Regular linear systems governed by neutral FDEs

The aim of this paper is to prove that a class of distributed parameter systems governed by neutral FDEs provides regular linear systems. Employing the well established theory of representation, transfer function and feedback of these later we then give new representations of the state and the output function of the neutral systems.     

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.     

Focal points and principal solutions of linear Hamiltonian systems revisited

In this paper we present a novel view on the principal (and antiprincipal) solutions of linear Hamiltonian systems, as well as on the focal points of their conjoined bases. We present a new and unified theory of principal (and antiprincipal) solutions at a finite point and at infinity, and apply it to obtain new representation of the multiplicities of right and left proper focal points of conjoined bases. We show that these multiplicities can be characterized by the abnormality of the system in a neighborhood of the given point and by the rank of the associated T-matrix from the theory of principal (and antiprincipal) solutions. We also derive some additional important results concerning the representation of T-matrices and associated normalized conjoined bases. The results in this paper are new even for completely controllable linear Hamiltonian systems. We also discuss other potential applications of our main results, in particular in the singular Sturmian theory.     

Julia operators and linear systems (nonuniqueness of linear system)

Complementation theory in Krein spaces can be extended for any self-adjoint transformation. There is a close relation between Julia operators and linear systems. The theory of Julia operators can be used to construct distinct Krein spaces which are the state spaces of extended canonical linear systems with given transfer function.     

Time varying linear systems with boundary conditions and integral operators. I. The transfer operator and its properties

The present paper is the first of a series of papers in which time varying linear systems with boundary conditions and integral operators with semi-separable kernels are studied. The interplay between the systems and the integral operators is one of the main features. A general theory of systems with boundary conditions is developed which includes a detailed study of minimality and minimal factorization. This first part has mainly an introductory character. It contains for systems with boundary conditions a systematic analysis of the concepts of transfer operator, realization, similarity, cascade connection and factorization. For discrete systems an analogous theory is developed.     

A New Method for the Analysis of Univariate Nonuniform Subdivision Schemes

This paper presents a new method for the analysis of convergence and smoothness of univariate nonuniform subdivision schemes. The analysis involves ideas from the theory of asymptotically equivalent subdivision schemes and nonuniform Laurent polynomial representation together with a new perturbation result. Application of the new method is presented for the analysis of interpolatory subdivision schemes based upon extended Chebyshev systems and for a class of smoothly varying schemes.     

Stability,stabilizability and detectability for Markov jump discrete-time linear systems with multiplicative noise in Hilbert spaces

In this article we discuss stability, stabilizability and detectability problems for Markov-jump discrete-time linear systems (MJDLSs) with multiplicative noise (MN) and countably infinite state space of the Markov chain. On the basis of a new solution representation formula, we give new deterministic characterizations of the stability and the detectability properties of MJDLSs with MN. These results are obtained using an operatorial approach and the properties of certain positive evolution operators defined on ordered Banach spaces of sequences of nuclear operators. Assuming detectability conditions and avoiding stochastic proofs, we prove that any global, nonnegative and bounded solution of the Riccati equation of control is stabilizing for the MJDLSs with MN and control. Finally, we apply our results to solve a linear quadratic optimal control problem. The theory is illustrated by an example.     

Asymptotics for solutions of periodic difference systemsSupported in part by Fondecyt 1030535, 1080034, DGI-MATH UNAP 2008 and Diufro 120521.

Using dichotomies and periodic conditions, we obtain asymptotic formulas for solutions of a difference system of Poincaré type with periodic coefficients. Some results about the theory of existence of periodic solutions for linear difference systems are presented. At the end, an open problem on the asymptotic spectral representation is proposed.     

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H^∞ control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô’s dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis.     

On Padé-type model order reduction of J-Hermitian linear dynamical systems

A simple, yet powerful approach to model order reduction of large-scale linear dynamical systems is to employ projection onto block Krylov subspaces. The transfer functions of the resulting reduced-order models of such projection methods can be characterized as Padé-type approximants of the transfer function of the original large-scale system. If the original system exhibits certain symmetries, then the reduced-order models are considerably more accurate than the theory for general systems predicts. In this paper, the framework of J-Hermitian linear dynamical systems is used to establish a general result about this higher accuracy. In particular, it is shown that in the case of J-Hermitian linear dynamical systems, the reduced-order transfer functions match twice as many Taylor coefficients of the original transfer function as in the general case. An application to the SPRIM algorithm for order reduction of general RCL electrical networks is discussed.     

