Optimally Controlled Moving Sets with Geographical Constraints

TL;DR

We study optimally controlled moving sets confined to a geographic region $V$, modeling eradiation of an invasive population with perimeter-based control costs and unit-speed natural growth. The authors develop a BV-based weak framework to formulate the Eradication, Minimum Time, and Optimization problems under geographic constraints, proving existence of solutions and deriving Pontryagin-type necessary conditions via needle perturbations and an adjoint variable $Y$. They establish a sufficient condition for optimality, and provide a detailed program to construct optimal set motions in various geometric settings, including symmetric and non-symmetric domains, discs, corners, and polygonal islands, often reducing the boundary dynamics to combinations of free arcs and circumferential controlled arcs with constant curvature. The work yields explicit, Dido-type optimal structures in several cases and presents local existence results for maximal free arcs, offering guidelines for explicit computation of optimal strategies in constrained environments and highlighting key open questions about regularity and global constructions.

Abstract

The paper is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a region $V\subset \R^2$ bounded by geographical barriers. If no control is applied, the contaminated set $Ω(t)\subset V$ expands with unit speed in all directions. By implementing a control, a region of area $M$ can be cleared up per unit time. Given an initial set $Ω(0)=Ω_0\subseteq V$, three main problems are studied: (1) Existence of an admissible strategy $t\mapstoΩ(t)$ which eradicates the contamination in finite time, so that $Ω(T)=\emptyset$ for some $T>0$. (2) Optimal strategies that achieve eradication in minimum time. (3) Strategies that minimize the average area of the contaminated set on a given time interval $[0,T]$. For these optimization problems, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions $t\mapsto Ω(t)$ are explicitly constructed in a number of cases. \end{abstract}

Figures (33)

• Figure 1: Left: a moving set, where the evolution is determined by assigning the inward normal speed $\beta$ at each boundary point. Right: the effort function $E(\beta)$ in (\ref{['E']}).
• Figure 2: The two invariants (\ref{['ko']}) and (\ref{['KO']}) in the case of an equilateral triangle with unit side. Left: one of the level sets of the function $\phi$ in (\ref{['KO']}) must go through the vertex $C$. Hence it will have length $\geq \sqrt 3/2$. Center: for any $\lambda\in [0, 1/2]$ we can cut a sector $V_\lambda$ with area $\lambda {{{\cal L}}^2}(V)$, so that its boundary is an arc with length $\leq\sqrt{3\sqrt 3/4\pi}$. Right: for $\lambda\in [1/2, 1]$ we can take $V_\lambda$ to be the complement of a sector with the same property.
• Figure 3: Left: in the case where $M>K(V)$, an eradication strategy can be constructed Right: if (\ref{['m2b']}) holds and $M<\kappa(V)$, then the area of the set $\Omega(t)$ can never become smaller than $\lambda^*{\cal L}^2(V)$.
• Figure 4: Two optimal solutions for the minimum time problem on a disc. For every $t\in \,]0,T[\,$, the boundary $\partial \Omega(t)$ should be an arc of circumference, crossing the boundary $\partial V$ perpendicularly.
• Figure 5: Two examples where the boundaries of the sets $\Omega(t)$ can be parameterized as $\xi\mapsto x(t,\xi)$, according to (A1)--(A3). Left: a case where $(t,\xi)\in W=[0,T]\times S^1$. Right: a case with geographical constraints, where $(t,\xi)\in W=[0,T]\times ]0,1[\,$.

Theorems & Definitions (33)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Example 3.1
• Theorem 3.1
• Theorem 4.1
• Remark 4.1
• Corollary 4.1
• Example 4.1
• Theorem 5.1