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1
Content available Comparison of direct and perturbation approaches to analysis of infinite-dimensional feedback control systems
Grabowski Piotr
International Journal of Applied Mathematics and Computer Science
|
2021
|
Vol. 31, no. 2
195--218
EN
For general boundary control systems in factor form some necessary and sufficient conditions for generation of an analytic exponentially stable semigroup are proposed in both direct and perturbation forms for comparison. The direct approach is applicable to operators with the numerical range satisfying certain additional conditions. In particular, it applies to operators similar to convexoids and therefore it generalizes previous results known for hyponormal operators. The perturbation result (indirect approach) is derived and formulated as an exponential stability robustness result using the frequence-domain considerations. It is expressed in terms of some estimates of the resolvent growth over the open right complex half-plane and compared with some recent results. The analysis is illustrated in detail with examples of an unloaded and loaded electric RC-transmission line with proportional negative feedback.
2
Content available remote Reversible Regular Languages : Logical and Algebraic Characterisations
Gastin Paul, Manuel Amaldev, Govind R.
Fundamenta Informaticae
|
2021
|
Vol. 180, nr 4
333--350
EN
We present first-order (FO) and monadic second-order (MSO) logics with predicates ‘between’ and ‘neighbour’ that characterise the class of regular languages that are closed under the reverse operation and its subclasses. The ternary between predicate bet(x, y, z) is true if the position y is strictly between the positions x and z. The binary neighbour predicate N(x, y) is true when the the positions x and y are adjacent. It is shown that the class of reversible regular languages is precisely the class definable in the logics MSO(bet) and MSO(N). Moreover the class is definable by their existential fragments EMSO(bet) and EMSO(N), yielding a normal form for MSO formulas. In the first-order case, the logic FO(bet) corresponds precisely to the class of reversible languages definable in FO(<). Every formula in FO(bet) is equivalent to one that uses at most 3 variables. However the logic FO(N) defines only a strict subset of reversible languages definable in FO(+1). A language-theoretic characterisation of the class of languages definable in FO(N), called locally-reversible threshold-testable (LRTT), is given. In the second part of the paper we show that the standard connections that exist between MSO and FO logics with order and successor predicates and varieties of finite semigroups extend to the new setting with the semigroups extended with an involution operation on its elements. The case is different for FO(N) where we show that one needs an additional equation that uses the involution operator to characterise the class. While the general problem of characterising FO(N) is open, an equational characterisation is shown for the case of neutral letter languages.
3
Content available The LQ/KYP problem for infinite-dimensional systems
Grabowski P.
Opuscula Mathematica
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2017
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Vol. 37, no. 1
21--64
EN
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.
4
Content available remote An Efficient Equivalence-checking Algorithm for a Model of Programs with Commutative and Absorptive Statements
Podymov V.
Fundamenta Informaticae
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2016
|
Vol. 147, nr 2/3
315--336
EN
For many program analysis problems it is useful to have means to efficiently prove that given programs have similar (equivalent) behaviors. Unfortunately, in most cases to prove the behavioral equivalence is an undecidable problem. A common way to overcome such undecidability is to consider a model of programs with an abstract semantics based on the real one, in which only some simple properties are captured, and to provide an efficient equivalence-checking algorithm for the model. We focus on two kinds of properties of data-modifying statements of imperative programs. Statements a and b are commutative, if the execution of sequences ab and ba lead to the same result. A statement b is (left-)absorptive for a statement a, if the execution of sequences ab and b lead to the same result. We consider propositional program models in which commutativity and absorption properties are caprtured (CA-models). Formally, data states for a CA-model are elements of a monoid over the set of statement symbols, defined by an arbitrary set of relations of the form ab = ba (for commutativity) and ab = b (for absorption). We propose an equivalence-checking algorithm for CA-models based on (what we call) progressive monoids. The algorithm terminates in time polynomial in size of programs. As a consequence, we prove a polynomial-time decidability for the equivalence problem in such CA-models.
5
Content available The lq-controller synthesis problem for infinite-dimensional systems in factor form
Grabowski P.
Opuscula Mathematica
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2013
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Vol. 33, no. 1
29-79
EN
The general lq-problem with infinite time horizon for well-posed infinite-dimensional systems has been investigated by George Weiss and Martin Weiss and by Olof Staffans with a complement by Kalle Mikkola and Olof Staffans. Our aim in this paper is to present a solution of a general lq-optimal controller synthesis problem for infinite-dimensional systems in factor form. The systems in factor form are an alternative to additive models, used in the theory of well-posed systems, which rely on leading the analysis exclusively within the basic state space. As a result of applying the simplified analysis in terms of the factor systems and an another derivation technique, we obtain an equivalent, however, astonishingly not the same formulae expressing the optimal controller in the time-domain and the method of spectral factorization. The results are illustrated by two examples of the construction of both the optimal control and optimal controller for some standard lq-problems met in literature: a control problem for a class of boundary controlled hyperbolic equations initiated by Chapelon and Xu, to which we give full solution and an example of the synthesis of the optimal control/controller for the standard lq-problem with infinite-time horizon met in the problem of improving a river water quality by artificial aeration, proposed by Zołopa and the author.
6
Content available remote Sierpiński’s rank for groups and semigroups
Mitchell J. D., Péresse Y.
Wiadomości Matematyczne
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2012
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T. 48, Nr 2
209--215
7
Content available remote Asymtotic stability of solutions to the equations of linear elasticity and thermoelasticity in viscoporus media
Głowiński P., Łada A.
