JOURNAL OF COMPUTER AND SYSTEM SCIENCES: 4, 3 8 - 4 9 (1970) A Convex Programming Problem in Banach Spaces and Applications to Optimum Control Theory R. CONTI University of Florence, Florence, Italy Received February 22, 1968 ABSTRACT A convex programming problem for a functional defined on a Banach space is solved, and necessary conditions are derived in the form of a maximum principle. Applications of the results are made to minimum final (or initial) distance and to minimum-effort problems connected with a control process described by a linear evolution equation. 1. INTRODUCTION Given a system consisting of k + 2 real Banach spaces X, Y, Z1 ,..., Zk ; one point yO ~ y; k convex sets Ci C Z~ (i = 1..... k); k + 1 continuous linear mappings, L : Y--~ X and L~:Zi--~X (i= 1,..., k), we consider the problem of finding solutions (y*, zl*,..., z~*) ~ Y • C 1 • "'" X Ck of k Ly + ~. ,Liz, = Ox (I.1) 1 Ox, the null element of X, such that [y. _ yO I = inf l Y -- yo I. (1.2) This is a convex programming problem consisting of minimizing the functional qo(y, z x ,..., z~) = l Y --3,ol on Y • Z x • --- • Zk with constraints represented by (1.1) and z/~ C i (i = l .... , k). We shall prove (See. 2) an existence theorem and give 38