Mathematics for Economists

Optimal control and calculus of variation

LetF



t,x,x



be a kernel for a calculus of variation problem. In this case

H=F(t,x,u)+pu.

By the optimality conditionHu= 0 =Fu^0 +pthat is

Fu^0 =Fx^0 =p, p=Hx^0 =Fx^0

That is

d

dtF

^0

x=

d

dtp=H

x^0 =Fx^0.