# Use polynomials to detach number of MPC controls from size of time-horizon?

Model Predictive Control (MPC) works by finding a set of controls $$u_0, \dots, u_n$$ that minimize a cost function $$J$$ over a time-horizon $$[t, t + T]$$. Typically when MPC is used on mobile robots, each control $$u_k$$ is a vector of velocity commands that are applied for the duration $$\delta$$ of a slice $$[t + \delta k, t + \delta (k + 1))$$ of the time-horizon. The consequence is that for a given time-horizon size $$T$$, we need to optimize over $$\lceil \frac{T}{\delta} \rceil$$ controls.

I'm looking into using MPC for path-following, and I had the idea of using $$n$$-degree polynomials to generate the components of $$u_k$$, then redefine $$J$$ in terms of the polynomial coefficients. For example, in the case of a differential platform, use one polynomial $$v(k) = v_t + a_1 k + a_2 k^2 \dots a_n k^n$$ to generate linear velocities, $$w(k) = w_t + b_1 k + b_2 k^2 \dots b_n k^n$$ to generate angular velocities, and have the MPC optimize over $$a_1, \dots, a_n, b_1, \dots, b_n$$. That way we have $$2n$$ independent variables regardless the values of $$\delta$$ and $$T$$; additionally, $$v(k)$$ and $$w(k)$$ could easily be differentiated to provide optimization constraints over acceleration, jerk etc.

This seemed like a reasonable idea to me, so I was certain I would find it already covered in the literature. However almost all articles I've seen so far stick to the approach of defining one independent control set per time slice; the only exception is this one, which instead optimizes over the space of targets for a pure-pursuit-like control law.

Am I missing something? Is there an obvious reason this approach wouldn't work (or at least not be worth the trouble) that I'm overlooking? Is it documented somewhere and I just wasn't able to find any references?