Jtdylktuy

I am studying a 3-agent distance-based formation control problem. The position of each agent is denoted $pmb{p}_1$, $pmb{p}_2$, $pmb{p}_3 in mathbb{R}^2$. The desired distance between $i$ and $j$ is denoted with $d_{ij}^*$. (Of course, the triangular inequality shall be satisfied for $d_{12}^*$, $d_{23}^*$ and $d_{31}^*$). The closed-loop system equation can be expressed as
$$dot{tilde{pmb{p}}} =
begin{bmatrix}
dot{tilde{pmb{p}}}_{12} \
dot{tilde{pmb{p}}}_{23} \
dot{tilde{pmb{p}}}_{31}
end{bmatrix}
=
-2Kpmb{A}
begin{bmatrix}
rho_{12} tilde{pmb{p}}_{12} \
rho_{23} tilde{pmb{p}}_{23} \
rho_{31} tilde{pmb{p}}_{31}
end{bmatrix}
$$
where $K$ is a scaler constant, the relative displacements $tilde{pmb{p}}_{ij} = pmb{p}_i - pmb{p}_j in mathbb{R}^2$,
the state vector
$$tilde{pmb{p}} = [tilde{pmb{p}}_{12}^mathrm{T}, tilde{pmb{p}}_{23}^mathrm{T}, tilde{pmb{p}}_{31}^mathrm{T}]^mathrm{T} in D={ tilde{pmb{p}} in mathrm{R}^6 ;|; tilde{pmb{p}}_{12} + tilde{pmb{p}}_{23} + tilde{pmb{p}}_{31} = 0 },$$
matrix
$$
pmb{A} =
begin{bmatrix}
2 & 0 & -1 & 0 & -1 & 0\
0 & 2 & 0 & -1 & 0 & -1\
-1 & 0 & 2 & 0 & -1 & 0\
0 & -1 & 0 & 2 & 0 & -1\
-1 & 0 & -1 & 0 & 2 & 0\
0 & -1 & 0 & -1 & 0 & 2
end{bmatrix}
$$
and
$$
rho_{ij}
= frac{2 left(||tilde{pmb{p}}_{ij}||^4 - d_{ij}^{*4}right)}{||tilde{pmb{p}}_{ij}||^4}.
$$

This system has a set of desired equilibria
$$E_d = { tilde{pmb{p}} in D ;|; ||tilde{pmb{p}}_{12}|| = d_{12}^*, ||tilde{pmb{p}}_{23}|| = d_{23}^* , ||tilde{pmb{p}}_{31}|| = d_{31}^* },$$

and a set of undesired equilibria
$$E_u = { tilde{pmb{p}} in D backslash E_d ;|; rho_{12}tilde{pmb{p}}_{12} = rho_{23}tilde{pmb{p}}_{23} = rho_{31}tilde{pmb{p}}_{31} }.$$

When I was trying to prove the closed-loop system stability, I found it necessary to prove the following proposition first:

If $tilde{pmb{p}}_{12}$, $tilde{pmb{p}}_{23}$ and $tilde{pmb{p}}_{31}$ are not collinear at $t=0$, then $tilde{pmb{p}}_{12}$, $tilde{pmb{p}}_{23}$ and $tilde{pmb{p}}_{31}$ are not collinear for any $t > 0$.

Or equivalently,

Set $Omega= { tilde{pmb{p}} in D ;|; tilde{pmb{p}}_{12}^mathrm{T} tilde{pmb{p}}_{23} < ||tilde{pmb{p}}_{12} || cdot || tilde{pmb{p}}_{23} || }$ is a positively invariant set.

I have been stuck here for a few weeks now. Unless an analytic solution to the differential equation can be found, I cannot think of a method to prove the proposition. I did some numerical simulations and no counterexample has been found so far. Any ideas on how this can be proved would be appreciated.