For a certain control mechanism for a model I'm trying to design, I'm trying to find a control parameter $c$ that can be used to ensure a physics body arrives at a height $y=.4$ at a time $t=7$. Based on the forces that act upon the body, I derived the set of differential conditions for the speed and position to be
$$ \frac{\partial }{\partial t} y\left(t\right)=v\left(t\right) $$
$$ \frac{\partial }{\partial t} v\left(t\right)=-C_\textrm{drag}\,{v\left(t\right)}^2-\frac{c}{5}+20\,\mathrm{b}-10 $$
Where $C_\textrm{drag}$ and $b$ are known constants.
Additionally, the constraints based on the known initial conditions and the boundary conditions can be written as:
$$ y(0) = 0,\ y(7)=.4 $$ $$ v(0) = 0 $$
However, a fourth condition can be found from a measurement that is carried out once which only results in the position of the body at a certain time, resulting in a fourth condition:
$$ y(t_\textrm{m}) = y_\textrm{m}$$
with
$$ 0<t_\textrm{p}<7,\ 0<y_\textrm{p}<.4 $$
The goal now is to determine the parameter $c$ that has been used. I have tried a number of approaches including both numerical solving using MATLAB as well as hand-calculations, but I'm getting stuck on the fourth condition.
My question then is whether or not this problem is well specified enough. Do the initial conditions constrain the problem enough to be solvable (or how do I determine this)? Are there any further steps I could take to determine a solution?
Thanks in advance for any assistance provided.
Just to simplify the notation a bit I will solve the following problem. Given a dynamical system,
\begin{align} \dot{y} &= v \\ \dot{v} &= -av^{2} + k \end{align} where $a\in(0,\infty)$ and $k\in\mathbb{R}$. To go back to you problem just substitue $a = C_{drag}$ and $k = -\frac{c}{5}+20b - 10$. The boundary conditions are, \begin{align} v(0) &= y(0) = 0 \\ y(7) &= \frac{4}{10} \\ y(t_m) &= y_{m} \end{align}
The solution to $\dot{v} = -av^{2} + k$ is: \begin{align} v(t) = \sqrt{\frac{k}{a}} \frac{e^{2\sqrt{ak}(t-q)}-1}{e^{2\sqrt{ak}(t-q)}+1}, \end{align} where $q$ is a free parameter.
Using the initial condition $v(0) = 0$ we get that $q=0$.
If we integrate $\dot{y}$ from $0$ to $7$ we know that it should equal to the boundary condition $y(7) = \frac{4}{10}$,
\begin{align} y(7) =\frac{4}{10} = \int_0^7 \dot{y} dt = \frac{1}{a} (-7 \sqrt{ak} - \log(2) + \log(1 + e^{14 \sqrt{ak}})). \end{align}
The term $k=-\frac{c}{5} + 20b-10$, so by solving for $k$ you can determine what value of $c$ you need to get the boundary condition. Notice that at this point you have used up all your free variables (q,c). So unless it just so happens that $y(t_m) = y_{m}$ we can't adjust $c$ to get this condition on top of the boundary conditions.