A moving object and observers in $\mathbb R^2$ with piecewise smooth shading set

We consider the motion of an object $t$ in the space ${\mathbb R}^2$, where a bodily bounded bounded set $G$ with piecewise smooth boundary hinders the motion and visibility. In a neighborhood of convex parts of the boundary, there are observers, which can hide from $t$ in a shade set $s(t)\subset {\mathbb R}^2\setminus G$ in the case of danger from $t$. We find characteristic properties of the trajectory $\mathcal T$ of the object that maximizes the value $\min\{\rho(t,s(t)):\ t\in {\mathcal T}\}$.