I have been working on a problem, and reached a part where I have a system of ODEs. Let $x(t) \in \mathbb R^n$ be a univariate function, and $\dot x(t)$ be its derivative. I will drop the $(t)$ for ease of notation.
Let $m < n$, and let $A$ be a full rank $m \times n$ matrix. Also, for $i = 1,\ldots,n-m$ let $L_i$ be a subset of $\{1,2,\ldots,n\}$ and $1 \le k_i \le n$, with $k_i \not\in L_i$. Let $x^{L_i} \in \mathbb R^{|L_i|}$ be the components of $x$ corresponding to the indices in $L_i$. The same for $\dot x$. The system of ODEs I am trying to solve has the following form:
\begin{align*} A\dot x &= 0 & \\ 2t \langle x^{L_i}, \dot x^{L_i}\rangle - \dot x^{k_i} &= -\Vert x^{L_i} \Vert && \mbox{$\forall i = 1,\ldots,n-m$} \end{align*}
I don`t know if my notation is confusing. If it is, please let me know. I am almost sure that I won't be able to solve it analytically, but a man can dream :)
Thanks for your time!
