Denoting (lnx 1)bx+d by '(x); the previous equation becomes
p2 + ('(x))3 = 0: (5.62)
Also, substituting 2p3 = (lnx 1)bx+d = '(x) in (5.46), the equation takes the following form:
!2
1 1u
sin = bx + lnx
2 dp3dp
3 + 2 (5.63)G 3b x
!
1 (lnx 1)bx + 2 d((lnx 1)bx+= bx + lnx + d + d ) 1 (5.64)
G d 3 b x
1 1)bx + ) 1
= bx + lnx + (lnx 1)bx + 2 (lnx (5.65)G 3 b x
1 1 2
usin(x) = lnx + bx + 1 : (5.66)G 3 3 bx
Note that this way, along the singular arc we can express our singular control in the feedback
as a function of x only. The graphs of the singular control usin and corresponding singular
curve are given in Figure 5.1 and Figure 5.2 correspondingly. Note that only a portion of the
control between u = 0 and u = a ( we used a = 15 in our simulations) satis…es the condition
of the problem and this will determine a corresponding admissible portion of the singular
curve on Figure 5.1 called the singular arc. Only the admissible parts of both graphs will
play a role in the numerical simulations presented in the next chapter.
This calculation shows that the formula for the singular arc and singular control are the
same as in [5]. Therefore the analysis of optimal controls done in [5] applies to this model
and it follows that the results are concatenations of 0asa0: Here 0 denotes an interval along
which no treatment is given, a stands for interval with full dose treatment whereas s denotes
an interval with treatment of varying partial doses represented by singular controls.
35
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