Chattering arcs and chattering controls

There are two different meanings of the wordchattering in control theory and optimization theory. Chattering arcs of the first kind are related to the notion of relaxation of the control (i.e., convexization of the maneuverability domain). Some sufficient conditions of equivalence of these notions are defined. Chattering arcs of the second kind appear before and after some optimal singular arcs, for instance, the intermediate thrust arcs of the optimal transfer problem of astrodynamics. The simplest examples of chattering arcs of the second kind appear in Fuller's problem, two cases of which are examined in detail. The conditions of chattering of the second kind are analyzed; they are related to the Kelley-Contensou optimality test of singular extremals, also known asGeneralized Legendre-Clebsch conditions; they lead to general solutions and not only to solutions restricted to particular terminal conditions; thus, the phenomenon of chattering is very important (fortunately, these solutions can generally be approximated very closely by simple piecewise continuous controls). Finally, some special and complex cases appear, some examples of which are analyzed.