Appl Math Optim 20:19-31 (1989) Applied Mathematics and Optimization © 1989Springer-VerlagNew York Inc. Sufficiency Conditions with Minimal Regularity Assumptions* Vera Zeidan Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Communicated by W. Fleming Abstract. The aim of this article is to provide second-order sufficiency criteria that extend known ones in [6] and [7] to the case where the control candidate and some of the data are merely essentially bounded, and/or the control set U is any convex subset of R m. In the classical setting, where a restriction on the velocity is imposed (~(t)~ U), it is shown that when U is compact the known strengthened Weierstrass condition is equivalent to the Weierstrass condition with strict inequality. I. Introduction 1.1. The Problem Given an interval [a, b], a point A in R ~, a subset U of R m, functions f and g mapping [a, b] x R" x R "~ to R n and R, respectively, and a function ~¢ from R n to R, the optimal control problem is defined to be: + b (C) Minimize J(X, U) "= ~(x(b)) ~a g(t, x(t), u(t)) dt over all pairs (x, u) of absolutely continuous functions x: [a, b] ~ R n and measurable func- tions u: [a, bier m such that Yc(t) =f(t,x(t), u(t)) a.e., x(a)=A,u(t)~U a.e.; such a pair is called feasible. * The research was supported by an NSERC Grant and by GNAFA of CNR, which are gratefully acknowledged.