[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84641-en":3,"doc-seo-84641-105":28,"detail-sidebar-cat-0-en-105":90},{"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},84641,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Robust Stabilization of Linear Markov-Jumping Hyperbolic PDEs with Boundary Input Delay","Robust stabilization is developed for 2×2 linear hyperbolic PDEs with Markov-jumping parameters and boundary input delay, addressing the coupled difficulties of stochastic mode switching and delayed actuation. A nominal delay-compensating backstepping controller is designed for a fixed nominal model, then transformed to the stochastic system, yielding additional mismatch perturbations. A mode-independent Lyapunov functional establishes pathwise exponential estimates, leading to mean-square exponential stability under an explicit small-mismatch condition. Simulation studies clarify how the conservativeness is interpreted numerically.","Robust Stabilization of Linear Markov-Jumping Hyperbolic PDEs with Boundary Input Delay  \nYihuai Zhang, Yidan Cao, Huan Yu, and Lu Liu  \narXiv :2607 .02081v1 [ ee ss . SY] 2 Jul 2026  \nAbstract—This paper studies the robust stabilization of 2 × 2 linear hyperbolic partial differential equations (PDEs) with Markov-jumping parameters and boundary input delay. The main challenge arises from the simultaneous presence of stochastic parameter variations and input delay, which complicates both the stability analysis and controller design. To address this issue, a nominal delay-compensating backstepping controller is first designed for a fixed nominal system. Applying the nominal transformation to the stochastic system yields a target system with additional perturbation terms induced by parameter mismatch. A mode-independent Lyapunov functional is then constructed to establish a pathwise exponential estimate, which directly implies mean-square exponential stability under an explicit small-mismatch condition. The proposed analysis provides a direct robustness certificate for nominal delay compensation without using mode-dependent Lyapunov functionals. Finally, we present simulation results and discuss how the conservative small-mismatch condition should be interpreted for the numerical example.  \nIndex Terms—Partial differential equations (PDEs), Backstepping, Markov-jumping parameters, Input delay, Mean-square exponential stability.  \nI. INTRODUCTION  \nHYPERBOLIC partial differential equations (PDEs) are  \nwidely used in engineering applications such as oil drilling [18], traffic flow [20], and gas pipelines [4] . For this class of systems, boundary control is especially relevant because the control action often enters through the boundary. Over the past decades, the backstepping method has become one of the most effective tools for the stabilization of linear hyperbolic PDEs, because it transforms the original system into an exponentially stable target system that can be analyzed using Lyapunov methods [13] .  \nIn many applications, however, the model coefficients are not deterministic. Transport speeds, source terms, and boundary couplings may vary because of uncertain operating conditions, disturbances, or abrupt mode changes. A convenient and mathematically tractable representation of such uncertainty is to model the time-varying coefficients as finite-state Markovjumping processes. This leads to stochastic hyperbolic systems whose analysis must account not only for the distributed  \nThis work was supported by the Research Grants Council of the Hong Kong Special Administrative Region of China under Project CityU/11212225 .(Corresponding author: Lu Liu.)  \nYihuai Zhang, Yidan Cao, and Lu Liu are with the Department of Mechanical Engineering, City University of Hong Kong. (email: [yihuai.zhang@cityu.edu.hk](yihuai.zhang@cityu.edu.hk), [yidancao4-c@my.cityu.edu.hk](yidancao4-c@my.cityu.edu.hk), [luliu45@cityu.edu.hk](luliu45@cityu.edu.hk)).  \nHuan Yu is with the Thrust of Intelligent Transportation, Hong Kong University of Science and Technology (Guangzhou) . (e-mail: huanyu@hkust[gz.edu.cn](gz.edu.cn)).  \ndynamics but also for random switching among different modes. Stochastic linear hyperbolic PDEs have been widely investigated [1], [5], [7], [14], [15] . In [19], the authors investigated the robust stochastically exponential stability and stabilization of uncertain linear hyperbolic PDEs with Markovjumping parameters by using linear matrix inequalities (LMIs) . Prieur [15] analyzed changes in the boundary conditions and derived sufficient conditions for the exponential stability of the switching system. LMIs were then applied to obtain the sufficient conditions for stochastic stabilization of traffic flow described by hyperbolic PDEs [22] . Furthermore, Zhao [25] investigated the output feedback stabilization of PDE-ODE cascade systems with stochastic jumps. By adopting the backstepping method, Auriol [3] demonstrated the meansqua","cbCaifJVZPxTVv0v","https://ap.wps.com/l/cbCaifJVZPxTVv0v","pdf",684027,1,"English","en",105,"# Introduction\n## Background on boundary control and backstepping\n## Stochastic hyperbolic systems with Markov-jumping parameters\n## Input delay modeling and prior robust results","[{\"question\":\"What system setting does the paper study?\",\"answer\":\"The paper studies robust stabilization for 2×2 linear hyperbolic PDEs with Markov-jumping parameters and boundary input delay.\"},{\"question\":\"How is the controller designed to handle the boundary input delay?\",\"answer\":\"A nominal delay-compensating backstepping controller is first designed for a fixed nominal system, then its nominal transformation is applied to the stochastic system to obtain a target system with mismatch perturbations.\"},{\"question\":\"What stability result is proved and under what condition?\",\"answer\":\"A mode-independent Lyapunov functional provides a pathwise exponential estimate, implying mean-square exponential stability under an explicit small-mismatch 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system setting does the paper study?","Question",{"text":74,"@type":75},"The paper studies robust stabilization for 2×2 linear hyperbolic PDEs with Markov-jumping parameters and boundary input delay.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How is the controller designed to handle the boundary input delay?",{"text":79,"@type":75},"A nominal delay-compensating backstepping controller is first designed for a fixed nominal system, then its nominal transformation is applied to the stochastic system to obtain a target system with mismatch perturbations.",{"name":81,"@type":72,"acceptedAnswer":82},"What stability result is proved and under what condition?",{"text":83,"@type":75},"A mode-independent Lyapunov functional provides a pathwise exponential estimate, implying mean-square exponential stability under an explicit small-mismatch 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