Mathematics for Economists

Optimal control

As there was no restriction onuthis problem is in fact the same as the

problem Z

T

0

x^2 (t)+



x

 2

(t)dt!min, x(^0 )=x 0.

The EulerñLagrange equation is

2 x=Fx^0 = d

dt

Fx^0 = d

dt

2 x= 2 x.

Asx(T)is free we need the transversality condition

0 =Fx^0



t,x(T),x(T)



= 2 x(T).

The characteristic polynomial of the second order linear equation is

λ^2  1 = 0 ,so the general solution of the EulerñLagrange equation is

x(t)=c 1 et+c 2 et.