I have a robot which moves abiding by the following equation: $$ \frac{d^2 \mathbf{x}}{dt} = -k \frac{d\mathbf{x}}{dt} + \mathbf{a}(t), $$ where $\mathbf{x}$ is a coordinate vector. It starts with $$ \frac{d\mathbf{x}}{dt} = v_0,$$ and $\mathbf{x} = (x0, y0, z0)$. I need to find vector valued acceleration function $\mathbf{a}(t)$, which gets robot into spot with coordinates $(0,0,0)$ the quietest way possible.
I'm not sure how to correctly write my target functional which to minimise.
It seems that my target functional is $$ \frac{\partial \int_{t=0}^{t=\tau}\frac{\mathbf{x}(t)}{dt}dt }{\partial \tau} = 0, $$ where $\frac{\mathbf{x}(t)}{dt}$ computed from solution of original differential equation with optimal $\mathbf{a}(t)$. Because if $\tau$ is optimal, then first derivative with respect to $\tau$ necessary zero. But I'm not sure I've got it right, so I would appreciate a few hints and pointers to the literature.