Vol. 20, No. 1, 1967

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Existence of optimal controls

Arthur William John Stoddart

Vol. 20 (1967), No. 1, 167–177

Let f = (f1,f2,,fn) be a mapping to En from a set D in E1 × En × Em; and f0 a real function on D. Consider a “control” function u from an interval I = [t0,t1] in E1 to En; and a “response” function x from I to En such that (t,x(t),u(t)) D for almost every t I,f0(t,x(t),u(t)) has an integral (finite or +) on I,f(t,x(t),u(t)) is integrable on I, and

x(t) = x(t0)+ t0f(s,x (s),u(s))ds

for all t I. In a class Γ of such control-response pairs (u,x), a pair (u,x) is called optimal (with respect to f0) if the “cost” functional

C (u,x) = (I)  f0(t,x,u)dt

has a minimum at (u,αj). Here we consider conditions sufficient for existence of such optimal pairs.

Mathematical Subject Classification
Primary: 49.00
Received: 8 December 1965
Published: 1 January 1967
Arthur William John Stoddart