5.5 Optimal singular arc
The minimum condition of Pontryagin Maximum Principle states that along optimal solu-
tion:
H = 0 (5.48)
H = h;fi+ uh;gi+ ku = 0: (5.49)
Since
h;gi+ k = 0 (5.50)
that would imply that
h;fi+ uh;gi+ ku = h;fi = 0: (5.51)
We also know that
h;[f;g]z(t)i = 0: (5.52)
Equations (5.51) and (5.52) form a system of two non linear homogeneous equations with
unknown 1 and 2: For this system to have a nontrivial solution, we must have that
   
 pln p 
jf;[f;g]j = q
Gp 
= 0: (5.53)  
2 bp  + dp
3 q Gbp  
Evaluating this further, we get
   
 pln p  q Gp 
= Gpq Gbp2 + Gbp2 p 5 ln + Gdp3q; (5.54)   
2 bp  + d p  q
3 q Gbp 
p 5Gpq Gbp2 + Gbp2 ln + Gdp
3q = 0 (5.55)q
Gbp2 p ln = Gpq + Gbp2 5Gdp3q (5.56)q
p Gpq + Gbp2 5Gdp3qln = (5.57)
q Gbp2
33
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