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Fejér Polynomials and Chaos

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Special Functions, Partial Differential Equations, and Harmonic Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 108))

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.

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Author information

Authors and Affiliations

  1. Odessa National Polytechnic University, 1 Shevchenko Ave., Odessa, 65044, Ukraine

    Dmitriy Dmitrishin

  2. Georgia Southern University, Statesboro, GA, 30458, USA

    Anna Khamitova & Alexander M. Stokolos

Authors
  1. Dmitriy Dmitrishin
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  2. Anna Khamitova
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  3. Alexander M. Stokolos
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Corresponding author

Correspondence to Alexander M. Stokolos .

Editor information

Editors and Affiliations

  1. DePaul University, Chicago, Illinois, USA

    Constantine Georgakis

  2. Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  3. Roosevelt University, Chicago, Illinois, USA

    Wilfredo Urbina

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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. https://doi.org/10.1007/978-3-319-10545-1_7

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