Mathematics

As it is well known, the time optimal control problem for a linear system with one-dimensional control is reduced to the certain algebraic system w.r.t. the optimal time and the switches of the optimal (bang-bang) control. In the particular “integrator” case $\dot x_1 = u, \dot x_k = x_{k−1}, k=2,...,n$, this system (of n equations w.r.t. n variables) is polynomial of a very special form. Such systems were studied much earlier within the moment problem theory. In particular, the time optimal problem for the integrator system is reformulated as the (in fact, nonclassical) power Markov moment problem on the minimal possible interval. The exact analytic solution for this problem and some generalizations will be presented. The lecture is mainly based on the results by V.I.Korobov and G.M.Sklyar (Kharkov National University) obtained since 1987.