Ok, let's start by considering a problem like the one that we did in
class. We'll suppose that we're mixing industrial-sized quantities of
cookie dough and are interested in controlling the quantity of
chocolate chips in the mixing vat (of volume V
) at all times.
The ideal quantity will be a function of time that we shall call
). We expect the chips to be pumped into the vat at
a rate r
), and dough is removed from the vat at a rate
: then a
) satisfies the differential
a'(t) + L a(t) / V = r(t).
However, it may be that the machine that pumps chips into the vat
isn't quite accurate, and instead of pumping them in at a rate
) pumps chips in at the rate q
instead. Then the actual quantity of chips in the vat won't satisfy
the differential equation above---instead, it will be a quantity
) that satisfies the differential equation
b'(t) + L b(t) / V = q(t).
Knowing that a
) is not equal to b
might ask how far off it is---in particular, if we know something
about the magnitude of the difference q
), we might ask if the difference between b
is small, or whether it might get arbitrarily large.
In these equations, the output is a(t) (or
b(t))---they are the response of the system to
the forcing, or input given by r(t)
(or q(t)). The difference in this output is
z(t) = b(t) -
a(t). If we subtract the two equations above, we
(b(t) - a(t))' + L
(b(t) - a(t)) / V =
(q(t) - r(t)), or
z'(t) + L z(t) / V =
(q(t) - r(t)).
Thus we have a differential equation for z
) that we
can actually solve. How big can this deviation get? We saw in class
that (at least for a nice function) if the difference
) - r
)) is bounded by some
number then we can be assured that the deviation z
must similarly remain bounded---and that it can't change
discontinuously as the difference in forcing is changed. This is what
the book calls continuity in the data
, and the book tells us
that this is a property of first-order differential equations
satisfying the conditions of the existence and uniqueness theorem.