Mathematics for Economists

Optimal control

Example

Solve the problem

ZT

0

x^2 (t)+u^2 (t)dt!min, x(t)=u(t)

x( 0 )=x 0.

The Hamiltonian is

H(t,x,u,p) = x^2 u^2 +pu.

Hx^0 =  2 x, Hu^0 = 2 u+p.

That is

dp

dt=^2 x(t), p(T)=^0 , u

(t)=p(t)/ 2 )dx

dt =p(t)/^2.