[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84451-en":3,"doc-seo-84451-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84451,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Bang-Bang Evasion Its Stochastic Optimality and a Terminal-Set-Based Implementation","Bang-bang evasion strategy for a planar endgame interception is formulated as an inherently stochastic optimal control problem with imperfect information and bounded accelerations. The guidance law is derived under the generalized separation theorem, using the posterior distribution of the engagement state. The work proves existence of an optimal evasion strategy and shows that at least one optimal solution is bang-bang, yielding a finite-dimensional formulation. A terminal-set-based evasion (TSE) closed-loop strategy is then proposed and evaluated in simulation, outperforming stochastic random-telegraph, Singer, and weaving approaches.","arXiv :2511 .21633v2 [ ee ss . SY] 13 Jul 2026  \nBang-Bang Evasion: Its Stochastic Optimality and a Terminal-Set-Based Implementation  \nLiraz Mudrik∗ and Yaakov Oshman†  \nTechnion—Israel Institute of Technology, Haifa, 3200003, Israel  \nWe address the problem of optimal evasion in a planar endgame engagement, where a target with bounded lateral acceleration seeks to avoid interception by a missile guided by a linear feedback law. Contrary to existing approaches, that assume perfect information or use heuristic maneuver models in stochastic settings, we formulate the problem in an inherently stochastic framework involving imperfect information and bounded controls. Complying with the generalized separation theorem, the control law factors in the posterior distribution of the state. We extend the well-known optimality of bang-bang evasion maneuversin deterministic settings to the realm of realistic, stochastic evasion scenarios. First, we prove that an optimal evasion strategy always exists, and that theset of optimal solutions includes at least one bang-bang policy, rendering the resulting optimal control problem finite-dimensional. Second, leveraging this structure, we propose the closed-loop terminal-set-based evasion (TSE) strategy, and demonstrate its effectiveness in simulation against a proportional navigation pursuer. Monte Carlo simulations show that the TSE strategy outperforms traditional stochastic evasion strategies based on random telegraph, Singer, and weaving models.  \n∗This work is part ofDr. Mudrik’s Doctoral Research, Stephen B. Klein Faculty of Aerospace Engineering; currently Postdoctoral Fellow, Department of Mechanical and Aerospace Engineering, Naval Postgraduate School, Monterey, CA 93943; [liraz109@gmail.com](liraz109@gmail.com. Member AIAA)[. Member AIAA](liraz109@gmail.com. Member AIAA) (Corresponding Author).  \n†Professor Emeritus, Stephen B. Klein Faculty of Aerospace Engineering; [yaakov.oshman@technion.ac.il](yaakov.oshman@technion.ac.il). Fellow AIAA.  \nNomenclature  \nx = engagement state vector  \n􀁢, 􀁢¤ = relative displacement and velocity normal to the initial LOS  \nq 􀀢 , q􀀩 = interceptor and target internal-dynamics state vectors  \n`􀀰􀀢 , `􀀰􀀩 = interceptor and target accelerations normal to the initial LOS  \n􀁄 􀀩, 􀁄ax = target acceleration command and its bound  \n􀁄 􀀢 , ˆ􀁄􀀢 = interceptor acceleration command and its target-side estimate F􀀺 , g 􀀢 , g􀀩 = discrete transition and input matrices  \nK􀀺 = interceptor guidance-gain matrix  \nC = terminal output-selection matrix  \n􀀵 , 􀁃􀀺􀀶􀀾 = terminal step and time-to-go, both random  \n􀀿 􀀵 (·) = probability mass function of the terminal step 􀀥􀀺􀀹 = mode probability of guidance law 􀀿 􀀹  \n􀁉, N = zero-effort miss and navigation gain  \na􀀽 (􀀸, 􀀹 ) , z􀀽􀀸, 􀀹 = input-to-terminal gain and command-independent terminal term 􀀭 􀀽􀀸, 􀀹 , 􀀆 􀀽􀀸, 􀀹 = mean and covariance of z􀀽􀀸, 􀀹  \n􀀨 􀀽 = terminal-set shaping function  \n􀀶 (·) , 􀀟 = terminal cost and performance index  \n􀁛 = uncertain interceptor-dynamics parameters  \n􀁧􀀢, 􀁧􀀩 = interceptor and target dynamics time constants 􀁬 􀀺 , 􀁡 􀀺 = process and measurement noise  \nΔ􀁃 = sampling interval  \nI. Introduction  \nAt the terminal phase of an interception scenario, a defending aerial vehicle must perform evasive maneuvers to avoid an attacking missile. This work considers such engagements in the commonly used planar lateral engagement, where the interceptor uses a state-feedback guidance law to home in on a maneuvering target. The optimal design of interceptor guidance laws has been extensively studied under deterministic settings, leading to canonical strategies such as proportional navigation (PN), augmented PN (APN), the optimal guidance law (OGL) [1], and the linear-quadratic differential game (LQDG) guidance law [2] . These strategies arise from linearized engagement dynamics and quadratic cost functionals and thus induce linear acceleration commands.  \nFrom the evader’s perspective, the optimal maneuver against such linear guidance","cbCaidrLs0gdYPav","https://ap.wps.com/l/cbCaidrLs0gdYPav","pdf",718938,1,31,"English","en",105,"# Introduction\n## Background in deterministic bang-bang evasion\n## Motivation for stochastic evasion under noisy sensing","[{\"question\":\"What evasion problem does the document address?\",\"answer\":\"It addresses optimal evasion in a planar endgame engagement where a target with bounded lateral acceleration avoids interception by a missile using linear feedback guidance.\"},{\"question\":\"How does the work incorporate uncertainty?\",\"answer\":\"It formulates the evasion as an inherently stochastic problem with imperfect information, so the control law depends on the posterior distribution of the state.\"},{\"question\":\"What is the proposed terminal-set-based evasion (TSE) strategy and how does it perform?\",\"answer\":\"The document proposes a closed-loop terminal-set-based evasion strategy and evaluates it in simulation against a proportional navigation pursuer. Monte Carlo results show TSE outperforms stochastic strategies based on random telegraph, Singer, and weaving models.\"}]",1784195697,78,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"bang-bang-evasion-its-stochastic-optimality-and-a-terminal-set-based-implementation","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/bang-bang-evasion-its-stochastic-optimality-and-a-terminal-set-based-implementation/84451/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 evasion problem does the document address?","Question",{"text":75,"@type":76},"It addresses optimal evasion in a planar endgame engagement where a target with bounded lateral acceleration avoids interception by a missile using linear feedback guidance.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the work incorporate uncertainty?",{"text":80,"@type":76},"It formulates the evasion as an inherently stochastic problem with imperfect information, so the control law depends on the posterior distribution of the state.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the proposed terminal-set-based evasion (TSE) strategy and how does it perform?",{"text":84,"@type":76},"The document proposes a closed-loop terminal-set-based evasion strategy and evaluates it in simulation against a proportional navigation pursuer. Monte Carlo results show TSE outperforms stochastic strategies based on random telegraph, Singer, and weaving models.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]