Matrix-valued Polynomials Research Articles

Stability and passivity analysis of delayed neural networks via an improved matrix-valued polynomial inequality

The stability and passivity of delayed neural networks are addressed in this paper. A novel Lyapunov–Krasovskii functional (LKF) without multiple integrals is constructed. By using an improved matrix-valued polynomial inequality (MVPI), the previous constraint involving skew-symmetric matrices within the MVPI is removed. Then, the stability and passivity criteria for delayed neural networks that are less conservative than the existing ones are proposed. Finally, three examples are employed to demonstrate the meliority and feasibility of the obtained results.

Improved Admissibility Criteria for Takagi–Sugeno Fuzzy Singular Systems With Time-Varying Delay

The article studies the admissibility stability issue of Takagi-Sugeno fuzzy singular(TFS) systems with time-varying delay. First, negative-definiteness constraints for nth-order matrix-valued polynomials are established based on Finsler's Lemma. A new reciprocally convex inequality(RCI) is developed according to the proposed negative-definiteness method. Then, an improved state decomposition Lyapunov-Krasovskii(L-K) functional is constructed based on the decomposed state method. The improved L-K functional is augmented by considering more information on TFS system states and the delay-dependent matrix. Based on the novel RCI and the L-K functional, a less conservative admissibility criterion is derived. Finally, a well-studied numerical example is performed to reveal the superiority and validity of the proposed admissibility criteria.

Bilinear matrix inequalities and polynomials in several freely noncommuting variables

Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.

Delay-Variation-Dependent Criteria on Extended Dissipativity for Discrete-Time Neural Networks With Time-Varying Delay.

This article is concerned with the extended dissipativity of discrete-time neural networks (NNs) with time-varying delay. First, the necessary and sufficient condition on matrix-valued polynomial inequalities reported recently is extended to a general case, where the variable of the polynomial does not need to start from zero. Second, a novel Lyapunov functional with a delay-dependent Lyapunov matrix is constructed by taking into consideration more information on nonlinear activation functions. By employing the Lyapunov functional method, a novel delay and its variation-dependent criterion are obtained to investigate the effects of the time-varying delay and its variation rate on several performances, such as performance, passivity, and performance, of a delayed discrete-time NN in a unified framework. Finally, a numerical example is given to show that the proposed criterion outperforms some existing ones.

A Generalized Reciprocally Convex Inequality on Stability and Stabilization for T–S Fuzzy Systems With Time-Varying Delay

This article investigates the stability and stabilization issues of Takagi–Sugeno (T–S) fuzzy systems with time-varying delay via a generalized reciprocally convex inequality (GRCI). The enhanced stability criteria with less conservatism are achieved by employing a novel GRCI and establishing an asymmetric Lyapunov–Krasovskii (L–K). The proposed reciprocally convex method encompasses some previous methods as exceptional examples by introducing a matrix-valued polynomial. Moreover, an asymmetric L–K functional with delay-product terms is proposed for the stability of the T–S fuzzy systems. Then, a stabilization method is produced based on the parallel distributed compensation. Finally, case studies are carried out to indicate the effectiveness and advantages of the stability and stabilization criteria.

Stability criteria for time-varying delay systems via an improved reciprocally convex inequality lemma

This paper investigates the stability analysis issue of time-varying delay systems via a novel reciprocally convex inequality lemma(RCIL). First, a novel reciprocally convex combination method is proposed by introducing a matrix-valued polynomial. Some previous methods are encompassed in the proposed method as special examples. Then, a newly asymmetric Lyapunov-Krasovskii(L-K) functional is presented via introducing some delay-product terms, which develop additional delay convexity information. Moreover, the proposed L-K functional and RCIL are amalgamated to develop new stability conditions. Finally, numerical examples and simulation experiments show the advancement of the proposed method.

Structure Identifiability of an NDS With LFT Parametrized Subsystems

Requirements on subsystems have been made clear in this paper for a linear time invariant (LTI) networked dynamic system (NDS), under which subsystem interconnections can be estimated from external output measurements. In this NDS, subsystems may have distinctive dynamics, and subsystem interconnections are arbitrary. It is assumed that system matrices of each subsystem depend on its (pseudo) first principle parameters (FPPs) through a linear fractional transformation (LFT). It has been proven that if in each subsystem, the transfer function matrix (TFM) from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the structure of the NDS is identifiable. Moreover, under some particular situations like there are no direct information transmission from an internal input to an internal output in each subsystem, a necessary and sufficient condition is established for NDS structure identifiability. A matrix valued polynomial (MVP) rank based equivalent condition is further derived, which depends affinely on subsystem (pseudo) FPPs and can be independently verified for each subsystem. From this condition, some necessary conditions are obtained for both subsystem dynamics and its (pseudo) FPPs, using the Kronecker canonical form (KCF) of a matrix pencil.

