Mathematics for Economists

Optimal control

So

d^2 p

dt^2 =^2

dx

dt=^2 p(t)/^2 =p(t).

It is a second order linear equation with characteristic polynomial

λ^2  1 = 0 .Using the general solution

p(t)=c 1 exp(t)+c 2 exp(t), p(T)= 0

On the other hand

dp

dt(^0 )=^2 x(^0 )=^2 x^0

c 1 = 2 x 0 eT/



eT+eT



, c 2 = 2 x 0 eT/



eT+eT



x(t) =^1

2

dp

dt

=x 0 e

Tt+etT

eT+eT