ИСТИНА

I plan to discuss a new approach to study non-smooth Hamiltonian systems. Namely, this approach is based on the fact that nilpotent approximations of such systems are nilpotent-convex problems of optimal control. The optimal synthesis in these problems forms a half flow on the phase space and hence can be studied from three different points of view: by methods of dynamical systems, by topological methods and by methods of convex analysis. This half flow has many nice properties and some of them can be restored in the original non-smooth Hamiltonian system. This approach gives very powerful results when the half flow in the corresponding nilpotent-convex problem has chaotic nature. Another interesting corollary comes from the Lefschetz formula which allows to prove existence of periodic trajectories of special kind.