ИСТИНА

It is designed the phase portrait of optimal synthesis (the g eometric behavior of the set of optimal trajectories) for a model problem with control taki ng values in a two-dimensional simplex, having singular extremals of the second order and modes of in finite accumulations of switches. The new phenomenon is the chaotic behavior of the set of traje ctories of optimal synthesis. The set of optimal non-wandering trajectories (NW) has the stru cture of Cantor’s set. The dynam- ics of the system is described by a topological Markov chain. The entropy and the Hausdorf dimension of the set NW is calculated. It is shown that simila r behavior is generic for piecewise smooth Hamiltonian systems in the vicinity of the intersect ion of three codimension 1 break- surface strata.