We show that given any μ > 1, an equilibrium x of a dynamic system

xn+1=f(xn) (1)

can be robustly stabilized by a nonlinear control

u=−∑j=1N−1εj(f(xn−j+1)−f(xn−j)), |εj| < 1, j=1,…,N−1, (2)

for f ′ (x) ∈ (−μ, 1). The magnitude of the minimal value N is of order √μ. 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.