Special Functions, Partial Differential Equations, and Harmonic Analysis

Volume 108 of the series Springer Proceedings in Mathematics & Statistics pp 49-75


Fejér Polynomials and Chaos

  • Dmitriy DmitrishinAffiliated withOdessa National Polytechnic University
  • , Anna KhamitovaAffiliated withGeorgia Southern University
  • , Alexander M. StokolosAffiliated withGeorgia Southern University Email author 

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We show that given any μ > 1, an equilibrium x of a dynamic system
$$\displaystyle{ x_{n+1} = f(x_{n}) }$$
can be robustly stabilized by a nonlinear control
$$\displaystyle{ u = -\sum _{j=1}^{N-1}\varepsilon _{ j}\left (f\left (x_{n-j+1}\right ) - f\left (x_{n-j}\right )\right ),\,\vert \varepsilon _{j}\vert < 1,\;j = 1,\ldots,N - 1, }$$
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.