JOURNAL

OF OPTIMIZATION

THEORY

AND APPLICATIONS:

Vol. 9, No. 3, 1972

T E C H N I C A L NOTE A Note on a Selector Theorem in Banach Spaces1 JON K. COLE2 Communicated by L. Cesari

A b s t r a c t . The existence of a strongly measurable function taking its values in a variable subset of a separable and reflexive Banach space was shown in Ref. t. Here, we show the collection of all such functions to be weakly compact in itself.

T h e o r e m 1. L e t Go: y---> cf(X) be a b o u n d e d mapping of a compact interval into the collection of nonempty, closed, bounded, and convex subsets of X, a separable and reflexive Banach space, and suppose that, if t~ --> t* a J, then (~ n=l

cl co 0 Go(h) C Go(t*).

(1)

i=n

T h e n , there is a strongly measurable function

u : J - - ~ x such that

u(t) e Go(t) for ever), t a jr T h e property of G Oin (1) is similar to Property (Q) of Cesari (Ref. 2). L e t U(Go) == {u [ u : J ---> X, u strongly measurable, u(t) ~ G0(t)}. Theorem

2.

U(Go) is weakly compact in itself.

P r o o f . Since U(Go) C B~(J, X), the reflexive (due to the reflexivity of X ) Banach space of Bochner square-integrable functions from J to X (Ref. 3, page 89), and since U(Go) is bounded, any sequence {us} from U(Go) has a subsequence weakly convergent to some u* E Bg,(J, X). So there is a sequence {sn} of convex sums of elements of {Un} so that i Paper received May 6, 1971. Assistant Professor, Department of Mathematics, St. John's University, Jamaica, New York. 214 © 1972PlenumPublishingCorporation,227 West 17thStreet,New York,N.Y. 10011.

JOTA: VOL. 9, NO. 3, 1972

(L) .Is [Is,(t) -- u*(t)][ ~ = l] s~ -- u* t]2 -+ 0 (Ref. 4, page 422). Hence, there is a subsequence {s~'} of {sn} that converges pointwise to u* a.e. on J. (Ref. 5, page 93). Each s~' is strongly measurable, and so is the limit u* (Ref. 3, page 74). Clearly, U(Go) is convex, and so each s,~'~ U(Go). Since each Go(t ) is closed, if s~((t) -4 u*(t), then u*(t) ~ Go(t ), as s,/(t) ~ Go(t ) for all n and atl t c J. So u*(t) c Go(t ) a.e. on J. Hence, u* ~ U(Go), after possible redefinition on a null subset of f. Remark. If a continuous linear process with controls in U(Go) has a minimizing sequence {u~}, then, under a wide variety of conditions on the cost functional, a weak limit point of {us} will be an optimaI control for the process.

References 1. CoI& J. K., A Selector Theorem in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 7, No. 3, 1971. 2. C•s,*/Rt, L., Existence Theorems for Multidimensional Lagrange Problems, Journal of Optimization Theory and Applications, Vol. 1, No. 2, 1967. 3. HILL~:, E., and PHILLIPS, R. S., Functional Analysis and Semi-Groups, The Waverly Press, Baltimore, Maryland, 1957. 4. DONFORD, N., and SCHWARTZ, J., Linear Operators, John Wiley and Sons (Interscience Publishers), New York, 1958. 5. ROYD~N, H., Real Analysis, The Macmillan Company, New York, 1968.