Optimal control, regularity
Assume thatx(T)is free. In this casep(T)=0 and ifp 0 =0 then
d
dtp(t) = H
x^0 (t,x(t),u(t),p(t))=
= 0 p(t)gx(t,x(t),u(t)).This is a linear di§erential equation with "initial condition"p(T)= 0 .As
p(t)0 is a solution and as it is a linear equation so it has a unique
solution(p 0 ,p(t))0 which is impossible.
We will ignore the regularity problem and we will always assume that
p 0 =1.