The vector p can be determined so that the coefficient of 6x vanishes: 
sui + al U pple 
= Be Be. © Bee oo 
Since du, is free, the usual argument that Su, is zero everywhere except 
on a small interval yields 
Po op stl oe = © (3.12) 
2 2 
Now du can be chosen so that du = O for t < t, and for t,,, <t. 
In this case, the only contribution to the cost difference (3.3) is that 
due to I hence 
(ee 
= 1 
ORS AG oy do 
Fig 
By (Boo Op, > 0; so 
> 0 (3.13) 
Let the Hamiltonian be defined by 
== 1 ap jp! 2 > AY © (3.14) 
where X is defined by 
KN, © © she @, = © 
4 
Mp, =O WE w, < YW 
J °5 
31 
