Optimal Synthesis Containing Chattering Arcs and Singular Arcs of the Second Order

Abstract

In most optimal control problems, in which the optimal controls can be explicitly constructed, they are piecewise analytic with a finite number of discontinuity points (called switching points). Meanwhile there is an old example, proposed by A.T. Fuller, in which the optimal controls have an infinite number of switchings on the finite-time interval. Such a phenomenon has been called “chattering”. Being published in 1961 [1], Fuller’s example aroused some interest but soon was forgotten. About twelve years later there was risen the second wave of interest in this phenomenon. Several examples with the optimal chattering trajectories have been found [2–7]. In the recent years we observe the third wave of interest due to the intensive attempts to understand a real content of the notion of regular synthesis, and especially due to the remarkable work of J. Cupka [8], who proved that in the case of general position for some class of discontinuous Hamiltonian systems in the dimensions more or equal than 12 there exists at least one chattering trajectory.