scholarly journals Solvability of the control problem of the asynchronous spectrum of linear almost periodic systems with a lower triangular representation of the averaging of coefficient matrix

2021 ◽  
Vol 65 (3) ◽  
pp. 263-268
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Almost Periodic ◽  
Average Value ◽  
The Matrix ◽  
Frequency Module ◽  
Necessary And Sufficient ◽  
And Control

A linear control system with an almost periodic matrix of coefficients and control in the form of the feedback linear in phase variables is considered. It is assumed that the feedback coefficient is almost periodic and its frequency module, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix. The system under consideration is studied in the case of a triangular average value of the matrix of coefficients. For the described class of systems, the control problem of the asynchronous spectrum with a target set of frequencies is solved. This task is to construct such a control from an admissible set that the system closed by this control has almost periodic solutions, a set of the Fourier exponents of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for the solvability of this problem are obtained.

scholarly journals Solvability criterion of the control problem of an asynchronous spectrum of linear almost periodic systems with the trivial averaging of the coefficient matrix

2020 ◽  
Vol 63 (6) ◽  
pp. 654-661
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Almost Periodic ◽  
Average Value ◽  
The Matrix ◽  
Solvability Criterion ◽  
Frequency Module ◽  
Necessary And Sufficient

A linear control system with an almost periodic matrix of coefficients and the control in the form of feedback linear in phase variables is considered. It is assumed that the feedback coefficient is almost periodic and its frequency module, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix. The system under consideration is studied in the case of a zero average value of the matrix of coefficients. For the described class of systems, the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies is solved. This task is as follows: to construct such a control from an admissible set so that the system closed by this control has almost periodic solutions, the set of Fourier exponents (frequency spectrum) that are contained in a predetermined subset; the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for solvability of the control problem of the asynchronous spectrum are obtained.

scholarly journals Necessary condition for solvability of the control problem of an asynchronous spectrum of linear almost periodic systems with trivial averaging of the coefficient matrix

2019 ◽  
Vol 55 (2) ◽  
pp. 176-181 ◽  
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Solvability Condition ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Periodic Systems ◽  
Necessary Condition ◽  
Almost Periodic ◽  
Real Numbers ◽  
Frequency Module ◽  
Target Set

We consider a linear control system with an almost periodic matrix of coefficients. The control has a form of feedback and is linear in phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i.e. the smallest additive group of real numbers, including all Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix.The following problem is formulated: choose such a control from an admissible set so that the closed system has almost periodic solutions, the frequency spectrum (a set of Fourier exponents) of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The problem is called the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies.The aim of the work aws to obtain a necessary solvability condition for the control problem of the asynchronous spectrum of linear almost periodic systems with trivial averaging of coefficient matrix The estimate of the power of the asynchronous spectrum was found in the case of trivial averaging of the coefficient matrix.

scholarly journals A sufficient condition for the unsolvability of the control problem of the asynchronous spectrum of linear almost periodic systems with the diagonal averaging of the coefficient matrix

2020 ◽  
Vol 56 (4) ◽  
pp. 391-397
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Periodic Systems ◽  
Sufficient Condition ◽  
Almost Periodic ◽  
Real Numbers ◽  
Average Value ◽  
Target Set ◽  
Special Case

We consider a linear control system with an almost periodic matrix of the coefficients. The control has the form of feedback that is linear on the phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency modulus of the coefficient matrix. The following problem is formulated: choose a control from an admissible set for which the system closed by this control has almost periodic solutions with the frequency spectrum (a set of Fourier exponents) containing a predetermined subset, and the intersection of the frequency modules of solution and the coefficient matrix is trivial. The problem is called as the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with the target set of frequencies. At present, this problem has been studied only in a very special case, when the average value of the almost periodic coefficients matrix of the system is zero. In the case of nontrivial averaging, the question remains open. In the paper, a sufficient condition is obtained under which the control problem of the asynchronous spectrum of linear almost periodic systems with diagonal averaging of the coefficient matrix has no solution.

scholarly journals On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

scholarly journals BOUNDARY CONTROL PROBLEM FOR ONE DEGENERATE HIBERBOLIC EQUATION

2019 ◽  
pp. 19-27
Author(s):  
А.Х. Аттаев
Keyword(s):  
Control Problem ◽  
Boundary Control ◽  
Sufficient Conditions ◽  
Analytical Form ◽  
Cauchy Data ◽  
Boundary Control Problem ◽  
Order Hyperbolic Equation ◽  
Boundary Controls ◽  
Necessary And Sufficient ◽  
Second Order Hyperbolic Equation

В работе изучается задача граничного управления для вырождающегося гиперболического уравнения второго порядка. Установлены необходимые и достаточные условия управляемости данными Коши за минимальный промежуток времени. Граничные управления предъявлены в явном аналитическом виде. The paper studies the boundary control problem for a degenerate second-order hyperbolic equation. Necessary and sufficient conditions are established for minimal time controllability over Cauchy data. Boundary controls are presented in an explicit analytical form.

The algebra of differential forms on a full matric bialgebra

1993 ◽  
Vol 114 (1) ◽  
pp. 111-130 ◽  
Author(s):  
A. Sudbery
Keyword(s):  
Vector Space ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Differential Forms ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Algebra Of Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Necessary And Sufficient

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

2021 ◽  
Vol 71 (6) ◽  
pp. 1375-1400
Author(s):  
Feyzi Başar ◽  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
The Matrix ◽  
Alpha Beta ◽  
Necessary And Sufficient ◽  
Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

Semidirect Product Compactifications

1983 ◽  
Vol 35 (1) ◽  
pp. 1-32
Author(s):  
F. Dangello ◽  
R. Lindahl
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Topological Semigroup ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Semigroups ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

Semigroup Compactifications of Direct and Semidirect Products

1983 ◽  
Vol 26 (2) ◽  
pp. 233-240 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Classical Result ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Products ◽  
Almost Periodic ◽  
Semidirect Products ◽  
Semigroup Compactifications ◽  
Necessary And Sufficient

AbstractA classical result of I. Glicksberg and K. de Leeuw asserts that the almost periodic compactification of a direct product S × T of abelian semigroups with identity is (canonically isomorphic to) the direct product of the almost periodic compactiflcations of S and T. Some efforts have been made to generalize this result and recently H. D. Junghenn and B. T. Lerner have proved a theorem giving necessary and sufficient conditions for an F-compactification of a semidirect product S⊗σT to be a semidirect product of compactiflcations of S and T. A different such theorem is presented here along with a number of corollaries and examples which illustrate its scope and limitations. Some behaviour that can occur for semidirect products, but not for direct products, is exposed

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