February 20, 2017. D. D. Kiselev (Russian Foreign Trade Academy, Moscow). An application of Galois theory to the optimal control.

Define the Zelikin-Lokutsievskiy polynomial f_n(x) with integer coefficients of degree n - 1 as follows

We show the irreducibility of f_(q-1)/2(x) over Q for any prime q > 3. We calculate the Galois group of the polynomial f_n(x), when the numbers p = n - 1, q = 2n + 1, r = 2n + 7 are prime and 889 is not a square modulo r. We also show under irreducibility hypothesis of the polynomial f_p+1(x) over Q for almost all primes p that there exists an infinite sequence of natural n, for which A_n - 1 is embeddable into Gal_Q(f_n(x)).

An example: for any natural k < 808 there exists an optimal control problem, the optimal control of which throws a dense winding of the k-dimensional torus in finite time.

List of talks at previous sessions of the seminar.