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Circle criterion and boundary control systems in factor form: input-output approach

100%
Grabowski P., Callier F. M.
International Journal of Applied Mathematics and Computer Science
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2001
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tom 11
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nr 6
1387-1403
EN
A circle criterion is obtained for a SISO Lur’e feedback control system consist- ing of a nonlinear static sector-type controller and a linear boundary control system in factor form on an infinite-dimensional Hilbert state space H previ- ously introduced by the authors (Grabowski and Callier, 1999). It is assumed for the latter that (a) the observation functional is infinite-time admissible, (b) the factor control vector satisfies a compatibility condition, and (c) the trans- fer function belongs to H∞ (Π+ ) and satisfies a frequency-domain inequality of the circle criterion type. We also require that the closed-loop system be well- posed, i.e. for any initial state x0 ∈ H the truncated input and output sig- nals uT , yT belong to L2 (0, T ) for any T > 0. The technique of the proof adapts Desoer-Vidyasagar’s circle criterion method (Desoer and Vidyasagar, 1975, Ch. 3, Secs. 1 and 2, pp. 37–43, Ch. 5, Sec. 2, pp. 139–142 and Ch. 6, Secs. 3 and 4, pp. 172–174), and uses the input-output map developed by the authors (Grabowski and Callier, 2001). The results are illustrated by two trans- mission line examples: (a) that of the loaded distortionless RLCG type, and (b) that of the unloaded RC type. The conclusion contains a discussion on improving the results by the loop-transformation technique.
2
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On sets related to maximal clones

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Susanti Y., Denecke K.
Discussiones Mathematicae - General Algebra and Applications
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2012
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tom 32
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nr 1
101-114
EN
For an arbitrary h-ary relation ρ we are interested to express n-clone Polⁿρ in terms of some subsets of the set of all n-ary operations Oⁿ(A) on a finite set A, which are in general not clones but we can obtain Polⁿρ from these sets by using intersection and union. Therefore we specify the concept a function preserves a relation and moreover, we study the properties of this new concept and the connection between these sets and Polⁿρ. Particularly we study $R_{a̲,b}^{n,k}$ for arbitrary partial order relations, equivalence relations and central relations.
3
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Minkowski difference and Sallee elements in an ordered semigroup

100%
Borowska D., Grzybowski J.
Commentationes Mathematicae
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2007
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tom 47
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nr 1
EN
In the manner of Pallaschke and Urbański ([5], chapter 3) we generalize the notions of the Minkowski difference and Sallee sets to a semigroup. Sallee set (see [7], definition of the family \(S\) on p. 2) is a compact convex subset \(A\) of a topological vector space \(X\) such that for all subsets \(B\) the Minkowski difference \(A -B\) of the sets \(A\) and \(B\) is a summand of \(A\). The family of Sallee sets characterizes the Minkowski subtraction, which is important to the arithmetic of compact convex sets (see [5]). Sallee polytopes are related to monotypic polytopes (see [4]). We generalize properties of Minkowski difference and Sallee sets to semigroup and investigate the families of Sallee elements in several specific semigroups.
4
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Fixed point properties for semigroups of nonexpansive mappings on convex sets in dual Banach spaces

75%
Lau A. T.-M., Zhang Y.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2018
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tom 17
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