5.1 Application of the Pontryagin Maximum Principle
Applying the Maximum Principle (Theorem 4) along with transversality condition (4.13) to
the problem OC stated above gives the following.
If u is an optimal control de�ned over the interval [0;T] with corresponding trajectory
(p;q)T; then there exists a constant 0  0 and an absolutely continuous co-vector,  :
[0;T] ! (R3) such that for all t 2 [0;T]; and (0;(t)) 6= (0;0)
 
_1 = @H p

=  (1 + ln@p 1 q)2 b 2pd ; (5.43 3 pq )
 
_2 = @H = @q 2 Gu +  + dp23 p

q1; (5.5)
with the following transversality conditions:
@p
1(T) = = 1; (5.6)@p
@p
2(T) = = 0; (5.7)@q
and the optimal control u minimizes the Hamiltonian H,
  p    
2H = 
0ku + 1 pln +  bp  + dp3 q Guq : (5.8)q 2
Proposition 10 The Hamiltonian H
       
2H = 
0ku + 1
ppln +  bp + d p
3 q Guq ; (5.9)q 2
is minimized along ((t);p;q(t)) over the control set [0;a] with minimum value given by 0.
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