Optimal control, minimum problems
One can handle a minimum problem by changing the sign of the goal
function. The Hamiltonian is
H= f+pg, dp
dt
= Hx^0 = ( fx+pgx).
Multiplying by 1 and introducingep= p
dp
dt
=( fx+pgx), dep
dt
= (fx+epgx)= Hex^0.
That isepis an solution of the adjoint equation of the maximum problem.
Zt 1
t 0
fdt!max, x=g.
u(t)is maximizingH= f+pg= (f+epg)and therefore there
is solutionepof the adjoint equation for whichu(t)is minimizing the
HamiltonianH=f+epg. In this case one can talk about a minimum
principle.
