Necessary optimality conditions for nonstandard control problems

Because relaxed controls form a convex and compact subset of a normed vector space, it is generally much easier to derive controllability results and corresponding optimization results for such problems. For standard and nonstandard unrelaxed control problems, these properties of relaxed controls are used to divide the essential part of the derivation into two steps: (a) a general open mapping theorem for appropriate perturbations of the relaxed problem, and (b) the proof that the corresponding set of unrelaxed controls is an “abundant” subset of the corresponding relaxed set. Zhu has derived necessary conditions for minimum (generalizing both the Pontryagin and the more general Kaskosz maximum principle) for optimization problems defined by nonconvex differential inclusions with endpoint and unilateral constraints. The authors summarize these results