## Cubic Spline Approximations

Hello,

I am working with an approximate optimal control problem, which can be expressed as a linear optimization problem with nonlinear constraints. Since time is considered continuous and my variables are time dependent, I use constraint generation to select a small but useful set of time points (between 0 and 1) where I enforce the constraints. My variables are also monotonically decreasing with time, so I express the derivatives of the variables as nonpositive cubic splines. Obviously, the more knot points I have and the more time points I consider in the constraint set, the more accurate my solution will be, but on the other hand the problem will take much longer to compute. What I am uncertain of is the following:

1. What is error bound for expressing the function as a cubic spline?
2. What is the best way to generate a set of mesh (knot) points where I enforce the spline?
3. What is the error bound on the selection of time points, where I use a line search over a discrete interval?

With regards to (1), my function is continuous, but I have no information on the second or third derivatives and it seems like the most cited error bounds require that information in addition to the modulus of continuity. So I am having trouble establishing the worst case for how bad the spline function can be at representing my function.

With regards to (2), I am fairly certain there must established mesh selection methods, but I am having a hard finding a well established/referenced system for control problems. Also the mesh selection will influence the approximation error, but as mentioned earlier I can not find a good reference for that.

With regards to (3), since the constraint generating function is a polynomial, I can get a very poor bound on the error between time points using Lipschitz. But this does not really take into account the fact that the function is a spline, which is fairly smooth and doesn't take into account the aspect that my problem is a control problem.

I hate to say I am not overly familiar with this area of mathematics, but any help on these questions or references that could point me in the right direction would be greatly appreciated. I am also new to this forum, so I apologize if I gave too much detail or insufficient detail for my problem.

Sam