizable system matrices. Then, using this criterion and other argumentation, we employ diagonal delayed feedback controls of conventional and Pyragas type to stabilize unstable steady states of the studied autonomous system. More precisely, we formulate explicit, non-improvable and immediately applicable conditions on time delay and feedback strength that enable such a stabilization. As an illustration, we stabilize the unstable steady states of the Rössler dynamical system considered under the standard choice of entry parameters when the uncontrolled system displays a chaotic behavior. Also, we consider a non-diagonal feedback control (whose rotational gain matrix, involving a feedback strength and phase, commutes with the Jacobi matrix of the uncontrolled system) and show its larger stabilization potential with respect to the appropriate diagonal control. The obtained results are tested by numerical experiments and confronted with the existing results. As a supplement, we provide MATLAB codes supporting theoretical conclusions.

Keywords

Delay differential equation; Stability and stabilization; Feedback control; Rössler dynamical system

Released

01.03.2020

Publisher

ELSEVIER B.V.

Location

RADARWEG 29, 1043 NX AMSTERDAM, NETHERLANDS

Pages from

1

Pages to

30

Pages count

30

URL

Documents

BibTex


@article{BUT162614,
  author="Jan {Čermák} and Luděk {Nechvátal}",
  title="On stabilization of unstable steady states of autonomous ordinary
differential equations via delayed feedback controls",
  annote="The paper discusses stabilizing effects of some time-delayed feedback controls applied to unstable
steady states of an autonomous system of ordinary differential equations. First, we derive explicit
delay-dependent stability conditions that are applicable to a family of time-delayed systems with
simultaneously triangularizable system matrices. Then, using this criterion and other argumentation,
we employ diagonal delayed feedback controls of conventional and Pyragas type to stabilize unstable
steady states of the studied autonomous system. More precisely, we formulate explicit, non-improvable
and immediately applicable conditions on time delay and feedback strength that enable such a stabilization.
As an illustration, we stabilize the unstable steady states of the Rössler dynamical system
considered under the standard choice of entry parameters when the uncontrolled system displays
a chaotic behavior. Also, we consider a non-diagonal feedback control (whose rotational gain matrix,
involving a feedback strength and phase, commutes with the Jacobi matrix of the uncontrolled system)
and show its larger stabilization potential with respect to the appropriate diagonal control. The
obtained results are tested by numerical experiments and confronted with the existing results. As
a supplement, we provide MATLAB codes supporting theoretical conclusions.
",
  address="ELSEVIER B.V.",
  chapter="162614",
  doi="10.1016/j.physd.2020.132339",
  howpublished="print",
  institution="ELSEVIER B.V.",
  number="1",
  volume="404",
  year="2020",
  month="march",
  pages="1--30",
  publisher="ELSEVIER B.V.",
  type="journal article in Web of Science"
}