I was dealing with 3-DOF attitude dynamics of rigid body in a geometrical framework and wanted to comment upon the following defined function $F$ at its maxima.

Consider $F \in \mathbb{R} : F(e_{\Omega}(t)) = \frac{1}{2}e_{\Omega}(t)^TJe_{\Omega}(t)$, where $e_{\Omega}(t) \in \mathbb{R}^3$ and $J$ is a positive definite matrix. Also, we have the following dynamics:

$$J\dot{e}_{\Omega} = -k_Re_R - k_{\Omega}e_{\Omega} $$

where $e_R(t) \in \mathbb{R}^3$, and constants $ k_R, k_{\Omega} \in \mathbb{R} > 0$. Hence, since $e_R$ and $e_{\Omega}$ are continuous (given/known), ${e}_{\Omega}$ is also differentiable, $i.e.\ {e}_{\Omega} \in {C}^1$.

We calculate the derivative of $F$ as follows: \begin{equation} \begin{aligned} \dot{F} &= e_{\Omega}^T J\dot{e}_{\Omega}\\ &= e_{\Omega}^T (-k_Re_R - k_{\Omega}e_{\Omega})\\ &= -k_R e_{\Omega}^Te_R - k_{\Omega}e_{\Omega}^T e_{\Omega} \end{aligned} \end{equation}

At the maximum, we have: \begin{equation} \begin{aligned} & &\dot{F} &= 0\\ & \Rightarrow & k_R e_{\Omega}^Te_R &= - k_{\Omega} \Vert e_{\Omega} \Vert^2 \\ & \Rightarrow & e_{\Omega}^Te_R &= - \frac{k_{\Omega}}{k_R} \Vert e_{\Omega} \Vert^2 \\ & Also, & \Vert e_{\Omega}^Te_R \Vert & \le \Vert e_{\Omega} \Vert \ \Vert e_R \Vert \\ & \Rightarrow & \Vert - \frac{k_{\Omega}}{k_R} \Vert e_{\Omega} \Vert^2 \Vert & \le \Vert e_{\Omega} \Vert \ \Vert e_R \Vert \\ & \Rightarrow & \frac{k_{\Omega}}{k_R} \Vert e_{\Omega} \Vert^2 & \le \Vert e_{\Omega} \Vert \ \Vert e_R \Vert \\ & \Rightarrow & \Vert e_{\Omega} \Vert & \le \frac{k_R}{k_{\Omega}} \Vert e_R \Vert \\ & \Rightarrow & \Vert e_{\Omega} \Vert & \le \frac{k_R}{k_{\Omega}} \qquad \qquad \{ \because \Vert e_R \Vert \le 1 \} \\ & \therefore & \Vert e_{\Omega} \Vert_{F_{max}} & \le \left( \frac{k_R}{k_{\Omega}} \right) \\ \end{aligned} \end{equation} Now, we choose $k_{\Omega}$ such that: \begin{equation} \begin{aligned} & \Vert e_{\Omega} \Vert_{F_{max}} & \le \left( \frac{k_R}{k_{\Omega}} \right) & \le \Vert e_{\Omega} \Vert_{max} \\ & \Rightarrow & \frac{k_R}{\Vert e_{\Omega} \Vert_{max}} &\le k_{\Omega} \\ \end{aligned} \end{equation}

After consulting this with my Prof., he said it's not even necessary that critical point exists, to which I asserted that then it would have to be a non-increasing function (non-decreasing is not possible according to another analysis) and maximum would exist at $t=0$.

In all other scenarios, global maximum of $F$ would be either at $t=0$ or at a local maxima at some $t$. He says he will be convinced only if I support my assertion and show this formally with references for the proofs (if needed).

Now, I thought this was elementary, but I don't know how to make it rigorous and formal, so would like some help.

My objective is to find a bound on $||e_{\Omega}||$ and choose $k_{\Omega}$ accordingly to bound it further by our chosen $||e_{\Omega}||_{max}$, given $||e_{\Omega}||_0 \le ||e_{\Omega}||_{max}$. Also I think I will be completely wrong if $||e_{\Omega}||$ doesn't attain its maxima at $F_{max}$, right?



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