The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

By critical point theory, a new approach is provided to studythe existence of periodic and subharmonic solutions of the secondorder difference equation where f C(R x R^m, R^m), f(t+M,z)+f(t,z) for any (t, z)R x R^mand M is a positive integer. This is probably the first timecritical point theory has been applied to deal with the existenceof periodic solutions of difference systems.     

CALIBRATED SUBMANIFOLDS OF R7 AND R8 WITH SYMMETRIES

The principal theory of this paper comprises a technique forconstructing associative, coassociative and Cayley submanifoldsof Euclidean space with symmetries, using first-order ordinarydifferential equations. Explicit examples of U(1)-invariantassociative cones in ⁷ and SU(2)-invariant Cayley 4-folds in⁸ are then produced using this method. Further examples of associative3-folds are presented, which are ruled, and other systems ofdifferential equations defining calibrated submanifolds in ⁷and ⁸ are given.     

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

Distributed control for cooperative systems involving parabolic operators with an infinite number of variables

^** Email: serraghm{at}yahoo.com The optimal control for cooperative systems involving parabolicoperators with an infinite number of variables is considered.First the existence and uniqueness of the states are proved;then the necessary and sufficient condition for the controlto be optimal is obtained by a set of inequalities. The controlin our problems is of distributed type and is allowed to bein the Hilbert space (L²(0, T, L²()))ⁿ.     

A Criticality Criterion for Thermal Explosions in Well-stirred Systems with Reactant Consumption: Addendum

OUR attention has been drawn to the fact that the criticalitycondition ^* = 1 of Adler & Herbert (1985), for well-stirredreactive systems, has been derived previously (Gray, 1975).It arises from an examination of trajectories in the temperaturereactant phase plane when a tube stability argument is employed.Using the criterion ^* = 1, values of the critical Semenov numberhave also been obtained numerically (Gray & Jones, 1981). Our work on criticality for systems with reactant consumptioncame about by trying to reconcile the inflection criterion ofBoddington et al. (1983), for finite B, with the correspondingmaximum criterion in the limit B . Our contribution was to showthat the Semenov number versus maximum temperature ^* curvehas a bifurcation at ^* = 1 for all B. Both Gray's work and ourown are attempting to resolve the same problem; the approachesare, however, quite distinct and complement each other     

Stability analysis of multirate sampled-data control systems

The paper considers the stability of finite-dimensional linearmultirate sampled-data control systems. A notion of internalstability is defined, based on continous-time singals. A completeproof is given for the equivalence between continuous-time internalstability and discrete-time internal stability. An input-outputproperty called L^v(1v ) hybrid stability is investigated formulatirate systems having exogenous continuous-time inputs andoutput, and hybrid stability is shown to be equivalent to internalstability, which is fundamental to a true development of multiratesampled-data L^v control design. The relationship between internalstability and input-output stability is also discussed in continuoustime.     

Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces

Email: vio{at}utgjiu.ro Received on September 12, 2007; Accepted on December 26, 2008 In this article, we discuss a quadratic control problem forlinear discrete-time systems with Markov perturbations in Hilbertspaces, which is linked to a discrete-time Riccati equationdefined on certain infinite-dimensional ordered Banach space.We prove that under stabilizability and stochastic uniform observabilityconditions, the Riccati equation has a unique, uniformly positive,bounded on N and stabilizing solution. Based on this result,we solve the proposed optimal control problem. An example illustratesthe theory.     

Non-linear Theory of Wave-front Propagation

Present address: Division of Mathematical Sciences, Perdue University, Lafayette, Indiana, U.S.A. Equations are derived which govern the mode of propagation andchange in strengths of wave fronts whose behaviours are controlledby quite general systems of quasi-linear hyperbolic equations.The general theory is applied to discuss conditions at the headof a sound wave of arbitrary initial form and strength and alsoat the head of gravity waves on sloping beaches. Conditionsunder which shocks and bores occur are derived. At expansivewaves the non-linearity produces "forgetfulness" at "large"times.     

Bounds on the solution of the time-varying linear matrix differential equation P(t) = AH(t) P(t) + P(t) A(t) + Q(t)

^** Email: pagilla{at}ceat.okstate.edu We derive upper and lower bounds for the trace of the solutionof the time-varying linear matrix differential equation (t)= A^H(t) P(t) + P(t) A(t) + Q(t). A practical numerical exampleis given to verify the bounds. The bounds obtained are usefulsince the considered equation is encountered in a number ofapplications in systems and control theory.     

Polynomial systems supported on circuits and dessins d'enfants

We study polynomial systems in which equations have as commonsupport a set of n + 2 points in ⁿ called a circuit. We finda bound on the number of real solutions to such systems whichdepends on n, the dimension of the affine span of the minimalaffinely dependent subset of , and the rank modulo 2 of . Weprove that this bound is sharp by drawing the so-called dessinsd’enfants on the Riemann sphere. We also obtain that themaximal number of solutions with positive coordinates to systemssupported on circuits in ⁿ is n + 1, which is very small comparedto the bound given by the Khovanskii fewnomial theorem.     

