Controllability of linear systems,differential geometry curves in grassmannians and generalized grassmannians,and riccati equations

We study the linear system =Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators e^t B. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ⁿ. We then view (t)=Im(e^t B) as a curve in appropriate Grassmannian and see that, in local coordinates, is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on that are equivalent to the controllability of the system. To get quantitiative results, we lift to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ⁿ). Explicit and simple expressions concerning the geometry of are computed in terms of the Lie algebra of GL(ⁿ), and these are related to the controllability of the system.James Wolper was a visiting professor in the Department of Mathematics at Texas Tech University while much of this research was conducted. He would like to express appreciation for the hospitality he received during his visit.     

Numerical solution of nth‐order integro‐differential equations using trigonometric wavelets

The main aim of this paper is to apply the trigonometric wavelets for the solution of the Fredholm integro‐differential equations of nth‐order. The operational matrices of derivative for trigonometric scaling functions and wavelets are presented and are utilized to reduce the solution of the Fredholm integro‐differential equations to the solution of algebraic equations. Furthermore, we get an estimation of error bound for this method. Copyright © 2011 John Wiley & Sons, Ltd.     

An obstruction to the simultaneous stabilization of two n-D plants

This paper calculates an obstruction to the simultaneous stabilization of twon-D plants. This obstruction is topological and lies in the (singular) cohomology of a subset of ⁿ that arises naturally in the problem. Given any family ofn-D plants, the vanishing of the corresponding cohomology classes of every pair of plants in the family is thus a necessary condition for the simultaneous stabilization of the family.     

Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n‐th order nonlinear differential equations, equations with a generalized ?‐Laplacian, and nonlinear partial differential systems.     

Spectral parameter power series for arbitrary order linear differential equations

Let L be the n‐th order linear differential operator Ly=?₀y⁽ⁿ⁾+?₁y^(n?1)+?+?_ny with variable coefficients. A representation is given for n linearly independent solutions of Ly=λry as power series in λ, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for n=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for λ=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of n‐th order initial value problems and spectral problems.     

Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth‐order integro‐differential equation

This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd.     

New algebraic conditions for controllability of neutral differential equations

The present paper is concerned with the study of the controllability of linear autonomous neutral functional differential equations in the state spaceR ⁿ×L ²([–h, 0],R ⁿ).Controllability conditions are based on an abstract evolution equation representation of the system. Useful algebraic criteria are derived. 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 zeros of solutions of linear differential equations

We consider linear differential equations with regular coefficients in $\text{¦ z ¦ < 1}$. We obtain sufficient conditions for all the solutions of these equations to vanish a given number of times at the most. First the results are obtained for differential equations of second order, then for differential equations of nth order, n > 2.     

Continuous transformations of differential equations with delays

The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer's transformationz(t)=g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of thenth order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions are considered.     

A new technique of using homotopy analysis method for solving high‐order nonlinear differential equations

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high‐order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth‐order nonlinear differential equation to a system of n first‐order equations. Second‐ and third‐ order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimates for the eigenvalues of Hill's equation and applications for the eigenvalues of the laplacian on toroidal surfaces

In this paper we give upper and lower bounds for each eigenvalue λ _n of Hill's differential equation. We apply the results to toroidal surfaces of revolution in order to get estimates for the eigenvalues of the Laplacian in terms of curvature expressions; they are sharp for the flat torus. As an example, we investigate the standard torus in IR³; here, the bounds depend on the radii only.     

Eigenvalues and diameter

Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively.     

Nonlinear functional differential equations with Properties A and B

In this paper, an nth order functional differential equation is considered for which the generalized Emden-Fowler-type equation

(0.1)     

Solving nonlinear fractional partial differential equations using the homotopy analysis method

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM‐Burgers, cubic Boussinesq, coupled KdV, and Boussinesq‐like B(m,n) equations with initial conditions, which are introduced by replacing some integer‐order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer‐order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Oscillations of higher order differential equations of neutral type

In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of nth order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.     

Reverse generalized Bessel matrix differential equation,polynomial solutions,and their properties

This paper is devoted to the study of reverse generalized Bessel matrix polynomials (RGBMPs) within complex analysis. This study is assumed to be a generalization and improvement of the scalar case into the matrix setting. We give a definition of the reverse generalized Bessel matrix polynomials Θ_n(A; B; z), , for parameter (square) matrices A and B, and provide a second‐order matrix differential equations satisfied by these polynomials. Subsequently, a Rodrigues‐type formula, a matrix recurrence relationship, and a pseudo‐generating function are then developed for RGBMPs. © 2013 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

New oscillation criteria for second order linear matrix differential equations with damping

By using the positive linear functional and the monotone subhomogeneous functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order linear matrix differential system
（P（t）X＇（t））＇ ＋ R（t）X＇（t） ＋ Q（t）X（t） =0, t ≥ to 〉 0
where P（t）, R（t）, Q（t） are n × n real continuous matrix functions, P（t） and R（t） are commutative. Theresults improve and generalize those given in some previous papers, which can be seen by the examples given at the end of this paper.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Finite element differential forms

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such differential equations by finite element methods, stable discretization depends on the development of spaces of finite element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of order zero, which are simply functions, these two families reduce to one, which is simply the well-known family of Lagrange finite element subspaces of the first order Sobolev space. For forms of degree 1 and of degree n − 1 (where n is the space dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element spaces developed over the last decades. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations

Summary. This paper studies a numerical method for second-order oscillatory differential equations in which high-frequency oscillations are generated by a linear time- and/or solution-dependent part. For constant linear part, it is known that the method allows second-order error bounds independent of the product of the step-size with the frequencies and is therefore a long-time-step method. Most real-world problems are not of that kind and it is important to study more general equations. The analysis in this paper shows that one obtains second-order error bounds even in the case of a time- and/or solution-dependent linear part if the matrix is evaluated at averaged positions.Mathematics Subject Classification (2000): 65L05, 65L70Acknowledgement I am grateful to Marlis Hochbruck and Christian Lubich for helpful discussions on the subject.