[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82209-en":3,"doc-seo-82209-105":28,"detail-sidebar-cat-0-en-105":82},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":13,"seo_description":14,"update_tm":26,"read_time":27},82209,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","On Robustness, Input-to-State Stability and Backstepping for Stochastic Differential Equations","We study conditions ensuring the stability of the origin of stochastic differential equations remains robust under small perturbations. Robustness is formulated in two complementary ways: preservation of stochastic stability under state-dependent parametric bounds that vanish at the origin but are positive elsewhere, and stochastic input-to-state stability (ISS) supporting nonzero perturbations globally. Using a Lyapunov function guaranteeing stability of the nominal system, the paper proves stochastic robustness and derives state-dependent perturbation scaling rules. It further establishes exponential implications under proportionally bounded perturbations, then develops stochastic integrator backstepping for pure-feedback forms based on these results.","On robustness, input-to-state stability and backstepping for stochastic differential equations  \nRobert H. Moldenhauer, Dragan Nei Fellow, IEEE, Mathieu Granzotto, Romain Postoyan Senior  \nMember, IEEE, and Andrew R. Teel Fellow, IEEE  \narXiv :2607 .09127v1 [ ee ss . SY] 10 Jul 2026  \nAbstract—We study conditions under which stability of the origin of stochastic differential equations is robust to small perturbations. We express robustness in two ways, firstly in the sense that stochastic stability is maintained under small parametric perturbations not exceeding a state-dependent bound vanishing at the origin but positive elsewhere, and secondly via stochastic inputto-state stability (ISS) which allows non-zero perturbations everywhere. We prove the former property assuming the existence of a Lyapunov function certifying stochastic stability of the nominal system. Under the same assumption, stochastic ISS holds under a suitable state-dependent perturbation scaling. Stochastic exponential stability is maintained under proportionally bounded perturbations and implies exponential ISS even without perturbation scaling. Finally, we propose a novel approach to stochastic integrator backstepping in pure-feedback form that uses the tools from our robustness analysis.  \nIndex Terms—Backstepping, input-to-state stability, robustness, stochastic differential equations, Lyapunov stability  \nThis work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.  \nI. INTRODUCTION  \nRobust stability is a property of dynamical systems that characterizes stability under perturbations and uncertainty. Oneway to express robustness is that stability is maintained under perturbations not exceeding a state-dependent bound vanishing at the equilibrium and positive everywhere else. For ordinary differential equations, it was shown in [11], using a smooth converse Lyapunov function theorem, that global asymptotic stability (GAS) of the origin exhibits this form of robustness under mild regularity conditions. An alternative way to  \nThis work was supported by the ARC under the Discovery Project DP210102600, by the ANR grant OLYMPIA ANR-23-CE48-0006, the AFOSR grant FA9550-25-1-0186 and the ARO grant W911NF-26-1- 0001.  \nR. Moldenhauer, D. Nei and M. Granzotto are with the Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, VIC 3010, Australia (e-mail: molden[hauer.r@student.unimelb.edu.au](hauer.r@student.unimelb.edu.au), [mathieu.granzotto@unimelb.edu.au](mathieu.granzotto@unimelb.edu.au), [dnesic@unimelb.edu.au](dnesic@unimelb.edu.au)) .  \nR. Moldenhauer and R. Postoyan are with the Universit de Lorraine, CNRS, CRAN, F-54000 Nancy, France (emails:{[name.surname](name.surname}@univ-lorraine.fr)[}](name.surname}@univ-lorraine.fr)[@univ-lorraine.fr](name.surname}@univ-lorraine.fr)).  \nA. Teel is with the Electrical and Computer Engineering Department, University of California, Santa Barbara, CA 93106 USA (e-mail: [teel@ece.ucsb.edu](teel@ece.ucsb.edu)) .  \nexpress robustness is through input-to-state stability (ISS), which allows non-vanishing perturbations at the equilibrium. In [13] it was shown that GAS of the origin for the nominal (unperturbed) system implies ISS when the input is scaled with a state-dependent function, which is referred to as weak ISS 1. The goal of this paper is to study these robustness concepts for stochastic differential equations (SDEs) and apply them to backstepping. For (discrete-time) stochastic difference inclusions, a very well-rounded treatment of converse theorems, robustness and ISS was achieved in [16–18] . There, smooth converse theorems and robustness are proved for the notions of GAS in probability (GASp) and recurrence under only mild regularity conditions. A stochastic version of the weak ISS result of [13] follows from [17] . For (continuous-time) SDEs the literature is more sparse","cbCairKZH2w5TbG2","https://ap.wps.com/l/cbCairKZH2w5TbG2","pdf",332681,1,"English","en",105,"# Introduction\n# Robustness and ISS Concepts\n# Lyapunov-Based Main Results\n# Stochastic Backstepping Method\n# Conclusion","[{\"question\":\"How does the paper connect exponential-type stability to input-to-state stability?\",\"answer\":\"Exponential stability is shown to persist under proportionally bounded perturbations, and this implies exponential input-to-state stability even without requiring a state-dependent perturbation scaling.\"}]",1784178823,20,{"code":4,"msg":29,"data":30},"ok",{"site_id":23,"language":22,"slug":31,"title":13,"keywords":32,"description":14,"schema_data":33,"social_meta":77,"head_meta":79,"extra_data":81,"updated_unix":26},"on-robustness-input-to-state-stability-and-backstepping-for-stochastic-differential-equations","",{"@graph":34,"@context":76},[35,52,67],{"@type":36,"itemListElement":37},"BreadcrumbList",[38,42,46,49],{"item":39,"name":40,"@type":41,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":43,"name":44,"@type":41,"position":45},"https://docshare.wps.com/document/","Document",2,{"item":47,"name":12,"@type":41,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":41,"position":51},"https://docshare.wps.com/document/on-robustness-input-to-state-stability-and-backstepping-for-stochastic-differential-equations/82209/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":22,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":39,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70],{"name":71,"@type":72,"acceptedAnswer":73},"How does the paper connect exponential-type stability to input-to-state stability?","Question",{"text":74,"@type":75},"Exponential stability is shown to persist under proportionally bounded perturbations, and this implies exponential input-to-state stability even without requiring a state-dependent perturbation scaling.","Answer","https://schema.org",{"og:url":50,"og:type":78,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":80,"canonical":50},"index,follow",{"doc_id":7,"site_id":23},{"code":4,"msg":5,"data":83},[84,88,92,96,101,106,111,114,118,121,125],{"id":20,"doc_module":4,"doc_module_name":44,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":45,"doc_module":4,"doc_module_name":44,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":44,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":44,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":44,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":44,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":44,"category_name":12,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":44,"category_name":116,"show_sort_weight":27,"slug":117},9,"Religion & Spirituality","religion-spirituality",{"id":27,"doc_module":4,"doc_module_name":44,"category_name":119,"show_sort_weight":27,"slug":120},"World Cup","world-cup",{"id":122,"doc_module":4,"doc_module_name":44,"category_name":123,"show_sort_weight":122,"slug":124},10,"Lifestyle","lifestyle",{"id":126,"doc_module":4,"doc_module_name":44,"category_name":127,"show_sort_weight":97,"slug":128},19,"General","general"]