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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 The LQ/KYP problem for infinite-dimensional systems
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.
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.
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.
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.
7
Content available remote Intrinsic ultracontractivity for Lévy processes
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.
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.
9
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.
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.
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.
12
Content available remote On subsemigroups of the L1/s
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.
13
Content available remote On residualities in the set of Markov continuous semigroups on C1
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 Cocycles of disjoint continuous iteration semigroups on the circle
EN
In this note we give the form ofcocycles of disjoint and continuous iteration semigroups on the unit circle.
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.
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 On the circle criterion for boundary control systems in factor form
EN
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.
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