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By Vangipuram Lakshmikantham, Mathematiker Indien USA

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It must be emphasized that these exceptional values of the wave number do not correspond to exceptional values for the exterior scattering p roblem but are the result of the particular reformulation of the exterior boundary value problem in terms of integral equations . Various means have been proposed to circumvent this dif­ ficulty (see e . g . [ 6 ] and [ 7 ] ) . Here , we follow [ 8 ] by introducing a second equation involving the normal derivative of the double layer potential which restores uniqueness for real wave numbers .

J . Math . Bio l . 6, 1 6 3-16 8 . Hsu, S . B . , and Chen g , K . S . (1980) . "Exploitat ive comp e t i t ion o f two microorganisms for two complementary nutrients in cont inuous culture s , " preprin t . Hutchinson , G . E . (1978) . " An Introduc t ion to Populat ion Ecology . " Yale Uni v . Pres s , New Haven , Conn . Lakshmikantham, V . , and Leela , S . (1969 ) . "Differen t ial and Integral Inequalitie s , Theory and Applicat ions , Vol . I . " Academic P ress , New York and London .

A. , Alvager , T . , and de Korvin , A. ( 1 9 79 ) . "A non-linear inte­ gral and a bang-bang theorem. " In "Applied Nonlinear Analysis" (V . Lakshmikantham, ed . ) , pp . 25-46 . Academic Pres s , New York . SOME OPTIMAL CONTROL PROBLEMS FOR THE HELMHOLTZ EQUATION'' Thomas S . U on the. vent,{,eth. ;t_h. day. I. INTRODUCTION In this paper, we review results obtained by the author in close collab­ oration with R. E . Kleinman [ 1 , 2 , 3 , 4 ] . The c lass of problems referred to in the title arise in the study of time harmonic electromagnetic and acoustic radiation and scattering .

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