Mathematics for Economists

Optimal control

By the maximum principleu(t)is maximizes

H(t,x,u,p(t))=u^2 x+(t (^2) )u,u (^2) [ 0 , (^1) ].
The optimization inuis independent ofx. AsHis convex inuthe
maximum is attained at the extremal points of[ 0 , 1 ].Henceu(t)is zero
or one. Ifu(t)=0 then H 0 =x.If u(t)=1 then
H 1 = 1 x+(t 2 )and in this case
H 0 H 1 , 0  1 +t 2 =t 1.
So
x
(t)=u(t)=



0 if t 2 ( 0 , 1 )

1 if t 2 ( 1 , 2 )

Hence the optimal solution is

x(t)=



0 if t 2 [ 0 , 1 ]

t1 if t 2 ( 1 , 2 ].