Optimal Assignment and Motion Control in Two-Class Continuum Swarms

TL;DR

This work addresses coordinating two-density classes (demand and resource) in a continuum swarm by formulating an optimal-control problem that trades off assignment cost $\mathcal{W}_2^2(R_t,D_t)$ and motion cost $\int_\Omega \|V_t\|^2 R_t$. In one dimension, the authors transform the problem into an infinite-dimensional, decoupled linear-quadratic tracking problem expressed in quantile space, enabling explicit solutions. They derive closed-form solutions for static and periodic demands by decoupling into scalar LQ tracking problems and show that optimal resource trajectories follow Wasserstein geodesics toward a nearest reachable demand, with clear interpretations in terms of transport maps and pushforwards. The results provide analytical benchmarks for centralized control of large swarms, offer insights into the role of motion costs in tracking, and point to future extensions to higher dimensions, unbalanced transport, and distributed implementations.

Abstract

We consider optimal swarm control problems where two different classes of agents are present. Continuum idealizations of large-scale swarms are used where the dynamics describe the evolution of the spatially-distributed densities of each agent class. The problem formulation we adopt is motivated by applications where agents of one class are assigned to agents of the other class, which we refer to as demand and resource agents respectively. Assignments have costs related to the distances between mutually assigned agents, and the overall cost of an assignment is quantified by a Wasserstein distance between the densities of the two agent classes. When agents can move, the assignment cost can decrease at the expense of a physical motion cost, and this tradeoff sets up a nonlinear infinite-dimensional optimal control problem. We show that in one spatial dimension, this problem can be converted to an infinite-dimensional, but decoupled, linear-quadratic (LQ) tracking problem when expressed in terms of the quantile functions of the respective agent densities. Solutions are given in the general one-dimensional case, as well as in the special cases of constant and periodically time-varying demands.

Figures (7)

• Figure 1: Depiction of the basic problem formulation. A time-varying demand distribution $D_t$ offloads tasks to a resource distribution $R_t$. The "task assignment" (depicted by the green lines) is the optimal instantaneous Kantorovich plan ${\@fontswitch\mathcal{K}}_t$ with "communication cost" $\mathcal{W}_2^2 ( R_t,D_t )$. $R_t$ is transported by the (control) velocity field $V_t$ to track $D_t$, optimally trading off the assignment cost and motion cost.
• Figure 2: (Top) A density $R$ is transported by a velocity field $V$ according to the continuity equation (\ref{['R_PF.eq']}). Equivalently, its CDF ${F_{\!{ R}}}$ (middle) is advected by $V$ according to the advection equation (\ref{['F_PF.eq']}). (Bottom) The corresponding quantile function ${Q_{\!{ R}}}(z,t)$ evolves independently at each $z$ with a derivative of $U(z,t) := V \!\left( {Q_{\!{ R}}}(z,t), t \right)$. Notice that Dirac masses in $R$ correspond to regions where ${Q_{\!{ R}}}$ is constant. Since Dirac masses must move with a single velocity, constant regions in ${Q_{\!{ R}}}$ must move with a single velocity as well.
• Figure 3: ( Top) The $q$-level-set partition ${\@fontswitch\mathcal{P}}_{\!{\rm q}}$ (Definition \ref{['partition_def']}) of the interval $[0,1]$ into level sets of a function $q:[0,1]\rightarrow \mathbb{R}$ (black). The partition is composed of the sets $P_a$, $P_b$ (finite intervals, corresponding to the levels $a$ and $b$ of $q$), and a continuum of singleton level sets $\left\{ P_i; ~i\in(a,b) \right\}$ of the values of $q$ between $a$ and $b$. ( Bottom) A density $\mu$, its ( middle) quantile function $Q_\mu$, and its average $\bar{Q}_\mu$ with respect to the partition ${\@fontswitch\mathcal{P}}_{\!{\rm q}}$. The average is ${\@fontswitch\mathcal{P}}$-piecewise-constant (Definition \ref{['PWC_P.def']}). ( Bottom) $\bar{\mu}$ is the density of the averaged quantile $\bar{Q}_\mu$. The Dirac masses, each of mass $\left| P_i \right|$, correspond to each $P_i$ of non-zero Lebesgue measure in the partition ${\@fontswitch\mathcal{P}}$.
• Figure 4: Decoupling in discrete-agent case. Each agent in the resource distribution (red) corresponds to a constant region in the quantile function. The vertical dotted lines show the separation of these constant regions via partitioning. A single scalar LQ tracking problem can be written for each element in this partition, with state $r_i = {Q_{\!{ R}}}(P_i)$ (red), control $u_i = U(P_i)$ (green), and tracking signal $d_i = {\bar{Q}_{\!{ D}}}(P_i)$ (black).
• Figure 5: Initial, intermediate, and final conditions of the resource $R$ (red), demand $D$ (blue), and their corresponding CDFs and quantile functions. The vertical dotted lines on the quantile function plots separate elements of the partition $\mathcal{P}$. The parameters $\alpha = 2$ and ${ T}=10$ were used for this particular example. Due to the relatively large value of ${ T} / {\alpha}$, the final distribution $R_{ T}$ ends up very close to $\bar{D}$, the nearest reachable distribution to $D$.

Theorems & Definitions (23)

• Definition 3
• Lemma 4: Santambrogio2015
• Lemma 5
• proof
• Lemma 6
• proof
• Definition 7: Pushforward
• Lemma 8: Santambrogio2015
• Proposition 9
• proof