{"title":"片断确定性马尔可夫过程的自适应平均控制","authors":"","doi":"10.1016/j.sysconle.2024.105894","DOIUrl":null,"url":null,"abstract":"<div><p>The main goal of this paper is to study the adaptive infinite-horizon average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) for the case in which there are no boundary jumps. It is assumed that the control acts continuously on the jump intensity <span><math><mi>λ</mi></math></span> and on the transition measure <span><math><mi>Q</mi></math></span> of the process, and that jump parameters (<span><math><mi>λ</mi></math></span> and <span><math><mi>Q</mi></math></span>), as well as the cost <span><math><mi>C</mi></math></span>, depend on an unknown parameter <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>. It is shown that the principle of estimation and control holds, that is, the strategy consisting of choosing, at each stage <span><math><mi>n</mi></math></span>, an action according to an optimal stationary policy, where the true but unknown parameter <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> is replaced by its estimated value <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span>, is average optimal, provided that the sequence of estimators <span><math><mrow><mo>{</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>}</mo></mrow></math></span> of <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> is strongly consistent, that is, <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> converge to <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> almost surely.</p></div>","PeriodicalId":49450,"journal":{"name":"Systems & Control Letters","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Adaptive average control for piecewise deterministic Markov processes\",\"authors\":\"\",\"doi\":\"10.1016/j.sysconle.2024.105894\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The main goal of this paper is to study the adaptive infinite-horizon average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) for the case in which there are no boundary jumps. It is assumed that the control acts continuously on the jump intensity <span><math><mi>λ</mi></math></span> and on the transition measure <span><math><mi>Q</mi></math></span> of the process, and that jump parameters (<span><math><mi>λ</mi></math></span> and <span><math><mi>Q</mi></math></span>), as well as the cost <span><math><mi>C</mi></math></span>, depend on an unknown parameter <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>. It is shown that the principle of estimation and control holds, that is, the strategy consisting of choosing, at each stage <span><math><mi>n</mi></math></span>, an action according to an optimal stationary policy, where the true but unknown parameter <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> is replaced by its estimated value <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span>, is average optimal, provided that the sequence of estimators <span><math><mrow><mo>{</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>}</mo></mrow></math></span> of <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> is strongly consistent, that is, <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup></math></span> converge to <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> almost surely.</p></div>\",\"PeriodicalId\":49450,\"journal\":{\"name\":\"Systems & Control Letters\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-08-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Systems & Control Letters\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0167691124001828\",\"RegionNum\":3,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"AUTOMATION & CONTROL SYSTEMS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Systems & Control Letters","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167691124001828","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
Adaptive average control for piecewise deterministic Markov processes
The main goal of this paper is to study the adaptive infinite-horizon average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) for the case in which there are no boundary jumps. It is assumed that the control acts continuously on the jump intensity and on the transition measure of the process, and that jump parameters ( and ), as well as the cost , depend on an unknown parameter . It is shown that the principle of estimation and control holds, that is, the strategy consisting of choosing, at each stage , an action according to an optimal stationary policy, where the true but unknown parameter is replaced by its estimated value , is average optimal, provided that the sequence of estimators of is strongly consistent, that is, converge to almost surely.
期刊介绍:
Founded in 1981 by two of the pre-eminent control theorists, Roger Brockett and Jan Willems, Systems & Control Letters is one of the leading journals in the field of control theory. The aim of the journal is to allow dissemination of relatively concise but highly original contributions whose high initial quality enables a relatively rapid review process. All aspects of the fields of systems and control are covered, especially mathematically-oriented and theoretical papers that have a clear relevance to engineering, physical and biological sciences, and even economics. Application-oriented papers with sophisticated and rigorous mathematical elements are also welcome.