General Orthogonal State Variables and Applications to Lyapunov Functions and Feedback Control Systems

A general set of orthogonal state variables is formulated forlinear time-invariant systems both in the time and the frequencydomains. A general control variable u(t) and general initialconditions v(0) are considered. The meaning of orthogonalityin frequency domain is demonstrated. The applications of generalorthogonal state variables to the evaluation of Lyapunov functionsat t =0 and the evaluation of a quadratic performance criterionof feedback control systems are discussed.     

Zeros of Entire Functions and Related Systems of Infinitely Many Nonlinearly Coupled Evolution Equations

Recent findings concerning the zeros of generic polynomials are extended to entire functions featuring infinitely many distinct zeros, and related systems of infinitely many nonlinearly coupled evolution ODEs and PDEs are identified, the solutions of which display interesting properties.     

Solution Bounds of the Continuous and Discrete Lyapunov Matrix Equations

A unified approach is proposed to solve the estimation problem for the solution of continuous and discrete Lyapunov equations. Upper and lower matrix bounds and corresponding eigenvalue bounds of the solution of the so-called unified algebraic Lyapunov equation are presented in this paper. From the obtained results, the bounds for the solutions of continuous and discrete Lyapunov equations can be obtained as limiting cases. It is shown that the eigenvalue bounds of the unified Lyapunov equation are tighter than some parallel results and that the lower matrix bounds of the continuous Lyapunov equation are more general than the majority of those which have appeared in the literature.     

Inversion of Analytic Functions via Canonical Polynomials: A Matrix Approach

An alternative to Lagrange inversion for solving analytic systems is our technique of dual vector fields. We implement this approach using matrix multiplication that provides a fast algorithm for computing the coefficients of the inverse function. Examples include calculating the critical points of the sinc function. Maple procedures are included which can be directly translated for doing numerical computations in Java or C. A preliminary version of this paper has been presented at AISC 2006.     

Sylvester与Lyapunov方程向后误差分析

利用矩阵Kronecker积的性质,研究Sylvester矩阵方程Ax YB=C与Lyapunov矩阵方程ATX XA=-Q(Q0)的向后误差,获得了这两类矩阵方程向后误差η(X,Y)与η(X)的精确表达式及其更易计算的上下界.这些结果是对有关文献相应结果的改进与补充.     

A New Matrix Approach For Solving Second-Order Linear Matrix Partial Differential Equations

The basic aim of this article is to present a novel efficient matrix approach for solving the second-order linear matrix partial differential equations (MPDEs) under given initial conditions. For imposing the given initial conditions to the main MPDEs, the associated matrix integro-differential equations (MIDEs) with partial derivatives are obtained from direct integration with regard to the spatial variable x and time variable t. Hence, operational matrices of differentiation and integration together with the completeness of Bernoulli polynomials are used to reduce the obtained MIDEs to the corresponding algebraic Sylvester equations. Using two well-known subspace Krylov iterative methods (i.e., GMRES(10) and Bi-CGSTAB) we provide two algorithms for solving the mentioned Sylvester equations. A numerical example is provided to show the efficiency and accuracy of the presented approach.

     

A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials {P_n}_n, with respect to a weight W, that are eigenfunctions of a second-order differential operator D. The weight W and the differential operator D were found in [12], using some aspects of the theory of the spherical functions associated to the complex projective spaces. We also find other second-order differential operator E symmetric with respect to W and we describe the algebra generated by D and E.     

A Galerkin Approach for Solving Matrix Equations with Hierarchical Matrices

We consider a new approach for computing solutions of certain matrix equations, for example AXA = C, AX + XB = C or AX = I. This approach is based on a variational formulation in the matrix space, employing the Frobenius inner product. Using the space of ℋ²-matrices as trial space leads to a sparse linear system that can be solved by iterative methods to compute an approximate solution of the underlying matrix equation. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

$H^2$-Stabilization of the Isothermal Euler Equations: a Lyapunov Function Approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm.To this...     

A Unified Approach to Construct a Class of Daubechies Orthogonal Scaling Functions

We use Lorentz polynomials to give an efficient way to prove Daubechies’ results on the existence of spline type orthogonal scaling functions and to evaluate a class of Daubechies scaling functions in a unified approach.     

Stability of Functional Equations in Matrix Random Normed Spaces: A Fixed Point Approach

In this paper, we prove the Hyers–Ulam stability of the Cauchy additive functional equation and the quadratic functional equation in matrix random normed spaces by using fixed point method.     

Vector Lyapunov Functions and Ultimate Poisson Boundedness of Solutions of Systems of Differential Equations

Various forms of uniform-ultimate Poisson boundedness of solutions and of ultimate Poisson equiboundedness of solutions are introduced. Sufficient conditions for various forms of uniform-ultimate Poisson boundedness and of ultimate Poisson equiboundedness of solutions are obtained by using the method of vector Lyapunov functions.     

Riemann–Hilbert Problems,Matrix Orthogonal Polynomials and Discrete Matrix Equations with Singularity Confinement

In this paper, matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not.     

Poisson Total Boundedness of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

We introduce the notions of Poisson total boundedness of solutions, partial Poisson total boundedness of solutions, and partial Poisson total boundedness of solutions with partly controlled initial conditions. We use the Lyapunov vector function method to obtain sufficient conditions for the Poisson total boundedness of solutions, the partial Poisson total boundedness of solutions, and the partial Poisson total boundedness of solutions with partly controlled initial conditions. As a consequence, we obtain sufficient conditions for the above-mentioned kinds of Poisson total boundedness of solutions based on the Lyapunov function method.

     

A Semigroup Approach to Stochastic Delay Equations in Spaces of Continuous Functions

We present a semigroup approach to stochastic delay equations of the form

Rational Functions Associated with the White Noise Space and Related Topics

Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida’s white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the Cauchy–Kovalevskaya product is replaced by the Wick product. D. Alpay thanks the Earl Katz family for endowing the chair which supports his research.     

A Product Integration Approach Based on New Orthogonal Polynomials for Nonlinear Weakly Singular Integral Equations

This paper provides with a generalization of the work by Chelyshkov (Electron. Trans. Numer. Anal. 25(7): 17–26, 2006), who has introduced sequences of orthogonal polynomials over [0,1] which can be expressed in terms of Jacobi polynomials. We develop a new approach of product integration algorithm based on these orthogonal polynomials including the numerical quadratures for solving the nonlinear weakly singular Volterra integral equations. The convergence analysis of the proposed scheme is derived and numerical results are given showing a marked improvement in comparison with recent numerical methods.     

Non-self-adjoint Dirac-type Systems and Related Nonlinear Equations: Wave Functions, Solutions, and Explicit Formulas

A version of the B?cklund-Darboux transformation, where Darboux matrix takes the form of the transfer matrix function from the system theory, for the non-self-adjoint Dirac-type system is considered. Related nonlinear Schr?dinger equations (coupled and multi-component), self-induced transparency equation, and non-Abelian sine-Gordon equation are treated. Explicit formulas for the wave functions and solutions are obtained.     

A Class of Orthogonal Refinable Functions and Wavelets

We give a construction, for any n 2, of a space S of spline functions of degree n – 1 with simple knots in (1/4)Z which is generated by a triple of refinable, orthogonal functions with compact support. Indeed, the result holds more generally by replacing the B-spline of degree n – 1 with simple knots at the integers by any continuous refinable function whose mask is a Hurwitz polynomial of degree n. A simple construction is also given for the corresponding wavelets.