On a Solution of the Multidimensional Truncated Matrix-Valued Moment Problem

We will consider the multidimensional truncated p times p Hermitian matrix-valued moment problem. We will prove a characterisation of truncated p times p Hermitian matrix-valued multisequence with a minimal positive semidefinite matrix-valued representing measure via the existence of a flat extension, i.e., a rank preserving extension of a multivariate Hankel matrix (built from the given truncated matrix-valued multisequence). Moreover, the support of the representing measure can be computed via the intersecting zeros of the determinants of matrix-valued polynomials which describe the flat extension. We will also use a matricial generalisation of Tchakaloff’s theorem due to the first author together with the above result to prove a characterisation of truncated matrix-valued multisequences which have a representing measure. When p = 1, our result recovers the celebrated flat extension theorem of Curto and Fialkow. The bivariate quadratic matrix-valued problem and the bivariate cubic matrix-valued problem are explored in detail.

Sum-of-squares hierarchies for binary polynomial optimization

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube $${{\mathbb {B}}^{n}=\{0,1\}^n}$$ . This hierarchy provides for each integer $$r \in {\mathbb {N}}$$ a lower bound $$\smash {f_{({r})}}$$ on the minimum $$f_{\min }$$ of f, given by the largest scalar $$\lambda $$ for which the polynomial $$f - \lambda $$ is a sum-of-squares on $${\mathbb {B}}^{n}$$ with degree at most 2r. We analyze the quality of these bounds by estimating the worst-case error $$f_{\min }- \smash {f_{({r})}}$$ in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed $$t \in [0, 1/2]$$ , we can show that this worst-case error in the regime $$r \approx t \cdot n$$ is of the order $$1/2 - \sqrt{t(1-t)}$$ as n tends to $$\infty $$ . Our proof combines classical Fourier analysis on $${\mathbb {B}}^{n}$$ with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds $$\smash {f_{({r})}}$$ and another hierarchy of upper bounds $$\smash {f^{({r})}}$$ , for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the q-ary cube $$\mathbb ({\mathbb {Z}}/ q{\mathbb {Z}})^{n}$$ . Furthermore, our results apply to the setting of matrix-valued polynomials.

Matrix-Valued hypergeometric Appell-Type polynomials

<abstract><p>In recent years, much attention has been paid to the role of special matrix polynomials of a real or complex variable in mathematical physics, especially in boundary value problems. In this article, we define a new type of matrix-valued polynomials, called the first Appell matrix polynomial of two complex variables. The properties of the newly definite matrix polynomial involving, generating matrix functions, recurrence relations, Rodrigues' type formula and integral representation are investigated. Further, relevant connections between the first Appell matrix polynomial and various matrix functions are reported. The current study may open the door for further investigations concerning the practical applications of matrix polynomials associated with a system of differential equations.</p></abstract>

Kummert's approach to realization on the bidisk

We give a simplified exposition of Kummert's approach to proving that every matrix-valued rational inner function in two variables has a minimal unitary transfer function realization. A slight modification of the approach extends to rational functions which are isometric on the two-torus and we use this to give a largely elementary new proof of the existence of Agler decompositions for every matrix-valued Schur function in two variables. We use a recent result of Dritschel to prove two variable matrix-valued rational Schur functions always have finite-dimensional contractive transfer function realizations. Finally, we prove that two variable matrix-valued polynomial inner functions have transfer function realizations built out of special nilpotent linear combinations.

Sufficient conditions for a class of matrix-valued polynomial inequalities on closed intervals and application to [formula omitted] filtering for linear systems with time-varying delays

This paper is concerned with sufficient conditions for a class of matrix-valued polynomial inequalities on closed intervals and their application to H∞ filtering for linear systems with time-varying delays. First, a class of higher degree matrix-valued polynomial inequalities are transformed into the first degree matrix-valued polynomial inequalities by introducing some slack matrices. As a result, a convex property is applied to derive sufficient conditions for a class of higher degree matrix-valued polynomial inequalities on closed intervals. Second, by choosing a Lyapunov–Krasovskii functional with a quadratic matrix-valued polynomial on a time-varying delay, and estimating its time-derivative as a third-degree matrix-valued polynomial on the time-varying delay, the proposed sufficient conditions are utilized to formulate a novel bounded real lemma on the existence of H∞ filters for time-delay systems. Third, some algorithms to filter design are provided and discussed in detail. It is pointed out that, if the filter gain is derived through a generalized inverse matrix method, it may be a fake solution, which is demonstrated through a liquid monopropellant rocket motor with a pressure feeding system.

