An Analysis of Fast-Slow Systems

This paper discusses the construction of an approximate solutionto the standard fast-slow system of equations. The usual representationin terms of a fast vector and a slow vector is shown to be satisfactoryonly when certain matrices are neither ill conditioned nor near-orthogonal.Applications of the results to certain control problems areinvestigated. This paper was written while on Leave of Absence at the Departmentof Mathematics, University of Salford, Salford.     

Remarks on Perturbations of the Equation

Perturbations of the equation are discussed. The structure of near-orthogonal matrices is shown, and implicationsfor certain control problems are pointed out. Examples are givenand the circumstances are noted in which the real part of aneigenvalue can be expected to change sign. This paper was written while the author was on a Leave of Absenceat the Department of Mathematics, University of Salford.     

Estimates of the region of attraction of continuous-time cascade systems

Xiao-Song Yang Department of Mathematics, Huazhong University of Science and Technology Wuhan, Hubei 430074, People's Republic of China Huimin Li Departments of Mathematics and Control and Engineering, Huazhong University of Science and Technology Wuhan, Hubei 430074, People's Republic of China Corresponding author. Email: yangxs{at}cqupt.edu.cn Received on May 1, 2006; Accepted on October 6, 2006 In this paper, we first present some new sufficient conditionsfor global asymptotic stability of continuous-time cascade systems.Then, we give some detailed results on the region of attractionof continuous-time cascade systems. Some examples are presentedto illustrate these results.     

On the Stability Analysis of Nonlinear Systems

The stability of nonautonomous delay differential equationsis studied via a combination of the global linerization andnonlinear variation of parameter techniques. Permanent address: Department of Mathematics, University ofElectronics Science & Technology of China, Chengdu, China.     

An Example of Only Linear Convergence of Trust Region Algorithms for Non-smooth Optimization

Most superlinear convergence results about trust region algorithmsfor non-smooth optimization are dependent on the inactivityof trust region restrictions. An example is constructed to showthat it is possible that at every iteration the trust regionbound is active and the rate of convergence is only linear,though strict complementarity and second order sufficiency conditionsare satisfied. Presented at the 1983 Dundee Conference on Numerical Analysis     

Pointwise Stabilization for a Chain of Coupled Vibrating Strings

For the system representing a chain of coupled vibrating strings,we show that the associated semigroup satisfies the assumptionof spectrum-determined growth, and hence obtain conditions forenergy to decay strongly or exponentially. We examine in detailthe three-string case, and our results include those obtainedby others for the two-string case. Permanent address: Beijing Institute of Information and Control,Beijing, China.     

Convergence of Methods for the Numerical Solution of the Korteweg--de Vries Equation

We prove that a family of methods for the numerical solutionof the Korteweg—de Vries equation is convergent. Thisfamily includes as particular cases some known finite differenceand finite element schemes. It is also found that the stabilityproperties of the methods vary significantly with the treatmentof the non-linear term. On leave from Shanghai University of Science and Technology,Shanghai, China.     

Differential algebraic control theory

Recent developments in differential algebraic control theoryare presented. The paper concentrates on the differential algebraiccontrollability, observability, elimination, realization, invertibility,and equivalence problem. Some other aspects can be found inthe references therein. The mathematical tool is differentialalgebra. This paper is an extended version of one presented at the IMAConference on Control: Modelling, Computation, and Information,UMIST, Manchester, September, 1992.     

A Finite-element Method for Solving Elliptic Equations with Neumann Data on a Curved Boundary Using Unfitted Meshes

This paper considers a finite-element approximation of a Poissonequation in a region with a curved boundary on which a Neumanncondition is prescribed. Piecewise linear and bilinear elementsare used on unfitted meshes with the region of integration beingreplaced by a polygonal approximation. It is shown, despitethe variational crimes, that the rate of convergence is stillorder (h) in the H¹ norm. Numerical examples show that the methodis easy to implement and that the predicted rate of convergenceis obtained. Supported by SERC postdoctoral fellowship RF/5830.     

Riccati equation for infinite-dimensional discrete bilinear systems

The problem of quadratic optimal feedback control for infinite-dimensionaldiscrete bilinear systems is associated with an algebraic Riccati-likeoperator equation. The existence and uniqueness of a nonnegativesolution to such a Riccati-like operator equation is establishedas the strong limit of a sequence of solutions to Lyapunov-likeoperator equations. This research was supported in part by CNPq (Brazilian NationalResearch Council) and FAPESP (Research Council of the Stateof sào Paulo).     

On Symmetric Minimum Norm Updates

The theory of the 1-inverse of a matrix is used to derive allsymmetric updates which satisfy a matrix equation. This generalsolution is then used to obtain the solution which is minimumin a symmetrically weighted Frobenius norm. The minimum normsolution can be used to derive the minimum norm updates usedin quasi-Newton methods for unconstrained optimization. This research was supported in part by NIH Grant No. AM17593.     

Modelling history-dependent parameters in the SMPS format for stochastic programming

G. Infanger Management Science and Engineering, Stanford University, Stanford, CA Email: Horand.Gassmann{at}dal.ca Received on 1 October 2004. Accepted on 31 January 2007. This paper proposes two extensions to the SMPS format for stochasticprograms to permit modelling of autoregressive-moving average(ARMA) processes. Sampling-based algorithms can thus proceedindependently of any underlying modelling system, increasingefficiency. An illustrative example demonstrates the power ofthe new constructs.     

