Mathematics for Economists

Optimal control

Theorem (Maximum principle)

If an(x(t),u(t))is an optimal solution of the problem then there is

some adjoint function p(t) such that u(t)is maximizes the Hamiltonian

at x(t)

u7!H(t,x(t),u,p(t)), u 2 U.

The adjoint function solves the di§erential equation

d

dt

p(t)=Hx^0 (t,x(t),u(t),p(t))

with some further conditions on the values of p at the time t 1 depending

on the boundary conditions on x(t 1 ).This further conditions on p are

called the transversality conditions:

(^1) x(t 1 )=x 1 Öxed then there is no restriction on p(t 1 ),
(^2) x(t 1 )is free, in this case p(t 1 )= 0.