Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side

Optimal spiral-like solutions near a singular extremal in a two-input control problem

“…Blowing up the singularity. To prove Theorem 3.2 we use the procedure of resolution of singularity for the Hamiltonian system (7) [54,51]. Consider the blowing up the singularity at the origin by the map B : z → (µ,z):…”

“…Let Q denote the cylinder {µ ∈ R} × Π, and Q 0 = Q ∩ {µ = 0}. It was proved [54] that B is a diffeomorphism from R 8 \ {0} onto Q ∩ {µ > 0}. In the coordinates (µ,z) the system (7) has the form:…”

“…Nonsingular trajectories can have an infinite number of control discontinuities in a finite time when entering on the singular arc (the chattering regime, or the Fuller phenomenon) [8,25,40,44,51,55,58]. Other singularities of solutions in the neighborhood of singular extremals were also revealed, e.g., optimal trajectories can have iterated Fuller singularities [9] or can be chaotic at finite time intervals [54].…”

“…For multidimensional controls, the structure of singular regimes is much richer and complex. Note that most of the results for problems with singular regimes and bounded multidimensional control were obtained when the control set is a multi-dimensional rectangle or a polyhedron [16,42,54]. The structure of optimal solutions for the control set, which is a convex set but not a polyhedron, was studied, e.g., in [15,29].…”

On properties of optimal controls for an inverted spherical pendulum

“…Often optimal trajectories consist of nonsingular and singular arcs and the concatenation structures of these arcs can be very irregular, for example, the chattering or the Fuller phenomenon (an infinite number of control discontinuities in a finite time interval) [17,20,22,26,33,34], iterated Fuller singularities [35], a chaotic behaviour of bounded pieces of optimal trajectories [36]. Such structure of optimal controls, rather complicated from a mathematical point of view, is very typical for controlling systems that possess singular regimes.…”