Mathematics for Economists

Optimal control, regularity

Example

Solve the problem

ZT

0

udt!max

x=u^2 , x( 0 )=x(T)= 0.

Asx0 andx( 0 )=x(T)=0 the only feasible solution isu= 0 ,so

there is an optimal solution. Ifp 06 =0 then the Hamiltonian is

H=u+pu^2 .As there is no restriction onuat the maximum

0 =Hu= 1 + 2 pu,

which is not satisÖed byu= 0 .IfH=p 0 
u+p
u^2 then

Hu=p 0 + 2 pu

is satisÖed byu=0 ifp 0 =0 andpis arbitrary.