We then have:  
@g(x(t 1))
k(t1) =  ; k = 1;:::;n: (4.21)@x
k
This yields to the transversality condition for the terminal payo¤ problem
 @g(x(t
 1))k(t1) = 0 ; k = 1;:::;n: (4.22)@x
k
Remark 6 Most problems we deal with are normal i.e 0 = 1 and condition 4.22 has the
following form  
@g(x(t 1))
k(t1) = ; k = 1;:::;n: (4.23)@x
k
4.4 Type of controls
The optimal control problem which we will be analyzing in this paper has the characteristic
that both the objective and the dynamics are linear with respect to the control u. If m
represents the minimum amount of drug and M the maximum amount,
= [m;M]; from
the minimum condition of the Maximum Principle this gives rise to the switching function
(t) such that @H@u = (t): The optimal control is given by:
8>
> m for (t) > 0>>
<
u(t) = M for (t) < 0 (4.24)>
>>>
: undefined for (t) = 0:
If (t) = 0 for only a …nite number of points, then we are dealing with a bang-bang type
of control. The points where (t) = 0 are called switching times. The bang-bang control
takes the following form:
8
>< m for (t) > 0
u(t) = (4.25)>
: M for (t) < 0:
21
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