A SECOND-ORDER NECESSARY CONDITION FOR OPTIMALITY IN THE GENERAL NONLINEAR CASE

Abstract

In this paper a second-order optimality condition is obtained for the general nonlinear problem with arbitrary boundary conditions and an integral-type minimized functional.Moreover, a second-order sufficient condition for controllability is derived.Bibliography: 3 titles.This paper is a direct continuation of [1].Here, the necessary condition for optimality contained in [1] is extended to the general nonlinear problem with arbitrary boundary conditions and minimized functional of integral type.Moreover, a second-order sufficient condition for controllability is derived as a corollary.The work consists of four sections.The problem is formulated and necessary notions introduced in §1.In §2 the basic result of the work, i.e. a second-order necessary condition for optimality (Theorem 1), is formulated, and simple consequences are noted.A sufficient condition for controllability, Proposition 2, is formulated in §3.Finally, the outline of the proof of Theorem 1 is presented in §4.The proof of this theorem, except for the changes related to boundary conditions and the functional being more general than that of [1], consists in essence of reducing the problem with the aid of sliding regimes to the case of linear control that was considered in [1].§ §1-3 can be read independently of [1].On the contrary, §4 depends on [1] in an essential way.I would like to express my deep gratitude to R. V. Gamkrelidze, under whose guidance this work was done.§1.Basic definitions Let there be given a (1 4-m)-dimensional, twice continuously differentiable function of two «-dimensional arguments x Q , x l E. R", \<7<n (*0.x l)J and the η-dimensional controlled equation x=f(x,u), x£R n ,