[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82970-en":3,"doc-seo-82970-105":30,"detail-sidebar-cat-0-en-105":83},{"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},82970,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Input-to-State Stability Implications in Contraction Theory","For nonlinear control systems on normed vector spaces, the work characterizes an incremental input-to-state stability (ISS) property where the overshoot constant multiplies both initial-condition and input contributions. Variational-system analysis proves two equivalent ISS-type bounds: one on the variational system and one incremental bound on the original system. An infinitesimal contraction condition via a Lyapunov-type function is likewise equivalent to an incremental Lyapunov condition, with differences tied to the input Lipschitz constant. Results hold under continuous differentiability, illustrated through sensitivity matrices and Lyapunov characteristic exponents, and extended to discrete-time systems.","arXiv :2607 .05640v2 [ ee ss . SY] 8 Jul 2026  \nInput-to-State Stability Implications in Contraction Theory ⋆  \nYu Kawanoa , Francesco Bullob ,  \na Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan  \nb Department of Mechanical Engineering and the Center for Control, Dynamical Systems, and Computation, University of  \nCalifornia, Santa Barbara, USA  \nAbstract  \nFor nonlinear control systems on normed vector spaces, we characterize an incremental input-to-state stability (ISS) type property in which the overshoot constant multiplies both the initial-condition and the input terms. Working through the associated variational system, we show that two properties are equivalent: an ISS-type bound on the variational system, and the incremental ISS-type bound on the original system. We further establish the equivalence between an infinitesimal contraction condition, expressed through a Lyapunov-type function, and an incremental Lyapunov condition. Each of these equivalent conditions yields a necessary condition and a sufficient condition for the ISS-type bounds, differing only in the input Lipschitz constant of the vector field. When the overshoot constant equals one, the infinitesimal contraction condition reduces to the standard norm-based contraction condition. We establish these implications under mere continuous differentiability of the vector field, and we illustrate the results through sensitivity matrices and Lyapunov characteristic exponents. Moreover, we develop similar implications for discrete-time systems.  \nKey words: Contraction Theory; Incremental Stability; Input-to-State Stability.  \n1 Introduction  \nContraction theory [5,7,14] provides a framework to study stability between the pair of trajectories of a system, known as incremental stability. Often, incremental stability is analyzed by using the associated variational system. For control systems, a central role is played by incremental input-to-state stability (ISS), also referred to as δISS. Contractivity and incremental ISS are closely related, and it is a classic problem to characterize the equivalence between these two properties.  \nThe study of this equivalence builds on the influential works [2,3] and [1], along with several recent contributions. Most references address the Riemannian or, more generally, Finslerian setting [1,7–11,14] for continuous-time systems; the normed vector space setting is considered in [4, 6] . In the classical Lyapunov setting, necessary and sufficient characterizations of incremental ISS are established in [2,3] .  \nThe contributions of this paper are as follows. First, as the main contribution, we establish a necessary condition and a sufficient condition for an incremental ISS-type property in the contractivity framework through the analysis of variational systems. These conditions differ only in the input Lipschitz constant of the vector field. To the best of the authors’ knowledge, no such characterization has been previously available. We further provide an equivalent incremental Lyapunov characterization. Second, unlike the aforementioned Finslerian setting, our analysis relaxes twice continuous differentiability of the vector field to continuous differentiability by showing that the transition  \n⋆ This paper was not presented at any IFAC meeting. Corresponding author Y. Kawano. Tel. +81-82-424-4210 . Email addresses: [ykawano@hiroshima-u.ac.jp](ykawano@hiroshima-u.ac.jp) (Yu Kawano), [bullo@ucsb.edu](bullo@ucsb.edu) (Francesco Bullo).  \nPreprint submitted to Automatica 10 July 2026  \nmatrix of the variational system coincides with the Gˆateaux derivative of the flow of the original system. A more detailed comparison with the literature can be found in Section 2 .4. Also, we illustrate some implications of our results for sensitivity functions [18] and Lyapunov characteristic exponents [17] . Finally, we develop similar ISS-type implications for discrete-time systems.  \n","cbCaifa3RazX0fu2","https://ap.wps.com/l/cbCaifa3RazX0fu2","pdf",427673,2,1,17,"English","en",105,"# 1 Introduction\n# 2 Input-to-State Stability Equivalences\n## 2.1 Variational Systems","[{\"question\":\"How do the conditions relate to input Lipschitz constants, and what happens when the overshoot constant equals one?\",\"answer\":\"The equivalent ISS-type characterizations differ only in the input Lipschitz constant of the vector field. When the overshoot constant equals one, the infinitesimal contraction condition reduces to the standard norm-based contraction condition.\"}]",1784184385,43,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"input-to-state-stability-implications-in-contraction-theory","",{"@graph":36,"@context":77},[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/input-to-state-stability-implications-in-contraction-theory/82970/",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","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How do the conditions relate to input Lipschitz constants, and what happens when the overshoot constant equals one?","Question",{"text":75,"@type":76},"The equivalent ISS-type characterizations differ only in the input Lipschitz constant of the vector field. 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