Problem formulation

Consider the system $$\dot{x}=f(x,u)$$where the state is $x \in \mathbb{X}\subset \mathbb{R}^{n}$, control input $u \in \mathbb{U}\subset \mathbb{R}^{m}$. The objective in coverage control is to design a policy $\gamma:\mathbb{X}\to \mathbb{U}$ for the system s.t. closed-loop trajectories gather information over a domain $\mathcal{D}\subset \mathbb{X}$. Let $$c=c(t,p)$$denote the ‘coverage level’ about a point $p \in \mathcal{D}$ at time $t$. Coverage increases through a sensing function $S:\mathbb{X}\times \mathcal{D}\to \mathbb{R}_{\ge 0}$ (positive when $p$ can be sensed from $x$ and 0 else), and coverage decreases at a rate $\alpha:\mathcal{D}\to \mathbb{R}_{\ge 0}$. Giving us the model: $$\dot{c}=S(x,p)(1-c)-\alpha(p)c.$$A point $p$ is said to be covered if $c(t,p)$ reaches a threshold $c^{*}$.

So, in summary: - $S(x,p)$ is the sensing function. This describes our quality of coverage of $p$ from our position $x$. - $\alpha(p)$ describes how the “coverage” of $p$ decays. - $c(t,p)$ describes the “coverage level” of $p$ at time $t$. - $c^{*}$ acts as an upper bound on the coverage level that can be obtained.

Toy Problem