Suppose I have a real-valued function $f(t)$ for $t\in[0,T]$ s.t. $f'''(t)$ is defined as piecewise constant values:
$$ f'''(t) = \begin{cases} k_0, & 0 < t \le t_0 \\ k_1, & t_0 < t \le t_1 \\ \cdots \\ k_n, & t_{n-1} < t \le t_n \end{cases} $$
subject to the following end-point constraints:
$$f(0)=0 \quad f(T)=X_T \quad f'(0) = 0 \quad f'(T) = V_T \quad f''(0) = 0 \quad f''(T) = 0$$
I want to find a new function $g(t)$ subject to the following constraints $\forall t \in [0,T]$:
$$g'(t) \le V_{max} \quad g''(t) \le A_{max} \quad g'''(t) \le J_{max}$$
Where $g(t)$ has the same end-point constraints as $f(t)$, with the assumption that $V_T \le V_{max}$.
I want to choose this function in such a way that $g(t)$ is as 'close' as possible to $f(t)$ over the interval, but I have flexibility in how to define this 'closeness' criterion. I'm looking for suggestions on a choice of criterion that will help make this problem easy to solve while still satisfying the spirit of the requirement. For instance, I could choose to minimize $\int_{0}^T{(g(t)-f(t))^2dt}$, but I'm not sure how to go about doing that or whether it's a good choice of criterion.
Right now I have a solution for my application that addresses this algorithmically, by using a discrete sampling of $f(t)$ and 'smoothing' it repeatedly via convolution with discrete filters until a discrete estimate of the derivatives satisfies all of the constraints. This sort-of works, but I'm like to know if there is a more rigorous way to approach this and find some optimal solution.
For those who are curious, the physical application here is a motion control system where I have an idealized motion path I'd like to follow, but need to impose constraints on the requested jerk, acceleration and velocity in order to prevent excessive torques in the motor.
UPDATE:
I think my definition of $f(t)$ means that it is composed piecewise of cubic polynomials. Is this right?
Suppose I model $g(t)$ in the same way? I can divide up my pieces s.t. there is a single $\Delta t = t_i - t_{i-1} \quad \forall i \in [1,n]$. I can choose this value at a resolution that will be useful for my purposes. I can then define $g'''(t)$ in the same way as $f'''(t)$, with different constants; let's call those $q_i$.
Now that the two functions are broken into pieces with the same intervals, I should be able to simplify this a bit. For ease of notation, let's define each piece like this:
$$ f_i'''(t) = k_i \quad g_i'''(t) = q_i $$
Now if I some have criterion function C I want to maximize, I think I can compute my overall value using:
$$ \sum_{i=1}^n \int_0^{\Delta t} C(f_i(t),g_i(t)) dt$$
I'm not sure where to proceed from here, though. It seems like any analytical expansion for $f_i(t)$ and $g_i(t)$ is going to be pretty complicated, even before I choose the criterion.
On a side note, my apologies for any imprecision or use of incorrect terminology; my last math class was quite a while ago.