Order and Junctions in Singular Optimal Control Problems.

Abstract

New definitions for the recently proposed (1980) concepts of singular problem order and singular arc order are formulated and discussed in order to emphasize the dependence of the former on the dynamics and of the latter on the specific singular extremal in question. The definitions are further discussed in relation to two fundamental classes of optimal control problems: the singular linear quadratic problem (SLQP) and singular problems with bilinear dynamics and quadratic cost integr and (SBQP). The SLQP always has arc order equal to problem order, and problem order unbounded whenever it exceeds state dimension. For the SBQP, two examples are presented which have singular subarcs with arc order greater than problem order. The problem of characterizing the continuity and smoothness properties of the optimal control at points, called junctions, joining nonsingular and singular subarcs of the control is reformulated to include the ideas of singular problem order and singular arc order. This approach has the advantage of indicating how key functions used in singular optimal control theory behave on the nonsingular as well as the singular sides of a wider class of junctions. In particular, junction conditions are developed for the case of singular arc order greater than singular problem order (q > p). These conditions are the first that also hold for dynamics nonlinear in control (p = 0). Previous results implicitly assumed that q = p > 0. The new results replace this assumption with r > 2q - 3p where r is the order of the least order discontinuous time derivative of the control at the junction. The q (GREATERTHEQ) p conditions are generalized in turn by introducing a kind of generalized singular arc order which is defined at isolated points (in particular, the endpoints), as opposed to nonempty subintervals of the singular interval, and a generalized Legendre-Clebsch (GLC) condition which holds at the same isolated points. The generalized arc order is described by a set of indices which indicate the least order nonzero time derivative of each member of a finite sequence of GLC expressions at the junction. Finally, results are proved for some classes of junctions which violate r > 2q - 3p and other similar restrictions. These results clarify a recent partial proof of McDanell's conjecture and suggest a new framework for considering junction conditions.