ИСТИНА

The main tool for solving deterministic optimal control problems is the Pontryagin Maximum Principle (PMP). It allows to reduce the control problem to a two-point boundary value problem of Hamiltonian dynamics. Suppose the control variable $u$ takes values in some set $\Omega$. Then the Hamiltonian function $H$ defining the dynamics is given as the maximum $H(q,p) = \max_{u \in \Omega}{\cal H}(q,p,u)$ over the control $u$ of the {\it Pontryagin function} ${\cal H}$, and the optimal control $\hat u(q,p)$, if it exists, is found among the maximizers. In general, the maximizer is unique in an open dense subset of the space of variables $q,p$ and depends smoothly on these variables. On this set, the Hamiltonian $H$ is smooth, whereas on its boundary the derivatives of $H$ experience discontinuities. Usually the Hamiltonian system as a whole is piece-wise smooth on the cotangent bundle. The cotangent bundle is divided into disjoint domains $A_1,\dots,A_k$ on which the Hamiltonian is given by smooth functions $H_1,\dots,H_k$, respectively. The dynamics is described by a system of ODEs with discontinuous right-hand side. We consider situations when the set $\Omega$ is a convex polyhedron, and the domains $A_i$ are those regions where the optimal control resides in a particular vertex $v_i$ of the polyhedron. The set of points where the derivatives of $H$ are discontinuous is a stratified manifold, and on each stratum the optimal control is confined to a particular face of the polyhedron $\Omega$. A trajectory of the Hamiltonian system evolving inside a smoothness domain is called {\it regular}. If a trajectory passes from one smoothness domain $A_i$ into another one $A_j$, then the corresponding optimal control will experience a jump from the vertex $v_i$ of the polyhedron $\Omega$ to the vertex $v_j$. This process is called {\it switching}, and the discontinuity hyper-surface is called {\it switching surface}. Typically trajectories intersect the switching surface transversally, in which case they are called {\it bang-bang trajectories}. It may happen, however, that a trajectory moves along the switching surface, in which case one speaks of a {\it singular trajectory}. Typically uniqueness of the solution does not hold in the vicinity of a singular trajectory, and many regular trajectories can join in the same point on a singular trajectory. This is possible because the right-hand side of the underlying ordinary differential equations experiences a discontinuity. All trajectories ending (or starting) at a fixed singular point form {\it integral vortex} of the point. For a singular trajectory lying on a switching hyper-surface one can define an {\it order}, in dependence on up to what maximum length of the Poisson brackets of the adjoining smooth pieces $H_i$ of the Hamiltonian vanish. If the order of the singular trajectory is even, then a regular trajectory cannot join it in a piece-wise smooth manner. In this case regular trajectories spiral around the singular trajectory and intersect the switching surface in an infinite number of points in finite time, in such a way that the joining point is the accumulation point of switchings. This phenomenon is called {\it chattering}, and is well-studied for the situation where exactly two smoothness domains meet at the singular trajectory in question. The situation where three smoothness domains $A_1,A_2,A_3$ meet at a manifold ${\cal S}_{123}$ of codimension 2 will be considered on the talk. This situation is equivalent to an optimal control problem with 2-dimensional control lying in a triangle. We observe an additional phenomenon appearing in this situation, namely, the chaotic behaviour of bounded parts of trajectories. This phenomenon was not yet seen in optimal control problems and is hence entirely new. Our findings are not limited to optimal control problems, but rather hold for a whole class of piece-wise smooth Hamiltonian systems with three smoothness domains joining at surface ${\cal S}_{123}$ containing singular trajectories of second order. It appears that integral vortexes of singular point has a chaotic nature: there exists a topological Markov chain $\Sigma_\Gamma$ such that the sequence of control switchings on a trajectory corresponds one to one to the trajectory through a point of $\Sigma_\Gamma$ under the right Bernoulli shift. The set of non-wandering trajectories has a fractal structure (as in Smale's horseshoe) and non-integer Hausdorff and box dimensions. Emphasize that all trajectories in an integral vortex fall into the singular point in finite time. The discovered phenomenon appears in the neighbourhood of a generic singularity. Namely, a theorem on the structural stability is proven. This phenomenon was discovered in the joined work with Mikhail Il'ich Zelikin and Roland Hildebrand.