[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85446-en":3,"doc-seo-85446-105":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},85446,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","On the Boundary of the Robust Admissible Set in State and Input Constrained Nonlinear Systems","Nonlinear control systems are studied under bounded disturbances and simultaneous state and input constraints. A robust admissible set is defined as the collection of initial states from which state and input constraints hold for all times against every admissible disturbance. The paper analyzes its boundary, separating a usable part on the state-constraint boundary and an interior barrier. At their intersection, the boundary is shown to be tangent to the state constraint set and to locally separate the set’s interior and exterior. Barrier dynamics are obtained from a saddle-point Hamiltonian principle linked to Pontryagin’s maximum principle. Results are illustrated via adaptive cruise control.","arXiv :2509 . 18825v2 [math .OC] 13 Jul 2026  \nOn the Boundary of the Robust Admissible Set in State and Input  \nConstrained Nonlinear Systems Franz Rußwurm1 , Jean L´evine2 , and Stefan Streif∗1  \n1 Automatic Control and System Dynamics, University of Technology Chemnitz, Germany  \n2 Unit´e Maths et Syst`emes, MINES Paris-PSL University, France  \nAbstract  \nIn this paper, we consider nonlinear control systems subject to bounded disturbances and to both state and input constraints. We introduce the definition of robust admissible set -the set of all initial states from which the state and input constraints can be satisfied for all times against all admissible disturbances. We focus on its boundary that can be decomposed into the usable part on the state constraint boundary and the barrier, interior to the state constraints. We show that, at the intersection of these two components, the boundary of the robust admissible set must be tangent to the state constraint set and separate the interior of the robust admissible set and its complement, a property that we call the ultimate locally separating hyperplane condition. Moreover, we prove that the barrier must satisfy a saddle-point principle on a Hamiltonian, based on Pontryagin’s maximum principle, whose final condition is precisely the ultimate locally separating condition, thus providing a set of differential equations made of the system and its adjoint for a direct construction of the barrier. Lastly, we illustrate our results by calculating the robust admissible set for an adaptive cruise control example. Keywords: nonlinear control, robust control, constrained systems, robust admissible set, bounded disturbances  \n1 Introduction  \nIn this paper, we consider nonlinear control systems, which are affected by disturbances and submitted to state and input constraints. For these systems, we aim at characterizing the robust admissible set, i.e. the set of all initial states for which a control input exists, such that the associated integral curve satisfies the state and input constraints for all times whatever the disturbances are.  \nOur approach extends the one presented in [15] for ordinary (single-valued) differential systems with disturbances, a version of the viability kernel theory introduced by [8] in the context of multi-valued differential systems. The so-called admissible set of [15] becomes here the robust admissible set and the part of its boundary running in the interior of the constraint set is still called the barrier.  \nThe concept of barrier was introduced by Isaacs [27] in the context of differential games. It plays here a central role not only because it separates the admissible states from the non admissible ones, but also because it constitutes a semi-permeable surface, namely a surface that can be one-way crossed, without possible return.  \nSuccessful applications have been made to systems without disturbances in several fields such as food production systems [7], power grids [6] and maintaining infection caps during epidemics [21], as well as some problems of Model Predictive Control [18], to cite just a few ones.  \nLet us first illustrate the topic of the present paper by the following example of adaptive cruise control. It was originally introduced in a different context in [4] and then extended to include disturbances in [42] . It is a simple model of two cars driving in a convoy, constrained to keep a safe distance between each other.  \n1.1 Introductory Example: Adaptive Cruise Control  \nWe consider an adaptive cruise control scenario involving two vehicles driving in a convoy, the leading one, whose speed is denoted by x1 and the follower, whose speed is denoted by x2 . The distance between them is denoted by x3 .  \n∗ Corresponding author: [stefan.streif@etit.tu-chemnitz.de](stefan.streif@etit.tu-chemnitz.de)  \nThe triple (x1 , x2 , x3 ), the system’s state, is supposed to satisfy the dynamical system  \nx˙1 (t) = a + d1 (t),  \nx˙2 (t) = −􀀀a0 + a1 x2 (t) + a2 x22(t)","cbCaii6PJw11ljx6","https://ap.wps.com/l/cbCaii6PJw11ljx6","pdf",399697,2,1,22,"English","en",105,"# Abstract\n# Introduction\n## Introductory Example: Adaptive Cruise Control\n## Some Historical Overview of the Problem","[{\"question\":\"What is the robust admissible set in this paper?\",\"answer\":\"It is the set of all initial states for which there exists a control input such that the state and input constraints are satisfied for all times, for every admissible bounded disturbance.\"},{\"question\":\"How is the boundary of the robust admissible set structured?\",\"answer\":\"The boundary is decomposed into a usable part on the state constraint boundary and a barrier that lies interior to the state constraints.\"},{\"question\":\"What is the “ultimate locally separating hyperplane condition”?\",\"answer\":\"At the intersection of the usable boundary component and the barrier, the robust admissible boundary must be tangent to the state constraint set and must separate the interior of the robust admissible set from its complement.\"}]",1784203595,55,{"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},"on-the-boundary-of-the-robust-admissible-set-in-state-and-input-constrained-nonlinear-systems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-the-boundary-of-the-robust-admissible-set-in-state-and-input-constrained-nonlinear-systems/85446/",4,{"url":51,"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","2026-07-24","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 is the robust admissible set in this paper?","Question",{"text":75,"@type":76},"It is the set of all initial states for which there exists a control input such that the state and input constraints are satisfied for all times, for every admissible bounded disturbance.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the boundary of the robust admissible set structured?",{"text":80,"@type":76},"The boundary is decomposed into a usable part on the state constraint boundary and a barrier that lies interior to the state constraints.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the “ultimate locally separating hyperplane condition”?",{"text":84,"@type":76},"At the intersection of the usable boundary component and the barrier, the robust admissible boundary must be tangent to the state constraint set and must separate the interior of the robust admissible set from its 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