Better Structure Restricted Derivatives

[pulsipher]

Currently, we provide 2 different ways to define derivatives over restricted domains.

model = InfiniteModel()
@infinite_parameter(model, t in [0, 1])
@infinite_parameter(model, x in [-1, 1])
@variable(model, y, Infinite(t, x))

dy1 = deriv(y, x)(0, x) # option 1
dy2 = deriv(y(0, x), x) # option 2

Here dy1 and dy2 are equivalent mathematically. However, programmatically dy1 is a restricted derivative and dy2 is a derivative of a restricted variable. We should choose one or the other to be created behind the scenes such that transformation techniques don't have to worry about both.

Note that this equivalence is not true if the restriction and derivative use the same infinite parameter:

dy3 = deriv(y, t)(0, x) # time derivative evaluated at t=0
dy4 = deriv(y(0, x), t) # time derivative of time restricted y(0, x) --> 0

So in this particular context, the transformation technique will only have to worry about dy3 since dy4 = 0.0.

Hence, it probably makes the most sense to only use restrictions of derivatives behind the scenes, since that covers both cases. So we would only have to worry about restricted derivatives (not derivatives of restrictions).