TY - JOUR

T1 - Fejér Polynomials and Control of Nonlinear Discrete Systems

AU - Dmitrishin, D.

AU - Hagelstein, P.

AU - Khamitova, A.

AU - Korenovskyi, A.

AU - Stokolos, A.

N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - We consider optimization problems associated with a delayed feedback control (DFC) mechanism for stabilizing cycles of one-dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function f: R→ R of the form x(k+1)=f(x(k))+u(k),where u(k)=(a1-1)f(x(k))+a2f(x(k-T))+⋯+aNf(x(k-(N-1)T)),with a1+ ⋯ + aN= 1. Following an approach of Morgül, we associate with each periodic orbit of f, N∈ N, and a1,.., aN an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of f, there exist N and a1, … , aN whose associated polynomial is Schur stable, and we find the minimal N that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fejér kernels found in classical harmonic analysis.

AB - We consider optimization problems associated with a delayed feedback control (DFC) mechanism for stabilizing cycles of one-dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function f: R→ R of the form x(k+1)=f(x(k))+u(k),where u(k)=(a1-1)f(x(k))+a2f(x(k-T))+⋯+aNf(x(k-(N-1)T)),with a1+ ⋯ + aN= 1. Following an approach of Morgül, we associate with each periodic orbit of f, N∈ N, and a1,.., aN an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of f, there exist N and a1, … , aN whose associated polynomial is Schur stable, and we find the minimal N that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fejér kernels found in classical harmonic analysis.

KW - Control theory

KW - Optimization

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85067990422&partnerID=8YFLogxK

U2 - 10.1007/s00365-019-09472-3

DO - 10.1007/s00365-019-09472-3

M3 - Article

AN - SCOPUS:85067990422

SN - 0176-4276

VL - 51

SP - 383

EP - 412

JO - Constructive Approximation

JF - Constructive Approximation

IS - 2

ER -