Extremal Trajectories, Small-time Reachable Sets and Local Feedback Synthesis: a Synopsis of the Three-dimensional Case

Abstract

In any optimization problem three basic questions have to be answered. Does an optimal solution exist? How can one restrict the candidates for optimality by way of necessary conditions? Is a candidate found in this way indeed optimal (in a local and/or global sense)? For problems on function spaces by now fairly general existence results are known see, for instance, Cesari [6]) which cover a wide range of realistic problem situations. The theories of necessary and sufficient conditions for optimality, on the other hand, lack a similar completeness of results. For optimal control problems, the Pontryagin Maximum Principle [11] gives first order necessary conditions for optimality. Several higher order conditions for optimality are known as well (cf. Krener [8], Knobloch [7] and the many references therein), but they mainly deal with special situations, like the generalized Legendre-Clebsch condition for singular arcs. Typically the necessary conditions will not suffice to single out the optimal control. In fact, in many cases there exists a significant gap between the structure of extremals (i.e. trajectories which satisfy the necessary conditions for optimality) and the structure of optimal trajectories in a regular synthesis. Roughly speaking, a regular synthesis consists of a family of extremals with the property that a unique extremal trajectory starts from every point of the state-space and which satisfies certain technical conditions which allow to prove that the corresponding feedback control is indeed optimal.