ИСТИНА

Recently a new phenomenon of chaotic entanglement of optimal trajecto- ries was discovered in optimal control problems which are ane in multivari- able control. Typically, chaotic behavior of a system is associated with the asymptotic properties of trajectories while time tends to innity. A distinc- tive feature of the considered phenomenon is that all the classic symptoms of chaos (such as non-zero entropy, fractal structure of the set of non-wandering points, etc.) are observed here on nite time intervals. The deep reason lies, apparently, in the presence of a discrete noncommu- tative group of almost symmetries. On the one hand it can be proved that the completion of the optimal process for a nite time is inevitable, and on the other hand, the presence of such a group prohibits the existence of regular synthesis. Theorem on structural stability is proved. In other words, small pertur- bations of the initial problem can not destroy chaos in the optimal synthesis. The talk will be divided into two parts. In the rst part, M. I. Zelikin will make a general introduction to the essence of the phenomenon on the example of the model optimal control prob- lem that serves as the key basic example. In the second part of the talk L. V. Lokutsievskiy will make an introduction to the basic ideas, enabling to detect and analyze this phenomenon.