Mathematics for Economists

Optimal control

Example

Let us consider the problem and try to apply Arrowîs condition.

2 π

Zb

a

x

r

1 +



x

 2

dt!min.

The Hamiltonian isH(t,x,u,p)=x

p

1 +u^2 +pu.This is a concave

function inu. So for the maximization

0 =Hu^0 =pxp u

1 +u^2

.

Atp(t)it is

p(t)=xp u

1 +u^2

which is not solvable for everyx,inuso the problem has no solution for

everyxso Arrowís condition is not applicable.