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1

Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces

Wairojjana Nopparat, Pakkaranang Nuttapol, Pholasa Nattawut

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

110--128

EN

In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

2

A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings

Wega Getahun B., Zegeye Habtu, Boikanyo Oganeditse A.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

152--166

EN

The purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

3

Ideal observability for bilinear discrete-time systems with and without delays in observation

Lhous M., Rachik M., Magri E. M.

Archives of Control Sciences

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2018

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Vol. 28, No. 4

601--616

EN

An ideal observability subspace expression is stated for bilinear abstract system with bounded operator in Hilbert spaces. The case of finite dimentional space is also treated. However, it’s noticed that the state ideal observability can never be fulfilled within an infinite dimensional phase space in the case of scalar output. The case of bilinear discrete-time system with delays in observation is also described. To illustrate this work some examples are presented.

4

Certain group dynamical systems induced by Hecke algebras

Cho I.

Opuscula Mathematica

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2016

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Vol. 36, no. 3

337--373

EN

In this paper, we study dynamical systems induced by a certain group [formula] embedded in the Hecke algebra H(Gp) induced by the generalized linear group Gp = GL2(Qp) over the p-adic number fields Qp for a fixed prime p. We study fundamental properties of such dynamical systems and the corresponding crossed product algebras in terms ol free probability on the Hecke algebra H(Gp).

5

Commutativity up to a factor in Banach algebras

Schmoeger C.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

895--900

EN

In this note we explore commutativity up to a factor ab = alfa ba for Hermitian or normal elements of a complex Banach algebra. Our results generalize results obtained for bounded linear operators on Hilbert spaces.

6

Dokładna rekonstrukcja stanu - teoria i przykłady zastosowania

Byrski W.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2003

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T. 7, z. 3

433-453

PL

Prezentowana jest ogólna teoria dokładnej rekonstrukcji skończenie wymiarowego wektora stanu w stacjonarnych układach liniowych, określonych w dowolnych przestrzeniach Hilberta. Dokładna rekonstrukcja stanu realizowana jest poprzez przetwarzanie sygnałów wejścia u i wyjścia y obserwowalnego systemu dynamicznego na zadanym przedziale czasowym T przez obserwator realizujący sumę iloczynu skalarnego pomiaru y i specjalnie dobranej funkcji obserwacji G1 oraz iloczynu skalarnego pomiaru u i specjalnie dobranej funkcji obserwacji G2 w wybranych przestrzeniach Hilberta. Funkcje G tworzące obserwator mają minimalną normę, co gwarantuje dodatkowe własności optymalności obserwatora w zadaniu minimalizacji błędu obserwacji stanu w obecności najbardziej niebezpiecznych znormalizowanych zakłóceń pomiarowych wejścia i wyjścia, przyjmowanych z kul jednostkowych.

EN

The general theory of exact reconstruction of finite dimensional state vector in time invariant linear systems is presented. The systems and signals are defined in Hilbert spaces. The exact reconstruction of the state is realized by processing of input u and output y signals in dynamic observable system on fix finite time interval T. The observer calculates the sum of inner product of y and special chosen observation function G1 and inner product of u and special observation function G2 in given Hilbert spaces. Observation functions G have minimal norm what guarantees the optimality properties of the observer in the task of minimization of observation error under noisy input/output measurement. Namely the most dangerous disturbances with bounded norm normalized to unit balls are assumed.

7

Exact method of solution of torsional vibrations problem for discontinuous viscoelastic shaft with oscillators

Cabański J.

Journal of Technical Physics

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2001

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Vol. 42, no 4

459-474

EN

In this paper a generalized, exact method for solving the problems of torsional vibrations of a non-continuous viscoelastic shaft with oscillators has been presented. The oscillators are attached to the shaft by means of the viscoelastic constraints. The vibrations of this compound system are described by the set of conjugated, partial and ordinary differential equations. The separation of the variables in the complex Hilbert space, the obtained results from the analysis of the boundary-value problems and the proved generalized orthogonality of complex eigenvectors have been used to obtain the solution for free vibrations of the system at the arbitrary initial conditions. Finally, by applying the harmonic moment of the forced vibrations, the problem of the shaft has been solved. In the herein presented method, the operational generalized principles [6] have been applied.