The multidimensional positive and nearly conservative Lagrange-Galerkin method. Part II: the use of C1 elements

In this paper we consider a monotonic and conservative variationon the direct Lagrange-Galerkin method due to Priestley, andaddress certain problems with both the original Lagrange-Galerkinmethod and this new method which disappear with the use of higher-orderC¹ basis functions. Here we concentrate on the use of Hermite-cubics,demonstrating their superior stability in the Lagrange- Galerkinmethod, and the adaptations of the recovery techniques neededto ensure monotonicity and near conservation. Current address: Schlumberger GeoQuest, Abingdon Business Park,Oxon OX14 10 UK.     

An analysis of the coupling of finite-element and Nystrm methods in acoustic scattering

In this paper we analyze the use of a combined Nystrm and finite-elementprocedure for approximating the scattered time-harmonic acousticfield produced when an incident wave interacts with a boundedinhomogeneity. The coupling technique uses a smooth artificialboundary and explicitly decouples the integral equation andfinite-element computations. We prove convergence of the method.One highlight of this analysis is that we prove Sobolev spaceerror estimates for the Nystrm scheme (which is usually analyzedin Hlder spaces). We also present some numerical results. E-mail: kirsch{at}am.uni-erlangen.de E-mail: monk{at}math.udel.edu     

Maximum Principles for Optimal Controls for Elliptic Variational Inequalities of the Second Kind

A maximum principle is obtained for optimal controls for someelliptic variational inequalities of the second kind by usingthe penalty method, Ekelans's variational principle, and lowersemicontinuity of some set-valued mappings. It has been shownthat this principle leads to some known optimality conditionsin many cases. It also yields new optimality conditions. Insome cases, it leads to an analogy of pontryagin's principle. Permanent address: Mathematics Department, Centre South Universityof Technology, China.     

Qualitative dynamics, root-loci and asymptotic structure of universal adaptive stabilizers

A problem of universal adaptive stabilization in Rⁿ is approachedin a qualitative manner to relate the form of the trajectoriesin the extended state space Rⁿ⁺¹ to the root locus of the associatedfixed-parameter linear system. Relationships are derived betweenthe values of the limit gain and the initial conditions. Numericalstudies are used to illustrate and support these ideas by computationof generic trajectories and attempted computation of certainnon-generic possibilities. The implications of the study formore general dynamic situations are outlined. These authors are also with the Department of Mathematics,University of Exeter, EX4 4QE.     

Characterization of Distributions through Maximization of {alpha}-Entropies

This paper provides characterizations of the Pareto distributionof the second kind, the Pearson type-VII distribution and thet-distribution. These characterizations have been obtained throughmaximization of -entropies. As 1, characterizations of the exponential,the Laplace, and the normal distributions are obtained. On sabbatical leave from the Department of Statistics, YarmoukUniversity, Irbid, Jordan.     

Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws

The theory of optimal recovery is applied to finite volume methodsfor the numerical solution of conservation laws in multiplespace dimensions. Classical polynomial EN0 (essentially non-oscillatory)algorithms can be interpreted as trivial recovery operatorsfor the point functional in the light of this theory. Thin platesplines are identified as optimal recovery functions in Beppo-Levispaces and can be seen as multi-dimensional analogues of cubicsplines. Recovery algorithms of ENO-type based on thin platesplines are developed and applied to test problems includingthe Euler equations of compressible gas dynamics. E-mail: Thomas.Sonar{at}ts.go.dlr.de     

Numerical Solution of Sigular Integral Equations Using Gauss-type Formulae I: I: Quadrature and Collocation on Chebyshev Nodes

Permanent address: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794. Of concern here are the Gauss—Chebyshev formulae for numericalsolution of singular integral equations of the Cauchy-type.The coefficient matrix of the linear sustem of algebraic equationscorresponding to the dominant term term is shown to have aninverse, which is expressed in a neat, closed form. Using thenorm of the inverse matrix, the effect of quadrature errorson the computed solution is estimated. Using Jackson's theoremson "best approximation", convergence of the procedure is provedunder favourable conditions. In the course of analysis, a numberof identities involving the zeros of Chebyshev polynomials offirst and second kind is obtained. ^*Dedicated to Professor I. N. Sneddon on his sixty-third birthday.     

On the Convergence of Some Approximate Methods of Conformal Mapping

A method was given in Ellacott (1978) for determining approximatelythe conformal mapping of a Jordan region on to a disc. Someresults on the convergence of this method are given, which canbe used to prove the result (conjectured in Ellacott, 1978)that if the boundary curve is analytic, then convergence isuniform. The corresponding result is also proved for the Bergmanand Szeg Kernel methods with polynomial basis functions. (Theresult is already known for the Szeg? Kernel method, but a differentproof is given.) Also discussed is the use of rational basisfunctions for the Bergman Kernel method. Currently visiting Forschungsinstitut fr Mathematik, ETH-Zentrum,CH-8092, Zurich.