I need help solving this. In the picture below a paper explains why that the jerk (third derivative of position) is already cost optimal. So this means that a sixth order polynomial is already jerk cost optimal.
In measurements of my minimization algorithm I want to obtain a minimal jerk root mean square (rms) value. But the optimized function I get is the same as a sixth order polynomial. So my hypothesis is that jerk also is rms minimal. I want to proof this mathematically.
In the picture below is the proof of cost, the proof uses $x(t)$ and $y(t)$:
Second part of cost jerk proof
Now I want to proof the same for rms but only for a single variable $\theta(t)$. This is what I already have but I can't proof it further.
$jerk: \dddot{\theta}(t)=\frac{d^3\theta(t)}{dt^3}$
H-function represents the rms of the function:
$H(\dddot{\theta}(t))=\sqrt{\frac{1}{T_2-T_1}\int_{T_1}^{T_2} [\dddot{\theta}(t)]^2 \,dx}$
I'm a engineering major and not math. So if possible please explain a bit.
Thank you for helping.