@article{c63e4325db764f30989bbf332f01d6b0,
title = "Extended spaces and the resolution topology",
abstract = "The extended L2e space used to define stability and instability for systems defined on an L2 space is shown to be the completion of the given L2 space in an appropriate topology. This, in turn, allows the extended space concept to be generalized to un arbitrary Hilbert resolution space by taking its completion in the appropriate resolution topology.",
author = "Avraham Feintuch and R. Saeks",
note = "Funding Information: 1. Introduction Over the past quarter century a number of researchers have attemptcd to formulate a theory of feedback systems in a norrned space setting. Although many approaches have been tried, these researchers have invariably encountered the same obstacle. Since every signal in a normcd space is bounded, any system which is modelled by an operator on that space is inherently , stable'. Moreover, every such signal 'asymptotically' approaches zero. One cannot, therefore, give a viable definition of stability in a norrned space 'setting, nor can one formulate the concepts of asymptotic tracking disturbance rejection (Callier and Desoer 1978, Vidyasagar et al, 1980). In the mid-1960s, two schools of thought developed. One approach was to embed the given norrned space into an appropriately constructed extension space which included unbounded signals, and then to define stability on the extended space (Callier and Desoer 1978, Willems 1971). The second approach was simply to define a stable system to be causal and bounded in the gh'en normed space (Damborg and Naylor 1970). Although there was considerable controversy over the relative merits of the two approaches, it was eventually shown that they were essentially equivalent, and a powerful stability theory has since been developed (Willems 1971). The precise relationship between the stable operators on an extension space and the causal bounded operators on a normed space, however, remains unclear, while a normed space formulation of the asymptotic tracking and dist.urba.nce rejection problems has yet to be given. The plll'pose of the present paper is to lay the founda.tions for an investigation of the feedback system design problem (Dcsoer et al: l!)77, Sacks and Murray 1979) in a Hilbert resolution space setting (Sacks 1973). These foundations include the formulation of an appropriate extended resolution space for a given abstract Hilbert resolution space and a precise charucterization of the relationship between the stable operators on the extended space and the causal bounded operators on the Hilbert space. Although we are Received 21 July 1980. t This research was supported in part by NSF Grant ENG-79-11315. :j: On leave from the Department of Mathematics, Ben Gurian University, Beer Sheva, Israel. § Department of Electrical Engineering, Texas Tech. University, Lubbock, Tx. 79409, U.S.A.",
year = "1981",
month = jan,
day = "1",
doi = "10.1080/00207178108922927",
language = "English",
volume = "33",
pages = "347--354",
journal = "International Journal of Control",
issn = "0020-7179",
publisher = "Taylor and Francis Ltd.",
number = "2",
}