{"title":"因果平移不变算子的算子值乘数定理及其在控制理论输入输出稳定性中的应用","authors":"Chris Guiver, Hartmut Logemann, Mark R. Opmeer","doi":"10.1007/s00498-024-00387-4","DOIUrl":null,"url":null,"abstract":"<p>We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from <span>\\(H^\\alpha ({\\mathbb {R}},U)\\)</span> to <span>\\(H^\\beta ({\\mathbb {R}},U)\\)</span> (fractional <i>U</i>-valued Sobolev spaces, <i>U</i> a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that <i>U</i> is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.</p>","PeriodicalId":51123,"journal":{"name":"Mathematics of Control Signals and Systems","volume":"35 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability\",\"authors\":\"Chris Guiver, Hartmut Logemann, Mark R. Opmeer\",\"doi\":\"10.1007/s00498-024-00387-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from <span>\\\\(H^\\\\alpha ({\\\\mathbb {R}},U)\\\\)</span> to <span>\\\\(H^\\\\beta ({\\\\mathbb {R}},U)\\\\)</span> (fractional <i>U</i>-valued Sobolev spaces, <i>U</i> a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that <i>U</i> is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.</p>\",\"PeriodicalId\":51123,\"journal\":{\"name\":\"Mathematics of Control Signals and Systems\",\"volume\":\"35 1\",\"pages\":\"\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematics of Control Signals and Systems\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1007/s00498-024-00387-4\",\"RegionNum\":4,\"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":"Mathematics of Control Signals and Systems","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1007/s00498-024-00387-4","RegionNum":4,"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}
引用次数: 0
摘要
我们证明了因果平移不变线性算子的算子值拉普拉斯乘数定理,它提供了从\(H^\alpha ({\mathbb {R}},U)\) 到\(H^\beta ({\mathbb {R}}. U)\的连续性特征、U)\) (分数 U 值 Sobolev 空间,U 是复希尔伯特空间)的转移函数(或符号)的有界性,它是复平面右半部分上的一个算子值全态函数。我们确定了这种有界性等价于转移函数边界函数的类似性质的充分条件。在 U 可分离的假设下,拉普拉斯乘数定理被用来推导傅立叶乘数定理。我们为一大类因果平移不变线性算子开发了一个新颖的输入输出稳定性框架,完善了现有的输入输出稳定性理论,从而将其应用于数学控制理论。此外,我们还展示了我们的工作是如何与问题解决线性系统理论和算子半群多项式稳定性结果相联系的。我们还详细讨论了几个实例。
Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability
We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from \(H^\alpha ({\mathbb {R}},U)\) to \(H^\beta ({\mathbb {R}},U)\) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.
期刊介绍:
Mathematics of Control, Signals, and Systems (MCSS) is an international journal devoted to mathematical control and system theory, including system theoretic aspects of signal processing.
Its unique feature is its focus on mathematical system theory; it concentrates on the mathematical theory of systems with inputs and/or outputs and dynamics that are typically described by deterministic or stochastic ordinary or partial differential equations, differential algebraic equations or difference equations.
Potential topics include, but are not limited to controllability, observability, and realization theory, stability theory of nonlinear systems, system identification, mathematical aspects of switched, hybrid, networked, and stochastic systems, and system theoretic aspects of optimal control and other controller design techniques. Application oriented papers are welcome if they contain a significant theoretical contribution.