[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81977-en":3,"doc-seo-81977-105":30,"detail-sidebar-cat-0-en-105":84},{"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},81977,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Stochastic Stability of Nonlinear MPPI via Contraction Theory and Control Lyapunov Functions","Model Predictive Path Integral (MPPI) control enables online updates on nonlinear systems using forward dynamics rollouts rather than gradients, linearizations, or convex optimization. This work proves a closed-loop stability certificate via a stability-inheritance argument. It assumes a deterministic nonlinear MPC policy certified by a CLF terminal cost and a contraction metric, then shows finite-sample MPPI approximates the reference with high probability, decomposing error into finite-temperature bias and Monte Carlo terms. MPPI inherits nominal contraction under a small-gain condition, yielding finite-horizon localized mean practical stability with an exponential decay toward residual floors, plus an ISS-type reformulation and finite-horizon design procedure.","Stochastic Stability of Nonlinear MPPI via Contraction Theory and Control  \nLyapunov Functions  \nHyung-Jin Yoon† and Hunmin Kim‡  \narXiv :2607 .06945v1 [ ee ss . SY] 8 Jul 2026  \nAbstract—Model Predictive Path Integral (MPPI) control is directly implementable on nonlinear systems because its online update requires only forward rollouts of the dynamics, not gradients, linearizations, or convex optimization. However, this algorithmic flexibility does not by itself provide a closed-loop stability certificate. This paper establishes such a certificate through a stability-inheritance argument. The result should be interpreted as an inheritance theorem, not as an existence theorem for stabilizing nonlinear MPC. The analysis proceeds in three steps. First, we assume that there exists a deterministic nonlinear MPC policy whose disturbance-free closed loop is certified by a Control Lyapunov Function (CLF) terminal cost and a contraction metric. This policy plays the same analytical role as the LQR controller in the companion LTI result [1]: it is not computed or used by MPPI online, but serves as a stabilizing reference whose robustness margin MPPI must approximate. Second, we show that finite-sample MPPI approximates this reference policy with high probability, with an error that decomposes into a finite-temperature bias floor and a Monte Carlo term that vanishes as the sample count grows. Third, we show that MPPI inherits the nominal contraction whenever a small-gain condition on the state-dependent approximation gain holds. The main result establishes finite-horizon, high-probability localized mean practical stability. For any prescribed horizon and confidence level, the closed-loop trajectory remains in a compact sublevelset with high probability, and the expected deviation from the equilibrium decays exponentially up to residual floors caused by MPPI approximation error, Gaussian process noise, and bad sampling events. The paper also provides an ISS-type restatement and an explicit finite-horizon design procedure for choosing the localization set, temperature, and minimum sample count.  \nI. INTRODUCTION  \nModel Predictive Path Integral (MPPI) control [2], [3] is a sampling-based receding-horizon method that has demonstrated strong empirical performance across roboticsand autonomous systems, including off-road navigation [4], legged locomotion, and aerial vehicles. At each time step, M random control perturbations are drawn, rolled out in parallel using only the forward model xi+1 = f (xi , ui), and combined via importance weighting to approximate the information-theoretic optimal control. This procedure requires no gradient of the cost or dynamics, no linearization,  \n†H.-J. Yoon is with the Department of Mechanical and Nuclear Engineering, Tennessee Technological University, Cookeville, TN, USA.  \n‡H. Kim is with the School of Engineering, Department of Electrical and Computer Engineering, Mercer University, Macon, GA, USA.  \nThis work was supported by internal funding at Tennessee Technological University. This is the second in a three-paper series on MPPI closed-loop stability. The companion paper [1] establishes exponential stability for LTI systems.  \nand no convexity assumption, making it directly applicable to nonlinear systems.  \nFormal closed-loop stability guarantees for MPPI, however, remain limited. A survey by Honda [5] notes:“Convergence and optimality results . . . do not directly imply closed-loop stability in the sense of classical MPC theory. . . Establishing stability guarantees for PI-MPC remains an open problem.”  \nThe companion paper [1] established exponential stability in expectation for the linear time-invariant (LTI) case, where the LQR controller provides an explicit stabilizing reference. The present paper addresses the nonlinear case. Because no universal stabilizing feedback exists for arbitrary nonlinear systems, the result is formulated as a stability-inheritance theorem: we assume the existence of ","cbCaitLfL2r21FYp","https://ap.wps.com/l/cbCaitLfL2r21FYp","pdf",631154,6,1,18,"English","en",105,"# Introduction\n## Problem Formulation","[{\"question\":\"How is the MPPI approximation error characterized and how does stability follow?\",\"answer\":\"The finite-sample approximation error splits into a finite-temperature bias floor and a Monte Carlo term that vanishes as sample count increases. Under a small-gain condition, the contraction margin absorbs the state-dependent approximation gain error, leading to finite-horizon localized mean practical stability.\"}]",1784177382,45,{"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":79,"head_meta":81,"extra_data":83,"updated_unix":28},"stochastic-stability-of-nonlinear-mppi-via-contraction-theory-and-control-lyapunov-functions","",{"@graph":36,"@context":78},[37,54,69],{"@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":53},"https://docshare.wps.com/document/stochastic-stability-of-nonlinear-mppi-via-contraction-theory-and-control-lyapunov-functions/81977/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72],{"name":73,"@type":74,"acceptedAnswer":75},"How is the MPPI approximation error characterized and how does stability follow?","Question",{"text":76,"@type":77},"The finite-sample approximation error splits into a finite-temperature bias floor and a Monte Carlo term that vanishes as sample count increases. Under a small-gain condition, the contraction margin absorbs the state-dependent approximation gain error, leading to finite-horizon localized mean practical stability.","Answer","https://schema.org",{"og:url":52,"og:type":80,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":82,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":85},[86,90,94,98,103,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":87,"show_sort_weight":88,"slug":89},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":91,"show_sort_weight":92,"slug":93},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Exam",70,"exam",{"id":99,"doc_module":4,"doc_module_name":46,"category_name":100,"show_sort_weight":101,"slug":102},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":99,"slug":130},19,"General","general"]