{"title":"非线性控制系统的组合算子连续反馈镇定","authors":"Bryce Chistopherson, B. Mordukhovich, F. Jafari","doi":"10.1051/cocv/2022022","DOIUrl":null,"url":null,"abstract":"The ability to asymptotically stabilize control systems through the use of continuous feedbacks is an important topic of control theory and applications. In this paper, we provide a complete characterization of continuous feedback stabilizability using a new approach that does not involve control Lyapunov functions. To do so, we first develop a slight generalization of feedback stabilization using composition operators and characterize continuous stabilizability in this expanded setting. Employing the obtained characterizations in the more general context, we establish relationships between continuous stabilizability in the conventional sense and in the generalized composition operator sense. This connection allows us to show that the continuous stabilizability of a control system is equivalent to the stability of an associated system formed from a local section of the vector field inducing the control system. That is, we reduce the question of continuous stabilizability to that of stability. Moreover, we provide a universal formula describing all possible continuous stabilizing feedbacks for a given system.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Continuous feedback stabilization of nonlinear control systems by composition operators\",\"authors\":\"Bryce Chistopherson, B. Mordukhovich, F. Jafari\",\"doi\":\"10.1051/cocv/2022022\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The ability to asymptotically stabilize control systems through the use of continuous feedbacks is an important topic of control theory and applications. In this paper, we provide a complete characterization of continuous feedback stabilizability using a new approach that does not involve control Lyapunov functions. To do so, we first develop a slight generalization of feedback stabilization using composition operators and characterize continuous stabilizability in this expanded setting. Employing the obtained characterizations in the more general context, we establish relationships between continuous stabilizability in the conventional sense and in the generalized composition operator sense. This connection allows us to show that the continuous stabilizability of a control system is equivalent to the stability of an associated system formed from a local section of the vector field inducing the control system. That is, we reduce the question of continuous stabilizability to that of stability. Moreover, we provide a universal formula describing all possible continuous stabilizing feedbacks for a given system.\",\"PeriodicalId\":50500,\"journal\":{\"name\":\"Esaim-Control Optimisation and Calculus of Variations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2022-03-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Esaim-Control Optimisation and Calculus of Variations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1051/cocv/2022022\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"AUTOMATION & CONTROL SYSTEMS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Control Optimisation and Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/cocv/2022022","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
Continuous feedback stabilization of nonlinear control systems by composition operators
The ability to asymptotically stabilize control systems through the use of continuous feedbacks is an important topic of control theory and applications. In this paper, we provide a complete characterization of continuous feedback stabilizability using a new approach that does not involve control Lyapunov functions. To do so, we first develop a slight generalization of feedback stabilization using composition operators and characterize continuous stabilizability in this expanded setting. Employing the obtained characterizations in the more general context, we establish relationships between continuous stabilizability in the conventional sense and in the generalized composition operator sense. This connection allows us to show that the continuous stabilizability of a control system is equivalent to the stability of an associated system formed from a local section of the vector field inducing the control system. That is, we reduce the question of continuous stabilizability to that of stability. Moreover, we provide a universal formula describing all possible continuous stabilizing feedbacks for a given system.
期刊介绍:
ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations.
Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines.
Targeted topics include:
in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory;
in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis;
in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.