Mathematics for Economists

Optimal control

Example

Solve the problem

Z 1

0

x(t)dt!max

x

(t)=x(t)+u(t), x(^0 )=^0 , u(t)^2 [^1 ,^1 ].

The Hamiltonian is

H(t,x,u,p)=x+p(x+u).

His concave in(x,u)so every solution is a maximum. As there is no

restriction onx( 1 )the adjoint equation is

p=H^0

x=(^1 +p), p(^1 )=^0