Necessary Conditions for Singular Extremals Involving Multiple Control Variables

Abstract

New necessary conditions are obtained from the second variation, via a transformed accessory minimum problem, for an important class of singular Bolza problems, which includes most of the singular optimal control problems that have been studied in recent years. This set of necessary conditions is a generalization of the classical Clebsch (Legendre) necessary condition. It is in a form easily used. For problems with multiple control variables, it is required that a certain matrix be symmetric; and if this symmetry property is satisfied, it then requires another matrix to be positive semidefinite. The positive semidefiniteness of the diagonal terms of the latter matrix imposes the same conditions as those obtained by other authors (Kelley, Kopp and Moyer). Should this generalized Clebsch condition be satisfied only in a semidefinite manner, then another similar set of necessary conditions can deduced, and so on. Three examples are studied. Firstly, we impose new necessary conditions on the variable thrust arcs of a rocket moving in a resisting medium in a vertical plane of a flat earth with two degrees of freedom, namely, the lift and thrust programs. Secondly, the doubly singular arcs of a problem in interplanetary guidance, formulated by Breakwell, are shown to be nonoptimal. Thirdly, a simple optimality condition is deduced for a class of identically singular optimal control problems, certain members of which were previously studied by Haynes, using an extension of the Green’s Theorem approach to higher dimension.