Mathematics for Economists

Optimal control, Bolza problem

ZT

0

f(t,x(t),u(t))dt+φ(x(T))!max

x(t)=g(t,x(t),u(t)),x( 0 )=x

0

By the fundamental theorem of the calculus

φ(x(T)) = φ(x( 0 ))+

ZT

0

φ^0 (x(t))x(t)dt=

= φ(x 0 )+

ZT

0

φ^0 (x(t))g(t,x(t),u(t))dt.

Asx 0 is Öxed one can write the goal function as

ZT

0

f(t,x(t),u(t))+φ^0 (x(t))g(t,x(t),u(t))dt!max.