A stability property for locally uniformly rotund renorming

It is proved that C(K,E) (the space of all continuous functions on a Hausdorff compact space K taking values in a Banach space E) admits an equivalent locally uniformly rotund norm if C(K) and E do so. Moreover, if the equivalent LUR norms on C(K) and E are lower semicontinuous with respect to some weak topologies, the LUR norm on C(K,E) can be chosen to be lower semicontinuous with respect to an appropriate weak topology. As a consequence we prove that if X and Y are two Hausdorff compacta and C(X), C(Y) admit equivalent (pointwise lower semicontinuous) LUR norms, then so does C(X×Y).     

Products of Commuting Boolean Algebras of Projections and Banach Space Geometry

New criteria and Banach spaces are presented (for example, GL-spacesand Banach spaces with property () that ensure that the Booleanalgebra generated by a pair of bounded, commuting Boolean algebrasof projections is itself bounded. The notion of R-boundednessplays a fundamental role. It is shown that the strong operatorclosure of any R-bounded Boolean algebra of projections is necessarilyBade complete. Also, for a Dedekind -complete Banach latticeE, the Boolean algebra consisting of all band projections inE is R-bounded if and only if E has finite cotype. In this situation,every bounded Boolean algebra of projections in E is R-boundedand has a Bade complete strong closure. 2000 Mathematics SubjectClassification 46B20, 47L10 (primary), 46B42, 47B40, 47B60 (secondary).     

Almost Frechet Differentiability of Lipschitz Mappings Between Infinite-Dimensional Banach Spaces

We give several sufficient conditions on a pair of Banach spacesX and Y under which each Lipschitz mapping from a domain inX to Y has, for every > 0, a point of -Fréchet differentiability.Most of these conditions are stated in terms of the moduli ofasymptotic smoothness and convexity, notions which have appearedin the literature under a variety of names. We prove, for example,that for > r > p 1, every Lipschitz mapping from a domainin an l_r-sum of finite-dimensional spaces into an l_p-sum offinite-dimensional spaces has, for every > 0, a point of-Fréchet differentiability, and that every Lipschitzmapping from an asymptotically uniformly smooth space to a finite-dimensionalspace has such points. The latter result improves, with a simplerproof, an earlier result of the second and third authors. Wealso survey some of the known results on the notions of asymptoticsmoothness and convexity, prove some new properties, and presentsome new proofs of existing results. 2000 Mathematical Subject Classification: 46G05, 46T20.     

Robust H{infty} dynamic output feedback control for 2D linear parameter-varying systems

James Lam Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Changhong Wang Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, People's Republic of China Corresponding author. Email: ligangwu{at}hit.edu.cn Email: james.lam{at}hku.hk Email: cwang{at}hit.edu.cn Received on June 28, 2006; Accepted on October 1, 2007 This paper is concerned with the problem of robust dynamicoutput feedback control for 2D discrete-time linear parameter-varyingsystems. Given a Fornasini–Marchesini local state-spacesystem with linear varying parameters, our attention is focussedon the design of full-order dynamic output feedback controller,which guarantees the closed-loop system to be asymptoticallystable and has a prescribed disturbance attenuation performance.A sufficient condition for the existence of a desired robustoutput feedback controller is established in terms of parameterizedlinear matrix inequalities, and the corresponding controllersynthesis is cast into a convex optimization problem which canbe efficiently handled by using standard numerical software.A numerical example is provided to illustrate the effectivenessof the proposed design method.     

H{infty} filtering for linear continuous-time systems with sampled measurements

The problem of H filtering of continuous-time systems with sampledmeasurements has been investigated. Both finite-and infinite-horizoncases have been considered. It has bas been shown that the sampled-dataH filtering problem can be solved in terms of a differentialRiccati equation with finite discrete jumps. When the systemis time-invariant with zero initial condition, and an infinitehorizon is concerned, the solution to the above problem is equivalentto a solution to a discrete-time algebraic Riccati equation.     

