Mathematics for Economists

Optimal control

Example

Solve the problem

2 π

ZT

0

x

p

1 +u^2 dt!min, x=u,x( 0 )=x 0 ,x(T)=xT.

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

p

1 +u^2 +pu.

dp

dt=

p

1 +u^2

Obviouslyp 6 = 0.

0 = Hu^0 =pxu

1 +u^2

+p,

p =

pxu

1 +u^2

=

xu

p =

xx

p

pp = x x.