A while ago I came up with an algorithm which can be used to numerically solve optimal control problems, which basically came down to discretizing the control input $u(t)$ and interpolating this to get a continuous function. I tried to optimize the cost function for each discrete value by using the golden search method and because of the interpolation this would mean I only have to integrate this over a small section in time. The fact that each discrete value of $u(t)$ only affects a small number of elements also allows to use parallel computing.For the initial problem, I tried to optimize, I used linear interpolation, since the cost function was only a function of $u(t)$ and $u'(t)$. However recently I tried to expand on this, such that I would also be able to use higher order derivatives of $u(t)$. For linear interpolation you only need the discrete points at the start and end of an element, $$f_{i}(\xi) = u_{i} (1 - \xi) + u_{i+1} \xi,$$where the $u(t)$ from $t_i$ to $t_{i+1}$ is defined as,$$u(t) = f_i\left(\frac{t - t_i}{t_{i+1} - t_i}\right).$$In order to ensure that $u'(t)$ is continuous (such that second derivative exists) I had to set the slope at the end of an element equal to the slope at the beginning of the next element. I decided on defining this slope using the central difference method, which meant using two extra points, one before and one after the element,$$f'_i(0) = \frac{u_{i+1} - u_{i-1}}{2},$$$$f'_i(1) = \frac{u_{i+2} - u_{i}}{2}.$$Using these additional constraints yields the following function for an element,$$f_i(\xi) = u_{i-1} \frac{-\xi^3 + 2\xi^2 - \xi}{2} + u_i \frac{3\xi^3 - 5\xi^2 + 2}{2} + u_{i+1} \frac{-3\xi^3 + 4\xi^2 + \xi}{2} + u_{i+2} \frac{\xi^3 - \xi^2}{2}.$$Instead of the central difference method I also could have used backward or forward difference method, which would require one less point to be used, but I would think that interpolation ought to be symmetrical. These points can also used in the central difference method for the second derivative. Using this as a constraint for the initial and final second derivative yields the following function for an element,$$f_i(\xi) = u_{i-1} \frac{2\xi^5 - 5\xi^4 + 3\xi^3 + \xi^2 - \xi}{2} + u_i \frac{-3\xi^5 + 15\xi^4 - 9\xi^3 - 2 \xi^2 + 2}{2} + u_{i+1} \frac{3\xi^5 - 15\xi^4 + 9\xi^3 + \xi^2 + \xi}{2} + u_{i+2} \frac{-\xi^5 + 5\xi^4 - 3\xi^3}{2}.$$However will this function have any disadvantages compared to the previous function? For example would the higher order polynomial make it harder to approximate sections of the cost function integral?
Comparison between of higher order interpolations
interpolation;polynomials;optimal control