The sum-of-squares hierarchy on the sphere and applications in quantum information theory

We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of sum-of-squares relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty and Wehner. Specifically, we show that the rate of convergence is no worse than O(d^2/ell ^2) in the regime ell = Omega (d) where ell is the level of the hierarchy and d the dimension, solving a problem left open in the recent paper by de Klerk and Laurent (arXiv:1904.08828 ). Importantly, our analysis also works for matrix-valued polynomials on the sphere which has applications in quantum information for the Best Separable State problem. By exploiting the duality relation between sums of squares and the Doherty–Parrilo–Spedalieri hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascués, Owari and Plenio.

Novel stability criteria for linear time-delay systems using Lyapunov-Krasovskii functionals with a cubic polynomial on time-varying delay

One of challenging issues on stability analysis of time-delay systems is how to obtain a stability criterion from a matrix-valued polynomial on a time-varying delay. The first contribution of this paper is to establish a necessary and sufficient condition on a matrix-valued polynomial inequality over a certain closed interval. The degree of such a matrix-valued polynomial can be an arbitrary finite positive integer. The second contribution of this paper is to introduce a novel Lyapunov-Krasovskii functional, which includes a cubic polynomial on a time-varying delay, in stability analysis of time-delay systems. Based on the novel Lyapunov-Krasovskii functional and the necessary and sufficient condition on matrix-valued polynomial inequalities, two stability criteria are derived for two cases of the time-varying delay. A well-studied numerical example is given to show that the proposed stability criteria are of less conservativeness than some existing ones.

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators.In Part 2 we verify that the group case SU(n+1) meets all the conditions that we impose in Part 1. For any k∈N0 we obtain families of orthogonal polynomials in n variables with values in the N×N-matrices, where N=(n+kk). The case k=0 leads to the classical Heckman-Opdam polynomials of type An with geometric parameter. For k=1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n≥2. We also give explicit expressions of the spherical functions that determine the matrix weight for k=1. These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n=1. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for (n,k) equal to (2,1) and (3,1).

Fatou–Julia dichotomy of matrix-valued polynomials

This article gives a precise description of the Fatou sets and Julia sets of matrix-valued polynomials in M ( 2 , C ) in terms of the corresponding polynomials in C . A normality criterion given by Aladro–Krantz ( [1] ) plays a crucial role in proving this result. Further, we construct Green functions and Böttcher-type functions for these matrix-valued polynomials.

The zeros of determinants of matrix-valued polynomials that are orthonormal on a semi-infinite or finite interval

Let the sequence of matrix-valued polynomials be orthonormal with respect to a nonnegative matrix-valued measure . Assuming that, for some , the support of is contained in the closed set , or , the zeros of the polynomials are shown to lie in the open set , or , respectively. Bibliography: 10 titles.

Bispectrality and Time–Band Limiting: Matrix-valued Polynomials

Abstract The subject of time–band limiting, originating in signal processing, is dominated by the miracle that a naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its eigenfunctions. Bispectrality is an effort to dig into the reasons behind this miracle and goes back to joint work with H. Duistermaat. This search has revealed unexpected connections with several parts of mathematics, including integrable systems. Here we consider a matrix-valued version of bispectrality and give a general condition under which we can display a constructive and simple way to obtain the commuting differential operator. Furthermore, we build an operator that commutes with both the time-limiting operator and the band-limiting operators.

Crouzeix's Conjecture Holds for Tridiagonal 3 x 3 Matrices with Elliptic Numerical Range Centered at an Eigenvalue

Crouzeix stated the following conjecture in [Integral Equations Operator Theory, 48 (2004), pp. 461-477]: For every n x n matrix A and every polynomial p, |p(A)| le 2 max_z in W(A)|p(z)|, where W(A) is the numerical range of A. We show that the conjecture holds in its strong, completely bounded form, i.e., where p above is allowed to be any matrix-valued polynomial, for all tridiagonal 3 x 3 matrices with constant main diagonal, [smallmatrix a b_1 0, c_1 a b_2, 0 c_2 a], a,b_k,c_k in C, or equivalently, for all complex 3 x 3 matrices with elliptic numerical range and one eigenvalue at the center of the ellipse. We also extend the main result of Choi in [Linear Algebra Appl., 438 (2013), pp. 3247-3257] slightly.

The zeros of determinants of matrix-valued polynomials that are orthonormal on a semi-infinite or finite interval

Пусть последовательность матричнозначных многочленов $(P_j)_{j=0}^{\infty }$ ортонормирована относительно неотрицательной матричнозначной меры $\sigma $. Если при некоторых $\alpha,\beta \in \mathbb{R}$ носитель матричнозначной меры $\sigma $ содержится в одном из замкнутых множеств $[\alpha, +\infty)$, $(-\infty, \beta]$, $[\alpha,\beta]$, то нули многочленов $(\det P_j)_{j=0}^{\infty }$ принадлежат соответственно открытым множествам $(\alpha, +\infty)$, $(-\infty, \beta)$, $(\alpha,\beta)$. Библиография: 10 названий.