Fejér Polynomials and Chaos

  • Dmitriy Dmitrishin
  • Anna Khamitova
  • Alexander M. Stokolos
Conference paper

DOI: 10.1007/978-3-319-10545-1_7

Volume 108 of the book series Springer Proceedings in Mathematics & Statistics (PROMS)
Cite this paper as:
Dmitrishin D., Khamitova A., Stokolos A.M. (2014) Fejér Polynomials and Chaos. In: Georgakis C., Stokolos A., Urbina W. (eds) Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol 108. Springer, Cham

Abstract

We show that given any μ > 1, an equilibrium x of a dynamic system
$$\displaystyle{ x_{n+1} = f(x_{n}) }$$
(1)
can be robustly stabilized by a nonlinear control
$$\displaystyle{ u = -\sum _{j=1}^{N-1}\varepsilon _{ j}\left (f\left (x_{n-j+1}\right ) - f\left (x_{n-j}\right )\right ),\,\vert \varepsilon _{j}\vert < 1,\;j = 1,\ldots,N - 1, }$$
(2)
for f(x) ∈ (−μ, 1). The magnitude of the minimal value N is of order \(\sqrt{\mu }.\) The optimal explicit strength coefficients are found using extremal nonnegative Fejér polynomials. The case of a cycle as well as numeric examples and applications to mathematical biology are considered.

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Dmitriy Dmitrishin
    • 1
  • Anna Khamitova
    • 2
  • Alexander M. Stokolos
    • 2
  1. 1.Odessa National Polytechnic UniversityOdessaUkraine
  2. 2.Georgia Southern UniversityStatesboroUSA