Pointwise Degeneracy Research Articles

Analysis of the pointwise completeness and the pointwise degeneracy of the standard and fractional descriptor linear systems and electrical circuits

The Drazin inverse of matrices is applied to analysis of the pointwise completeness and the pointwise degeneracy of the descriptor standard and fractional linear continuous-time and discrete-time systems. It is shown that: 1) The descriptor linear continuous-time system is pointwise complete if and only if the initial and final states belong to the same subspace. 2) The descriptor linear discrete-time system is not pointwise complete if its system matrix is singular. 3) System obtained by discretization of continuous-time system is always not pointwise complete. 4) The descriptor linear continuous-time system is not pointwise degenerated in any nonzero direction for all nonzero initial conditions. 5) The descriptor fractional system is pointwise complete if the matrix defined by (36) is invertible. 6) The descriptor fractional system is pointwise degenerated if and only if the condition (41) is satisfied. Considerations are illustrated by examples of descriptor linear electrical circuits.

Geometric Phase Effect in Attosecond Stimulated X-ray Raman Spectroscopy

Conical intersections (CIs) are diabolical points in the potential energy surfaces generally caused by point-wise degeneracy of different electronic states, and give rise to the geometric phases (GPs) of molecular wave functions. Here we theoretically propose and demonstrate that the transient redistribution of ultrafast electronic coherence in attosecond Raman signal (TRUECARS) spectroscopy is capable of detecting the GP effect in excited state molecules by applying two probe pulses including an attosecond and a femtosecond X-ray pulse. The mechanism is based on a set of symmetry selection rules in the presence of nontrivial GPs. The model of this work can be realized for probing the geometric phase effect in the excited state dynamics of complex molecules with appropriate symmetries, using attosecond light sources such as free-electron X-ray lasers.

The pointwise completeness and the pointwise degeneracy of linear discrete-time different fractional order systems

The pointwise completeness and the pointwise degeneracy of linear discrete-time different fractional order systems

Application of the Drazin inverse to the analysis of pointwise completeness and pointwise degeneracy of descriptor fractional linear continuous-time systems

Abstract The Drazin inverse of matrices is applied to the analysis of pointwise completeness and pointwise degeneracy of fractional descriptor linear continuous-time systems. It is shown that (i) d...

Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders

Abstract Descriptor and standard linear continuous-time systems with different fractional orders are investigated. Descriptor systems are analyzed making use of the Drazin matrix inverse. Necessary...

The pointwise completeness and the pointwise degeneracy of fractional descriptor discrete-time linear systems

The pointwise completeness and the pointwise degeneracy of fractional descriptor discrete-time linear systems

Punktowa zupełność oraz punktowa degeneracja wybranej klasy układów dyskretnych singularnych niecałkowitego rzędu

Punktowa zupełność oraz punktowa degeneracja wybranej klasy układów dyskretnych singularnych niecałkowitego rzędu

Pointwise completeness and pointwise degeneracy of positive fractional descriptor continuous-time linear systems with regular pencils

Abstract Pointwise completeness and pointwise degeneracy of positive fractional descriptor continuous-time linear systems with regular pencils are addressed. Conditions for pointwise completeness and pointwise degeneracy of the systems are established and illustrated by an example.

Pointwise completeness and pointwise degeneracy of fractional descriptor continuous-time linear systems with regular pencils

Abstract Pointwise completeness and pointwise degeneracy of the fractional descriptor continuous-time linear systems with regular pencils are addressed. Conditions for the pointwise completeness and pointwise degeneracy of the systems are established and illustrated by an example.

Pointwise completeness and pointwise degeneracy of 2D standard and positive Fornasini‐Marchesini models

PurposeThe purpose of this paper is to formulate and solve the pointwise completeness and pointwise degeneracy of standard and positive 2D Fornasini‐Marchesini and 2D Roesser models.Design/methodology/approachA procedure is proposed for investigation of the pointwise completeness and pointwise degeneracy of 2D standard and positive Fornasini‐Marchesini and Roesser models.FindingsNecessary and sufficient conditions for pointwise completeness and pointwise degeneracy of standard and positive Fornasini‐Marchesini and Roesser models are established.Originality/valueThe paper proposes a method for investigation of the pointwise completeness and pointwise degeneracy of the standard and positive Fornasini‐Marchesini and Roesser models.

Pointwise completeness and pointwise degeneracy of standard and positive hybrid linear systems described by the general model

Pointwise completeness and pointwise degeneracy of standard and positive hybrid linear systems described by the general model

Computation of the state transition matrix for a class of delay-differential systems†

Systems which can be modelled by a linear constant coefficient delay-differential equation of the form [xdot](t) = A0x(t) + A1,.x(t— h) are studied, where,x(r)∊Rn is the instantaneous state at time t, h > 0 is the delay and matrices A0 and A1 are of appropriate dimensions. A method for the computation of the state transition matrix in the closed form for this class of systems is given. We then analyse the form of the State transition matrix for the delay-differential systems with finite spectrum. Finally, the problem of pointwise degeneracy for such systems is discussed.

Function Space Controllability of Linear Retarded Systems: A Derivation from Abstract Operator Conditions

Controllability of linear retarded systems is investigated by using the abstract representation of such systems given by $\dot x = \tilde Ax + \tilde Bu$, where x belongs to Hilbert space $R^n \times L_2 ([ - h,0],R^n )$ denoted as $M_2 $, and $\tilde A$ generates a $C_0 $-semigroup. It is shown that useful, practically verifiable conditions can be obtained by this approach. The following problems are investigated: approximate controllability in the space $M_2 $ and its subspace, $L_2 $, exact Euclidean $(R^n )$ controllability, spectral controllability, feedback stabilizability and a relation between pointwise degeneracy and function space controllability. Starting from the abstract functional analytic framework, the analysis is carried down to the matrix theory level, through the crucial intermediate role of the theory of entire functions.

On pointwise degeneracy, controllability and minimal time control of linear dynamical systems with delays

The first part of this paper surveys controllability and degeneracy results for linear time-invariant delay systems and discusses their relationship to minimal time, delay feedback control. In the second part a now construction for augmented delay feedback which achieves function space null controllability is presented. New results on the pointwise degeneracy of dynamic systems with pure delays are also stated.

On Quasiequilibrium States and Range Degeneracy for Linear Semigroup Systems

For an abstract free-motion system x(t) = S(t)x(0), t ≥ 0 characterized by a semigroup of linear operators S(t) on a linear space X-fundamental properties of quasiequilibrium states (QEQ) and degeneracy directions (DED) are examined. The main theorem establishes duality between these notions. The paper can be viewed as generalization of pointwise degeneracy phenomenon known for functional-differential systems and may have also considerable implications in the structure theory of infinite-dimensional input-output systems.

Pointwise degeneracy of linear, time-invariant, delay-differential equations

Pointwise degeneracy of linear, time-invariant, delay-differential equations