[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82215-en":3,"doc-seo-82215-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82215,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Control Laguerre Tessellation Semi-discrete Optimal Transport Over Control Systems","Control Laguerre Tessellation (CLT) studies optimal transport of optimally controlled agents from an absolutely continuous compactly supported source measure to a discrete target consisting of finitely many Dirac masses. The ground cost is induced by each agent’s optimal control motion. Under the twist condition, the optimal transport map is characterized almost everywhere by a Laguerre tessellation of the state space, and CLT is illustrated for linear control systems with minimum-energy and minimum-time objectives.","Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems  \nRipon C. Sarker, Abhishek Halder  \narXiv :2607 .09139v1 [math .OC] 10 Jul 2026  \nAbstract—We study the optimal transport of optimally controlled agents from a compactly supported absolutely continuous source to a discrete target measure. The ground cost for the transport is induced by the optimal cost of the agents’motion. When this ground cost satisfies the twist condition, the optimal transport map is given almost everywhere in terms of a Laguerre tessellation of the state space. We refer to this control-theoretic generalization of Laguerre tessellation as Control Laguerre Tessellation (CLT), and illustrate it for two ground costs induced by linear controlled agents with minimum energy and minimum time objectives.  \nI. INTRODUCTION  \nConsider a large population of indistinguishable active agents, i.e., identical control systems, with absolutely continuous normalized 1 population measure µ supported over a compact state space X ⊆ Rn. Suppose we want to optimally transport this population to a finitely supported target measure ν = Pri=1 νi δyi with fixed r ∈ N ≥2, where δyi denotes the Dirac delta at yi ∈ Rn ∀i ∈ JrK := {1,..., r}, and (ν1 ,...,νr) ∈ ∆r−1 (standard simplex) ⊂ R0. We assume that the target states are distinct, i.e., yi  yj for i  j. The vector2 ν := (ν1 , ... ,νr) can be interpreted as either normalized capacities or relative importance of the target states y 1 , . . . , yr . Fig. 1 shows an SDOT instance with n = 2, r = 5 .  \nThe large population of active agents comprising the source measure µ may represent micron-sized chiplets controlled by electric field for micro-assembly [1]–[3], or magnetic nanoparticles for targeted drug delivery [4]–[6] . In economic resource allocation, µ may represent a population (e.g., people, children in a city) and ν may represent localized resource (e.g., coffee shops, elementary schools) with given capacities. Such applications naturally lead to the semidiscrete optimal transport (SDOT) problem [7, Ch. 5]:  \nT:X→m{iyi}i = z| eT\\# µ=ν ZX c (x, T(x))dµ (1)  \nwhere T\\# µ denotes the pushforward of µ via the transport map T, and c : X ×{yi }ri=1 →7 R ≥0 is a known ground cost induced by the active agents. The triple µ,ν, c comprise the problem data for (1) .  \nFor x ∈ X, i ∈ JrK, the c (x, yi) quantifies the cost of transporting unit amount of mass from the source state x  \nRipon C. Sarker and Abhishek Halder are with the Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA,{rcsarker,[ahalder](ahalder}@iastate.edu)[}](ahalder}@iastate.edu)[@iastate.edu](ahalder}@iastate.edu).  \nThis research was partially supported by NSF award 2111688 .  \n1Here RX dµ = 1 .  \n2We use the boldfaced ν for a vector on the standard simplex, and the unboldfaced ν for the corresponding discrete measure.  \nFig. 1: An instance of SDOT for n = 2, r = 5 . The source measure µ (dark = high, light = low values) is supported on X = [−1, 1]2 , and ν = (ν1 ,...,ν5 ) = (0 .2 , 0.2 , 0.2 , 0.3 , 0. 1) .  \nto the target state yi. The term “semi-discrete” refers to that the source measure µ is absolutely continuous w.r.t. the Lebesgue measure on X while the target measure ν is discrete (weighted sum of Dirac masses) . SDOT has found applications in fluid dynamics [8],[9], machine learning [10],[11], mean field games [12], and robotic coverage control [13], [14] .  \nThe purpose of this work is to explore the solution of (1) when c is induced by an optimal control cost. Our motivation comes from applications such as micro-assembly and drug delivery mentioned earlier, where optimality of the individual agent’s controlled trajectories are desired in addition to the collective transport guarantees. Specifically, we consider the individual agents to be identical control systems with initial states distributed in the support of µ . The agents move to minimize certain performance objective (e.g., energy, time), a","cbCaifGhFcWT1Mfk","https://ap.wps.com/l/cbCaifGhFcWT1Mfk","pdf",3090157,1,6,"English","en",105,"# Introduction\n## Semi-discrete optimal transport setup\n## Twist condition and Laguerre cells\n## Discrete Monge–Ampère equation and numerical solution","[{\"question\":\"What does “semi-discrete” mean in the studied optimal transport problem?\",\"answer\":\"The source measure is absolutely continuous on the state space, while the target measure is discrete, represented as a weighted sum of Dirac masses.\"},{\"question\":\"How is the ground cost constructed in this work?\",\"answer\":\"The ground cost between a source point and a target location is induced by the optimal cost of an individual agent’s controlled motion.\"},{\"question\":\"What role does the twist condition play in determining the optimal transport map?\",\"answer\":\"When the ground cost satisfies the twist condition, the optimal transport map is uniquely determined almost everywhere and admits a geometric description via Laguerre cells, leading to a Laguerre tessellation of the state 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does “semi-discrete” mean in the studied optimal transport problem?","Question",{"text":74,"@type":75},"The source measure is absolutely continuous on the state space, while the target measure is discrete, represented as a weighted sum of Dirac masses.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How is the ground cost constructed in this work?",{"text":79,"@type":75},"The ground cost between a source point and a target location is induced by the optimal cost of an individual agent’s controlled motion.",{"name":81,"@type":72,"acceptedAnswer":82},"What role does the twist condition play in determining the optimal transport map?",{"text":83,"@type":75},"When the ground cost satisfies the twist condition, the optimal transport map is uniquely determined almost everywhere and admits a geometric description via Laguerre cells, leading to a Laguerre tessellation of the state 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