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On sets related to maximal clones

100%

Susanti Y. , Denecke K.

Discussiones Mathematicae - General Algebra and Applications

2012

tom 32

nr 1

101-114

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.

The LQ/KYP problem for infinite-dimensional systems

100%

Grabowski P.

2017

tom Vol. 37, no. 1

21--64

Our aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur’e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system.

On the circle criterion for boundary control systems in factor form

100%

Grabowski P.

tom Vol. 23

25-47

In this paper we return to the origins of the circle criterion initiated by Irwin Sandberg nearly forthy years ago. A version of the Leray-Schauder alternative is applied to get an existence of an abstract Hammerstein output equation for the closed-loop system. This existence result completes Sandberg's method based on using the Banach fixed-point theorem. It is shown that the assertion of the circle criterion can be strengthened by adding a characterization of an asymptotic behaviour of the state trajectories. Results are being compared with a recent version of the circle criterion for boundary control systems in factor form. Some prospects for further studies are also suggested.

Dynamical model of propagation of pollutants in a river

75%

Żołopa E. , Grabowski P.

2008

tom T. 12, z. 2

169-182

In this paper a dynamical model of propagation of pollutants in a river with M point controls in the form of aerators and K point measurements is being transformed to an abstract model on a suitably chosen Hilbert space. Our model belongs to the class of abstract models of the factor-type. It is shown that the semigroup generated by the state operator A has a property of decaying in a finite-time, the observation operator is admissible, and the system transfer function is in the space H∞ (C+, L(CM, CK)). In the final part we also formulate the LQ problem with infinite-time horizon.

W artykule przekształcany jest model dynamiczny rozprzestrzeniania się zanieczyszczeń w rzece, z punktowymi sterowaniami w postaci M aeratorów i K punktami pomiarowymi, do modelu abstrakcyjnego na odpowiednio dobranej przestrzeni Hilberta. Model abstrakcyjny jest typu sfaktoryzowanego. Pokazano, że półgrupa generowana przez operator stanu A ma własność zanikania w skończonym czasie, operator obserwacji jest dopuszczalny i transmitancja systemu należy do przestrzeni H∞ (C+, L(CM, CK)). W końcowej części pracy formułuje się problem liniowo-kwadratowy z nieskończonym horyzontem czasowym.

On an interpretation of globularity of higher graphs

75%

Bartosik J.

Zeszyty Naukowe Politechniki Świętokrzyskiej. Elektryka

2005

tom Z. 42

39-46

In [7] a common structure of various "stepwise" decomposition has been described. Such decompositions are represented by the so called n-hypergraphs being generalizations of the well known globular sets (see [2] or [3]). In this paper an example of such n-hypergraphs is presented. The example is well known from the theory of (object oriented) programming, data bases and linear algebra.

W pracy [7] została opisana pewna wspólna struktura rozmaitych "krok po kroku". Rozkłady takie są reprezentowane przez tzw. n-hipergrafy będące uogólnieniami zbiorów globularnych (patrz [2] lub [3]). W pracy pokazano przykład takiego n-hipergrafu znanego z teorii (obiektowego) projektowania baz danych i algebry liniowej.

On residualities in the set of Markov continuous semigroups on C1

63%

Kuna B.

2006

tom Vol. 39, nr 2

439-453

We show that the set of all stochastic strongly continuous semigroups on C1 such that limt-oo |||T(t) - Qx*||| = 0, where Qx* is one-dimensional projection for some state X*, is norm open and dense. Moreover this set forms a norm dense Gb if a state X* is strictly positive.