Let me see if I can describe this problem adequately without pictures.
Let's say I have two variables which both linearly affect velocity, a and b. As a increases, velocity increases linearly, and vice versa. Same goes for b- as it increases, velocity increases linearly, and vice versa. Now let's say that velocity over time has already been determined, and is a nice smooth spline, call it v(t). We also know that a given time, v = f(a,b), where f is some basic linear function determining v from a and b. Finally, we have some cost function c(a,b,t) which defines cost at a particular time and a and b value.
What I'm trying to do is plot a cost-minimizing spline where each point determines a and b at a particular time, with the hard condition that f(a,b,t) = v(t) for all times, and the soft condition that we try to minimize c(a,b,t). If you flatten this to two dimensions, with a on one axis, and b on the other, for a particular time, you'll see that in order to satisfy the hard constraint, that there is some line in the a-b plane we have to be on, but where on that line we should place a point depends on the cost function.
If the cost function was simple, this would be a relatively easy problem to solve- at each t, just determine a and b such as to minimize cost, and be done. However, it is possible for cost to change suddenly at retain boundaries (e.g. for t >= 5, costs of a < 0.6 increase dramatically), and I'd want my spline to anticipate that and start increasing a before we get to t = 5, so as to smooth things out.
Where it's breaking down for me is that all spline formulas that I can find require fixed points in n-space. They may not travel through those points, but they do require them. My case does not require the spline travel through a specific point in [a,b,t] but does require they pass through a line for a particular t value (and the rest is a minimization). Is there some way to simplify this problem to a basic spline one by looking at derivatives, etc.?
This paper describes how to solve a similar problem, but it seems to require the spline to be a best fit through points, not lines. http://www.cs.berkeley.edu/~ravir/dspline.pdf
Thanks for any help you can provide.
