Proof. Proof follows from [5]
5.3 The switching function
The Hamiltonian H for our problem is given as
p   2H = u(k Gq
2)p1 ln +  bpq  + dp3 : (5.11)q 2
Therefore, minimizing the Hamiltonian is equivalent to minimizing the following expression
which is linear for v 2 [0;a]
v(k Gq2): (5.12)
This is exactly the situation described earlier in Section 4.4 and the switching function (t)
for our problem will have the form
(t) = k Gq(t)2(t) (5.13)
and consequently, 8
>> 0 if (t) > 0>
><
u(t) = a if (t) < 0 : (5.14)>
>>>:
undefined if (t) = 0:
Even though this optimal control looks like a "bang-bang" type of control, it is too soon to
draw this conclusion. If the times when (t) = 0 are not just isolated points when the control
switches its values but (t) = 0 on an interval, the minimum condition does not give us an
optimal solution. In this case, we will deal with the so-called singular controls introduced in
Section 4.4. However, it is obvious that when (t) = 0; all of its derivatives must also vanish
on that interval. Thus, in order to properly understand the behavior of the singular control
we must then analyze the derivatives of (t): The concept of Lie brackets provides us then
with a fairly easy tool for the computation of the derivatives of the switching function. For
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