Demonstratio Mathematica
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2009
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Vol. 42, nr 4
757-779
EN
The systems of evolution equations modelling elasticity and thermoelas-ticity of viscoporous bounded media are considered. The existence of co-semigroups of contractions defining solutions to the systems is proved. The asymptotic vanishing of energies of solutions when t -> (infinity) is explained.
8
Content available remote Intrinsic ultracontractivity for Lévy processes
Grzywny T.
Probability and Mathematical Statistics
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2008
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Vol. 28, Fasc. 1
91--106
EN
We prove the intrinsic ultracontractivity for semigroups generated by a large class of symmetric Lévy processes killed on exiting a bounded and connected Lipschitz set under some conditions about the behavior of the Lévy measure in the neighborhood of the origin.
9
Content available The motion planning problem and exponential stabilization of a heavy chain. Part II
Grabowski P.
Opuscula Mathematica
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2008
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Vol. 28, no. 4
481-505
EN
This is the second part of paper [8], where a model of a heavy chain system with a punctual load (tip mass) in the form of a system of partial differential equations was interpreted as an abstract semigroup system and then analysed on a Hilbert state space. In particular, in [8] we have formulated the problem of exponential stabilizability of a heavy chain in a given position. It was also shown that the exponential stability can be achieved by applying a stabilizer of the colocated-type. The proof used the method of Lyapunov functionals. In the present paper, we give other two proofs of the exponential stability, which provides an additional intrinsic insight into the exponential stabilizability mechanism. The first proof makes use of some spectral properties of the system. In the second proof, we employ some relationships between exponential stability and exact observability.
10
Content available Dynamical model of propagation of pollutants in a river
Żołopa E., Grabowski P.
Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie
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2008
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T. 12, z. 2
169-182
EN
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.
PL
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.
11
Content available Well-posedness and stability analysis of hybrid feedback systems using Shkalikov's theory
Grabowski P.
Opuscula Mathematica
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2006
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Vol. 26, no. 1
45-97
EN
The modern method of analysis of the distributed parameter systems relies on the transformation of the dynamical model to an abstract differential equation on an appropriately chosen Banach or, if possible, Hilbert space. A linear dynamical model in the form of a first order abstract differential equation is considered to be well-posed if its right-hand side generates a strongly continuous semigroup. Similarly, a dynamical model in the form of a second order abstract differential equation is well-posed if its right-hand side generates a strongly continuous cosine family of operators. Unfortunately, the presence of a feedback leads to serious complications or even excludes a direct verification of assumptions of the Hille-Phillips-Yosida and/or the Sova-Fattorini Theorems. The class of operators which are similar to a normal discrete operator on a Hilbert space describes a wide variety of linear operators. In the papers [12, 13] two groups of similarity criteria for a given hybrid closed-loop system operator are given. The criteria of the first group are based on some perturbation results, and of the second, on the application of Shkalikov's theory of the Sturm-Liouville eigenproblems with a spectral parameter in the boundary conditions. In the present paper we continue those investigations showing certain advanced applications of the Shkalikov's theory. The results are illustrated by feedback control systems examples governed by wave and beam equations with increasing degree of complexity of the boundary conditions.
12
Content available Some analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations. Part. 1
Pytel-Kudela M., Prykarpatsky A.K.
Opuscula Mathematica
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2006
|
Vol. 26, no. 1
131-136
EN
The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bilinear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed.
13
Content available remote On subsemigroups of the L1/s
Jabłoński W.
Demonstratio Mathematica
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2006
|
Vol. 39, nr 4
815-823
EN
In this paper we consider subsemigroups of the group L1/s such that the r–th parameter of xs is the function of the remaining ones. Moreover we generalize some results concerning the existence of certain form subsemigroups of the group L1/s.
14
Content available remote On residualities in the set of Markov continuous semigroups on C1
Kuna B.
Demonstratio Mathematica
|
2006
|
Vol. 39, nr 2
439-453
EN
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.
15
Content available remote L'equation de translation sur le demi-groupe des elements non negatifs d'un groupe ordonne et archimedien
Mach A., Moszner Z.
Demonstratio Mathematica
|
2006
|
Vol. 39, nr 4
825-836
16
Content available remote Cocycles of disjoint continuous iteration semigroups on the circle
Ciepliński K.
Demonstratio Mathematica
|
2006
|
Vol. 39, nr 3
519-522
EN
In this note we give the form ofcocycles of disjoint and continuous iteration semigroups on the unit circle.
17
On an interpretation of globularity of higher graphs
Bartosik J.
Zeszyty Naukowe Politechniki Świętokrzyskiej. Elektryka
|
2005
|
Z. 42
39-46
EN
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.
PL
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.
18
Content available remote Von inversen Operatoren erzeugte Einparameter-Halbgruppen
Scheffold E.
Demonstratio Mathematica
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2004
|
Vol. 37, nr 4
893--904
19
Content available remote The null-exact controllability of the generalized wave problem governed by once integrated semigroups
Kowalski T., Sadkowski W.
Demonstratio Mathematica
|
2004
|
Vol. 37, nr 3
745--760
EN
The existence and the uniqueness of an integrated solution of the generalized wave problem were proved by using once integrated semigroups. Also a null-exact controllability of the generalized wave system was obtained.
20
Content available remote Groupoids, their representations and imprimitivity systems
Pysiak L.
Demonstratio Mathematica
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2004
|
Vol. 37, nr 3
661--670
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