[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83761-en":3,"doc-seo-83761-105":30,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83761,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","L1 Optimal Control of Continuous-Time Stochastic Positive Systems","Presents an L1-optimal control problem for linear nonnegative costs under multiplicative Itô diffusion dynamics with elementwise linear input constraints. Establishes forward invariance of the positive orthant for the stochastic system and proposes a simulation method consistent with this invariance. Treats both finite-horizon and discounted infinite-horizon stochastic L1-optimal control, yielding explicit solutions via a vector-valued ODE (finite horizon) and a vector-valued algebraic equation with static feedback (infinite horizon), matching the deterministic Linear Regulator framework.","arXiv :2607 .03952v1 [math .OC] 4 Jul 2026  \nL 1 Optimal Control of Continuous-Time Stochastic  \nPositive Systems  \nAlba Gurpeguia , Takashi Tanakab , Anders Rantzera  \na Department of Automatic Control, Lund University, Ole R¨omers v¨ag 1, Lund, 223  \n63, Sweden  \nb School of Aeronautics and Astronautics, Elmore Family School of Electrical and Computer  \nEngineering, Purdue University, 701 W. Stadium Ave., West Lafayette, IN 47907-2045, United States  \nAbstract  \nWe present an L 1-optimal control problem class with linear nonnegative costs subject to multiplicative Itˆo diffusion processes with elementwise linear input constraints. Forward invariance of the positive orthant is established for the considered stochastic dynamics, and a simulation method consistent with this invariance property is proposed. Both finite-horizon and discounted infinitehorizon stochastic L 1-optimal control problems are considered. These problems admit explicit solutions characterized by a vector-valued ordinary differential equation in the finite-horizon case and by an algebraic equation in the infinitehorizon case. Notably, the optimal value function and feedback policy coincide with those of the corresponding deterministic problem, demonstrating robustness to multiplicative stochastic uncertainty. A portfolio example illustrates our results.  \nKeywords: Stochastic Control, Positive Systems, Large-scale systems, Optimal Control, Robust Control.  \nStochastic optimal control is a fundamental area of control theory with several engineering applications [2] . A classical example is the Linear Quadratic Gaussian problem (LQG) [12], which minimizes a quadratic cost for linear systems with Gaussian noise and admits closed-form solutions. While L2-optimal control of stochastic systems has proven to be effective in a wide range of applications, including robotics, aerospace, and process control [26, 5, 11], the L 1-norm is often a more appropriate choice in the presence of actuation constraints and sparse resource allocation [19], for instance in minimum-fuel control problems [22] or maximum hands-off control [20] . Recent work on deterministic L 1-optimal control has introduced a novel optimal control problem class that admits explicit solutions in discrete [16] and continuous time [8] . We refer to this framework as the Linear Regulator problem (LR), which is characterized by linear costs, positive linear dynamics and elementwise linear constraints on  \nEmail addresses: [alba.gurpegui_ramon@control.lth.se](alba.gurpegui_ramon@control.lth.se) (Alba Gurpegui), [tanaka16@purdue.edu](tanaka16@purdue.edu) (Takashi Tanaka), [anders.rantzer@control.lth.se](anders.rantzer@control.lth.se) (Anders Rantzer)  \n0 This work is partially funded by the Wallenberg AI, Autonomous Systems and Software Program (WASP), the European Research Council under the grant agreement No 101199738, DARPA grant HR0011-25-3-0210 and AFOSR grant FA9550-25-1-0347 .  \ninputs.  \nA relevant feature of the LR framework is the positivity of the dynamics. In deterministic continuous-time positive systems, forward invariance of the positive orthant follows directly from the Metzler structure of the state matrix [17, 6] . Beyond this classical setting, the analysis and control of positive systems have been studied for singular and impulsive dynamics with time delays [24, 25, 23] . In contrast, for Itˆo diffusion processes the diffusion term may drive trajectories outside the positive orthant even when the drift term in the equation satisfies the corresponding positivity conditions. Classical results on stochastic invariance and viability are given in [4, 18], including a stochastic version of the Nagumo viability theorem for stochastic differential equations with Lipschitz or monotone dynamics [3] . Related questions regarding positivity, stability [27] and the L∞ and L 1 performance [28, 21] have also been studied for stochastic systems with Markov and semi-Markov jump processes. In this paper w","cbCaisZDlpLqFSQy","https://ap.wps.com/l/cbCaisZDlpLqFSQy","pdf",1530091,4,1,14,"English","en",105,"# Abstract\n## Problem setup and main results\n## Notation\n# Stochastic Positive Systems\n## Positivity and Metzler structure\n## Stochastic invariance background","[{\"question\":\"What kind of stochastic dynamics and constraints are considered in the L1 optimal control problem?\",\"answer\":\"The paper considers multiplicative Itô diffusion processes with linear nonnegative costs and elementwise linear input constraints scaled through bounded state-dependent inputs.\"},{\"question\":\"How does the paper guarantee that system trajectories stay in the positive orthant?\",\"answer\":\"It proves forward invariance of the positive orthant for the class of Itô diffusions under the stated bounded-input conditions, and then builds a positivity-preserving discretization/simulation method consistent with this property.\"},{\"question\":\"What form do the explicit optimal solutions take for finite and infinite horizons?\",\"answer\":\"For the finite-horizon problem, the solution is characterized by a vector-valued ordinary differential equation and a time-varying switching policy; for the discounted infinite-horizon case, it is characterized by a vector-valued algebraic equation and an optimal static feedback policy.\"}]",1784190264,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"l1-optimal-control-of-continuous-time-stochastic-positive-systems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/l1-optimal-control-of-continuous-time-stochastic-positive-systems/83761/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What kind of stochastic dynamics and constraints are considered in the L1 optimal control problem?","Question",{"text":75,"@type":76},"The paper considers multiplicative Itô diffusion processes with linear nonnegative costs and elementwise linear input constraints scaled through bounded state-dependent inputs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper guarantee that system trajectories stay in the positive orthant?",{"text":80,"@type":76},"It proves forward invariance of the positive orthant for the class of Itô diffusions under the stated bounded-input conditions, and then builds a positivity-preserving discretization/simulation method consistent with this property.",{"name":82,"@type":73,"acceptedAnswer":83},"What form do the explicit optimal solutions take for finite and infinite horizons?",{"text":84,"@type":76},"For the finite-horizon problem, the solution is characterized by a vector-valued ordinary differential equation and a time-varying switching policy; for the discounted infinite-horizon case, it is characterized by a vector-valued algebraic equation and an optimal static feedback policy.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]