Properties of the weighted logarithmic matrix norms

Mingzhu Liu Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Email: ghu{at}hit.edu.cn, ghuca{at}yahoo.ca Email: mzliu{at}hit.edu.cn Received on January 31, 2006; Accepted on December 9, 2006 In this paper, we are concerned with the properties of the weightedlogarithmic matrix norms. A relation between the elliptic logarithmicmatrix norm and the weighted logarithmic matrix norm is given.Based on Lyapunov equations, two weighted logarithmic matrixnorms are constructed which are less than 1-logarithmic matrixnorm and -logarithmic matrix norm, respectively. Then, an iterativescheme is presented to obtain the logarithmically -efficientmatrix norm. Numerical examples are given to illustrate theresults.     

Simultaneous output feedback stabilization of time-delay systems

^** Email: magdi_mahmoud{at}cic-cairo.com^*** Email: hnounou{at}uaeu.ac.ae This paper develops new results to the problem of simultaneousfeedback stabilization using static output feedback (SOF) controlfor a finite collection of linear dynamical systems with unknownstate delay. We establish tractable synthesis conditions fornominal SOF controllers and subsequently extend the resultsto resilient SOF to cope with possible additive controller gainperturbations. In both cases, the design of SOF controllersis based on guaranteed cost and approaches. Previous relatedresults are recovered and all the developed results are convenientlycast in the format of linear matrix inequalities. A numericalexample is presented.     

Outline and Amalgamated Triple Systems of Even Index

This paper proves that a triangulated graph is an outline triplesystem of even index if and only if it is an amalgamated triplesystem of even index. 2000 Mathematical Subject Classification:primary 05B07; secondary 05C99.     

Robust H{infty} control of uncertain systems with time-varying state and control input delays

In this paper, a new method for robust H controller of uncertainsystems with time-varying state and control input delays isproposed. A sufficient condition, which is dependent on themaximum value of the time derivative of time-varying delay,is also presented for the existence of H controller. The proposedcontroller not only stabilizes closed-loop uncertain systemsbut also guarantees a prescribed H norm bound of the closed-looptransfer matrix from the disturbance to controlled output. Furthermore,we propose a new robust stability criterion to decide the allowedmaximum bound of plant uncertainty which still can be stabilizedby proposed robust controller. An example given illustratesthe availability of the proposed design method.     

Robust disturbance attenuation of fluid dynamic systems with norm-bounded uncertainties

In this paper, we consider the problem of robust control of fluid dynamic systems in an Hsense. The system under studyis nonlinear in state and linear in inputs with parameter uncertaintiesin all entries. The controller is designed, via Hamilton–Jacobiequation and Riccati equation approaches, such that the resultingclosed-loop systems is robust stable and has a robust Hperformance.In particular, it is shown that the robust Hcontrol problemis solvable for a nonlinear system (governed by Navier–Stokes equation) in the neighbourhood of the origin if a robust Hcontrolproblem is solvable for a corresponding linearized system.     

A Criterion for Finite Topological Determinacy of Map-Germs

Let f, g: (Rⁿ, 0) (R^p, 0) be two C map-germs. Then f and gare C⁰-equivalent if there exist homeomorphism-germs h and lof (Rⁿ, 0) and (R^p, 0) respectively such that g = l f h^–1.Let k be a positive integer. A germ f is k-C⁰-determined ifevery germ g with j^k g(0) = j^k f(0) is C⁰-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C⁰-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle J^k(N,P) with fiber J^k(n, p). Let z in J^k(n, p) and let f be suchthat z = j^kf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let W^k(N, P) be the subbundle of J^k(N, P) with fiber W^k(n, p).Mather has constructed a finite Whitney (b)-regular stratificationS^k(n, p) of J^k(n, p) – W^k(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if S^k(N,P) denotes the corresponding stratification of J^k(N, P) –W^k(N, P) and f C(N, P) is a C map such that j^kf is multitransverseto S^k(N, P), j^kf(N) W^k(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rⁿ, 0) (R^p, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U R^p such that j^kf(U – 0) W^k (Rⁿ, R^p = and suchthat j^kf is multitransverse to S^k (Rⁿ, R^p) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate j^kf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.     

Newton's method on Riemannian manifolds: Smale's point estimate theory under the {gamma}-condition

^** Email: cli{at}zju.edu.cn^*** Email: wjh{at}zjut.edu.cn The -conditions for vector fields on Riemannian manifolds areintroduced. The -theory and the -theory for Newton's methodon Riemannian manifolds are established under the -conditions.Applications to analytic vector fields are provided and theresults due to Dedieu et al. (2003, IMA J. Numer. Anal., 23,395–419) are improved.     

