On damping problem for control system with global aftereffect on a time tree: stochastic interpretation

Recently, with the help of the idea of global delay, N.N. Krasovsky's problem of calming a controlled system with aftereffect was transferred to the so-called quantum graphs, while the case of an arbitrary tree was covered. In turn, this led to the concept of a time graph, when the variable parametrizing the edges is identified with time. At the same time, each inner vertex can be understood as the moment of branching of the process, which gives several different scenarios for its further course. This directly imposes conditions for continuous gluing of trajectories at internal vertices. However, as in spatial networks, Kirchhoff-type conditions can also occur here. Namely, they will be satisfied with such a trajectory of the process throughout the tree, which is optimal taking into account all possible scenarios at once. The talk discusses the stochastic interpretation of a controlled system on a tree. In particular, the replacement of coefficients in an ordinary equation on an interval by discrete random processes with discrete time leads to a system on a tree.