guidance, flight mechanics and trajectory optimization

The Hsmiltonian for the problem is:
H = P’i =/+Art&)
But, the Pontryagin Msximum Principle states that the function e(t) which
minimizes the coast, #(t> , is that function which causes the Hamiltonian
be a maximum at each instant of time. Therefore, this value of 8 can be
to
found by setting the derivative
d/4
dH/dB equal to zero:
or
dB
=-$aodjnB tGao cbo~ =o
The Pi are the co-state variables and obey the differential equations:
or
q-0,
Since, the boundary condition on Pl is:
or
And since P, =
pczp=o
0, the solution for Pl (t) is:
/9 (t) = 0
Solving the differential equations for P3 and P4 with P1 = 0 gives
and the steering law becomes:
or letting
tLwe = c, f c- t
77
(4.38)