Results and open questions on the boundary control of moving sets

TL;DR

The paper surveys boundary-control of moving sets motivated by invasive species, connecting PDE travel-front control to moving-set dynamics via the inward normal speed $\beta$ on the boundary $\partial\Omega(t)$ and a cost that balances the contaminated area with the control effort. It introduces an optimal set-motion framework in which the boundary motion is driven by the inward velocity and the cost is expressed through the boundary effort $E(\beta)$, linked to traveling-front optimization. It establishes existence results for optimal set motion under convexity and BV framework and derives necessary conditions via an adjoint $Y$ and a pointwise optimality law $\lambda(t) E(\beta)-Y(t,\xi) \beta = \min_{\beta\ge\beta^*}\{\lambda(t) E(\beta)-Y(t,\xi) \beta\}$, analyzing eradication and minimum-time problems under geographical constraints with invariants $\kappa(V)$ and $K(V)$. It also discusses a sharp-interface limit connecting the parabolic model to the moving-set formulation and outlines open questions on boundary regularity and optimal strategies on general domains. The work highlights geometric strategies, Dido-type problems, and a program to extend results to general domains with BV- regulatory frameworks.

Abstract

These notes provide a survey of recent results and open problems on the boundary control of moving sets. Motivated by the control of an invasive biological species, we consider a class of optimization problems for a moving set $t\mapsto Ω(t)$, where the goal is to minimize the area of the contaminated set $Ω(t)$ over time, plus a cost related to the control effort. Here the control function is the inward normal speed, assigned along the boundary $\partial Ω(t)$. We also consider problems with geographical constraints, where the invasive population is restricted within an island. Existence and structure of eradication strategies, which entirely remove the invasive population in minimum time, is also discussed.

Figures (23)

• Figure 1: A graph of the function $f$ in (\ref{['CRD']}).
• Figure 2: Approximating the function $u(t,\cdot)$ with the characteristic function of a set $\Omega(t)$. Here $\beta$ is the speed of the boundary in the inward normal direction.
• Figure 3: Without any control, the parabolic equation (\ref{['A1']}) yields a traveling wave profile with speed $\beta^*<0$, and the invasive population keeps expanding.
• Figure 4: By adding a large negative source term $\alpha u$, one can achieve a traveling profile of any speed $\beta >\beta^*$. In particular, when $\beta >0$ the invasive population shrinks in size.
• Figure 5: By constructing a feedback control $\alpha(U)$ which is strictly positive on the interval $[a,b]$, one obtains a trajectory $\gamma$ of (\ref{['T3']}) which connects the unstable manifold $\Gamma^u$ through $(0,0)$ with the stable manifold $\Gamma^s$ through $(1,0)$. This yields a traveling profile for (\ref{['cpe']}) with speed $\beta$.

Theorems & Definitions (14)

• Theorem 2.1
• Theorem 2.2
• Remark 2.1
• Theorem 3.1
• Theorem 4.1
• Theorem 5.1
• Remark 5.1
• Proposition 6.1
• Example 6.1
• Example 6.2