▁address ing ▁dynamic ▁problems , ▁especially ▁as ▁to ▁economic ▁growth ▁equilibrium ▁and ▁stability ▁of ▁economic ▁systems , ▁of ▁which ▁a ▁text book ▁example ▁is ▁optimal ▁consumption ▁and ▁saving . ▁A ▁cru cial ▁distinction ▁is ▁between ▁determin istic ▁and ▁st ochastic ▁control ▁models . ▁ ▁Other ▁applications ▁of ▁optimal ▁control ▁theory ▁include ▁those ▁in ▁fin ance , ▁invent ories , ▁and ▁production ▁for ▁example . ▁ ▁Function al ▁analysis ▁ ▁It ▁was ▁in ▁the ▁course ▁of ▁prov ing ▁of ▁the ▁existence ▁of ▁an ▁optimal ▁equilibrium ▁in ▁his ▁ 1 9 3 7 ▁model ▁of ▁economic ▁growth ▁that ▁John ▁von ▁Ne umann ▁introduced ▁functional ▁analyt ic ▁methods ▁to ▁include ▁topology ▁in ▁economic ▁theory , ▁in ▁particular , ▁ ▁fixed - point ▁theory ▁through ▁his ▁general ization ▁of ▁Br ou wer ' s ▁fixed - point ▁theorem . ▁Following ▁von ▁Ne umann ' s ▁program , ▁Kenneth ▁Ar row ▁and ▁G ér ard ▁Deb reu ▁form ulated ▁abstract ▁models ▁of ▁economic ▁equilib ria ▁using ▁convex ▁sets ▁ ▁and ▁fixed – point ▁theory . ▁ ▁In ▁introdu cing ▁the ▁Ar row – Deb reu ▁model ▁in ▁ 1 9 5 4 , ▁they ▁proved ▁the ▁existence ▁( but ▁not ▁the ▁uniqu eness ) ▁of ▁an ▁equilibrium ▁and ▁also ▁proved ▁that ▁every ▁Wal ras ▁equilibrium ▁is ▁P are to ▁efficient ; ▁in ▁general , ▁equilib ria ▁need ▁not ▁be ▁unique . ▁In ▁their ▁models , ▁the ▁(" pr imal ") ▁vector ▁space ▁represented ▁quantities ▁while ▁the ▁" du al " ▁vector ▁space ▁represented ▁prices . ▁ ▁In ▁Russia