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Gavin's DiffEq Class Clarification: Feb 18

Question: What is this ``input,'' ``output,'' and ``z'' in our sensitivity analysis?
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 a(t). We expect the chips to be pumped into the vat at a rate r(t), and dough is removed from the vat at a rate L: then a(t) satisfies the differential equation
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 r(t) pumps chips in at the rate q(t) instead. Then the actual quantity of chips in the vat won't satisfy the differential equation above---instead, it will be a quantity b(t) that satisfies the differential equation
b'(t) + L b(t) / V = q(t).
Knowing that a(t) is not equal to b(t), we might ask how far off it is---in particular, if we know something about the magnitude of the difference q(t) - r(t), we might ask if the difference between b and a 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 get

(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(t) 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 (q(t) - r(t)) is bounded by some number then we can be assured that the deviation z(t) 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.

Gavin's DiffEq Clarification 000218
Last Modified: Sun Feb 20 12:51:11 CST 2000
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