On Russell's method of controllability via stabilizability

Russell has observed that a linear system is controllable provided it is stabilizable in both positive and negative time. We give a version of this result valid for nonlinear systems, and illustrate its use by giving new proofs of two classical results from control theory, the first involving bounded perturbations of controllable linear systems, and the second involving controllability of linear systems by bounded controls.This research was supported by the Natural Sciences and Engineering Research Council of Canada.     

Completely J —positive linear systems of finite order

Completely J — positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J — positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J — symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J — symmetric realizations are discussed.     

Calculation formulas of linear geometrical spreading for ray tracing in a 3D block inhomogeneous gradient medium

Suitable recurrent formulas are derived for direct programming of linear geometrical spreading of a central field of seismic rays in a 3D block gradient medium. These formulas are needed to organize shooting for area observation systems. For recalculation formulas across an interface, a new representation which involves a special nonorthogonal projection operator is found. This operator allows for additive separation of terms that depend only on the ray curvature, boundary curvature, and variable character of the velocity ratio along the boundary. Formulas which express partial derivatives of the eikonal via linear and angular geometrical spreadings are presented.     

How to impose physically coherent initial conditions to a fractional system?

In this paper, it is shown that neither Riemann–Liouville nor Caputo definitions for fractional differentiation can be used to take into account initial conditions in a convenient way from a physical point of view. This demonstration is done on a counter-example. Then the paper proposes a representation for fractional order systems that lead to a physically coherent initialization for the considered systems. This representation involves a classical linear integer system and a system described by a parabolic equation. It is thus also shown that fractional order systems are halfway between these two classes of systems, and are particularly suited for diffusion phenomena modelling.     

Enveloping semigroups in topological dynamics

This is a survey of the theory of enveloping semigroups in topological dynamics. We review the, already classical, theory of enveloping semigroups, due mainly to Robert Ellis, and then proceed to describe some new connections which were discovered in the last few years between three seemingly unrelated theories: of enveloping semigroups, of chaotic behavior, and of representation of dynamical systems on Banach spaces.     

Receding Horizon Robust Control for Nonlinear Systems Based on Linear Differential Inclusion of Neural Networks

In this paper, we present a new receding horizon neural robust control scheme for a class of nonlinear systems based on the linear differential inclusion (LDI) representation of neural networks. First, we propose a linear matrix inequality (LMI) condition on the terminal weighting matrix for a receding horizon neural robust control scheme. This condition guarantees the nonincreasing monotonicity of the saddle point value of the finite horizon dynamic game. We then propose a receding horizon neural robust control scheme for nonlinear systems, which ensures the infinite horizon robust performance and the internal stability of closed-loop systems. Since the proposed control scheme can effectively deal with input and state constraints in an optimization problem, it does not cause the instability problem or give the poor performance associated with the existing neural robust control schemes.     

Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems

In this paper, we consider the existence and multiplicity of solutions of second-order Hamiltonian systems. We propose a generalized asymptotically linear condition on the gradient of Hamiltonian function, classify the linear Hamiltonian systems, prove the monotonicity of the index function, and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear Hamiltonian systems by global analysis methods such as the Leray-Schauder degree theory, the Morse theory, the Ljusternik-Schnirelman theory, etc.     

Caractérisations de la représentation de Burau

This paper is devoted to characterizations of the (reduced) Burau representation of the Artin braid group, in terms of rigid local systems. We prove that the Burau representation is the only representation of the Hecke algebra for which some local system associated to every linear representation of the braid group is irreducible and rigid in the sense of Katz. We also use previous results to give a characterization of the corresponding Knizhnik-Zamolodchikov system.     

三对角线性系统的P—稳定

本文分析了三对角线性系统的P-稳定性，得到三个关于三对角线性系统是P-稳定的充要条件．文中结论部分地回答了文［1，2］所提出的公开问题