ИСТИНА

The concept of rough dynamical systems was introduced by A.A.Andronov and L.S.Pontryagin. They analyzed family of phase curves of a $C^1$-vector field on a two-dimensional disk when the field does not vanish at the disk boundary and has no tangency with it and found the necessary and sufficient conditions such that for a field satisfying these conditions and any vector field sufficiently $C^1$ -close to it the families of phase curves of these fields are translated one into another by a homeomorphism being close to the identity. Later on Ì.Peixoto proved that rough vector fields are generic on any closed orientable surface. An analogous problem for dynamic inequalities was formulated by Myshkis in 1964. Such a problem naturally includes analysis of local controllability properties. Structural stability of generic control systems on closed orientable surfaces was proved in 1991, and for generic dynamic inequality this problem is open up to now, while the stability of local controllability properties of generic dynamic inequality with locally bounded derivatives was proved in 1995 and structural stability of generic simplest dynamic inequality on $S^2$ is also proved in 2007. The talk is devoted to these results and some related ones in other areas of mathematics.