I'm trying to design a trajectory for a course. I have 2 equations, $y_1$ and $y_2$ on the interval $0$ to T. I can choose any equations for these providing that I satisfy the following conditions: $y_1(0) = y_2(0) = 0$, $y_1(T) = y_2(T) = 1$, $\dot{y_1}(0) = \dot{y_2}(0) = \dot{y_1}(T) = \dot{y_2}(T) = 0$, and lastly: $\lim_{t\to0}\frac{\dot{y_2}(t)}{\dot{y_1}(t)} = \lim_{t\to T}\frac{\dot{y_2}(t)}{\dot{y_1}(t)} = 0$.
At first what I did was choose an arbitrary polynomial for $y_1$ and reason that since $lim_{x\to0}\frac{x^2}{x}$ is $0$, that my $y_2$ should have $\dot{y_2} = \dot{y_1}^2$. I could not find such a $y_1$ that would allow $y_2$ to satisfy the conditions on the bounds. So then I tried sinusoidal functions also to no avail.
Was I going down the right path with the polynomial efforts? Is there a sinusoidal function I'm missing? Should I be trying exponential functions? Is there any path or technique that could allow me to solve this? Thank you.
If you apply L'Hôpital's rule then it follows that the only additional constraints are that $\ddot{y}_2(0) = \ddot{y}_2(T) = 0$ and $\ddot{y}_1(0) = \ddot{y}_1(T) \neq 0$.
Using this you should be able to find a solution of the form:
$$ y_1(t) = a_1\, t^2 + b_1\, t^3, $$
$$ y_2(t) = a_2\, t^3 + b_2\, t^4 + c_2\, t^5. $$