H{infty}-norm and frequency-response computations--sensitivity and a reliable algorithm

It is shown that the Hoo norm and the frequency response ofa linear time-invariant state-space system can be computed byusing frequency sampling along with bounds for the frequencyresponse between the sampled frequencies. A sensitivity androunding-error analysis proves that the algorithm is numericallystable in a restricted sense.     

The Z-transform and linear multistep stability

^** Email: mapjjc{at}maths.bath.ac.uk^*** Corresponding author. Email: ath{at}maths.bath.ac.uk^**** Email: hl{at}maths.bath.ac.uk This paper makes systematic use of control-theoretic methodssuch as the -transform, small-gain theorems and frequency-domainstability criteria in the analysis of the stability behaviourof linear multistep methods. Some of the results in Nevanlinna'swork are recovered and a number of new boundedness and asymptoticproperties of solutions of numerical schemes are obtained. Inparticular, we give a careful and detailed analysis of the nonlinearstability properties of strictly zero-stable methods.     

Finite-dimensional H{infty} control of flexible beam equation systems

In this paper, we consider H control problem with measurementfeedback for flexible beam equation systems. The aim is to constructa finite-dimensional H controller with a given level for theflexible beam equation system. For that purpose, we first formulatethe system as an infinite-dimensional system in l² and derivea finite-dimensional reduced-order system for the infinite-dimensionalsystem. Then, an H controller with level d less than is constructedfor the reduced-order model. The finite-dimensional controllertogether with a residual mode filter plays a role of a finite-dimensionalH controller with level for the original flexible beam equationsystem, if the order of the residual mode filter is chosen sufficientlylarge.     

Stabilization and H{infty} control of two-dimensional Markovian jump systems

This paper extends the results obtained for one-dimensionalMarkovian jump systems, to investigate the problems of stochasticstabilization and H control of two-dimensional (2D) systemswith Markovian jump parameters. The mathematical model of 2Djump systems is established upon the well-known Roesser model,and sufficient conditions are obtained for the existence ofdesired controllers in terms of linear matrix inequalities,which can be readily solved by available numerical software.These obtained results are further extended to more generalcases whose system matrices also contain parameter uncertaintiesrepresented by either polytopic or norm-bounded approaches.A numerical example is provided to show the applicability ofthe proposed theories.     

Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix

^** Email: mengi{at}cs.nyu.edu^*** Email: overton{at}cs.nyu.edu Two useful measures of the robust stability of the discrete-timedynamical system x_k+1 = Ax_k are the -pseudospectral radius andthe numerical radius of A. The -pseudospectral radius of A isthe largest of the moduli of the points in the -pseudospectrumof A, while the numerical radius is the largest of the moduliof the points in the field of values. We present globally convergentalgorithms for computing the -pseudospectral radius and thenumerical radius. For the former algorithm, we discuss conditionsunder which it is quadratically convergent and provide a detailedaccuracy analysis giving conditions under which the algorithmis backward stable. The algorithms are inspired by methods ofByers, Boyd–Balakrishnan, He–Watson and Burke–Lewis–Overtonfor related problems and depend on computing eigenvalues ofsymplectic pencils and Hamiltonian matrices.     

The Mathieu Group M11 and the Modular Curve X(11)

In this paper, we prove that the modular curve X(11) over afield of characteristic 3 admits the Mathieu group M₁₁ as anautomorphism group. We also examine some aspects of the geometryof the curve X(11) in characteristic 3. In particular, we showthat every point of the curve is a point of inflection, thecurve has 110 hyperflexes and there are no inflectional trianglesand 11232 inflectional pentagons, of which 144 are self-conjugate.The hyperflexes correspond to the supersingular elliptic curves.We comment on the relationship of Ward's quadrilinear invariantfor M₁₂ to our work and announce for the first time the equationsfor Klein's A-curve of level 11. We also comment on the relationof our work to some unpublished work of Bott and Tate. 1991Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10,14N10, 20B25, 20C34.