The Convergence of Operator Approximations at Turning Points

Present address: Department of Mathematics, University of Reading, Reading RG6 2AX. We consider the convergence of solution curves of approximationsto parameter-dependent operator equations of the form G(, x)= 0. Provided G_x(, x) remains non-singular this problem is cateredfor by a simple extension to standard theory. In this paper,however, attention is concentrated on solution curves throughcertain singular points (⁰, x⁰), and the main result is thatconvergence depends on consistency and stability results forthe linear eigenvalue problem G_x(⁰, x⁰)0 = 0.     

Truncation Errors for Infinite Linear Systems

An upper and a lower bound for the ¹-norm of the dzerence betweensolutions of a countable infinite linear ditlerential systemand a corresponding finite truncated system are obtained. These bounds are uniformly bounded or decrease exponentiallyin time, unlike the evaluations of Chew, Skivakumar, & Williams(1980). The hypotheses allow the infinite system to be written as adifferential equation in ¹. An upper bound for the truncationerror of infinite algebraic systems which can be written aslinear equations in is also derived.     

Differentiated service delivery in cooperative IP-based broadcast and mobile telecommunications networks

Email: angelo.centonza{at}brunel.ac.uk Corresponding author. Email: thomas.owens{at}brunel.ac.uk Email: john.cosmas{at}brunel.ac.uk^¶ Email: y.h.song{at}brunel.ac.uk Received on 26 November 2005. Accepted on 20 March 2007. The evolution of wireless systems has led recently to the deploymentof cooperative network infrastructures where networks basedon different technologies cooperate together to offer innovativeservices which the networks individually could not offer. Cooperativenetwork infrastructures are hybrid systems. Examples of hybridsystems are already in use in areas as diverse power systemsand locomotion systems. It is very important to apply efficientsystem management techniques to hybrid systems that produceadvantageous business case scenarios for each participatingsystem and efficient use of the available resources. In thispaper, scenarios are investigated where cooperation betweenan integrated project (IP)-based broadcast network and an IP-basedmobile telecommunications network has the potential to allowservices to be provided to mobile users which would not be economicalto offer over either an IP-based broadcast network or an IP-basedmobile telecommunications network alone. The paper presentsa novel technique for determining which network to use to deliversuch services at a given point in time. The application of thistechnique in appropriate scenarios has the potential to generateadditional income for both the IP-based broadcast network operatorand the IP-based mobile telecommunications network operatorin an infrastructure where the operators cooperate to offerinnovative services. The paper explains the construction ofutility functions for cooperative IP-based broadcast and mobiletelecommunications networks. These utility functions are thenused to provide results that enable the efficiency of the managementof the network to be assessed in terms of the utility volumegenerated by the innovative services provided.     

Local-global double algebras for slow H{infty} adaptation; the case of l2 disturbances

In this paper, a common algebraic framework is described forthe frozen-time analysis of stability and H optimization inslowly time-varying systems, based on the notion of a normedalgebra on which local and global products are defined. Relationsbetween local stability, local (near) optimality. local coprimefactorization, and global versions of these properties are sought.The framework is valid for time-domain disturbances in l² (Z)^R. The first part of the paper elaborates the double-algebra conceptfor Volterra operators which approximately commute with theshift. The main algebraic properties and norm inequalities aresummarized. Local conditions for global inveribility are obtained.Classical frozen-time stability conditions are incorporatedin relations between local and global spectra. The paper continuesby establishing an explicit formula linking global and localsensitivity for systems with stable plants, where local sensitivityis a Lipschitz continuous function of data. Frequency-domainestimates of time-domain sensitivity norms. which become accurateas rates of time-variation approach zero, are obtained. Notionsof adaptive versus nonadaptive (robust) control are introduced.It is shown that adaptive control can achieve better sensitivitythan optimal nonadaptive control. It is demonstrated by an examplethat, in general, H-optimal interpolants do not depend Lipschitz-continuouslyon data. However, -suboptimal interpolants of the AAK central(maximal entropy) type are shown to satisfy a tractable Lipschitzcondition.     

An egg-yolk principle and exponential integrability for quasiregular mappings

Quasiregular mappings f:ⁿⁿ are a natural generalization of analyticfunctions from complex analysis and provide a theory which isrich with new phenomena. In this paper we extend a well-knownresult of Chang and Marshall on exponential integrability ofanalytic functions in the disk, to the case of quasiregularmappings defined in the unit ball of ⁿ. To this end, an ‘egg-yolk’principle is first established for such maps, which extendsa recent result of the first author. Our work leaves open aninteresting problem regarding n-harmonic functions.     

Partition deformations in hierarchical hybrid control systems

Hierarchical hybrid control (HHC) (Caines & Wei, submitted)needs to be applicable to systems in which the dynamics of thesystem, the nature of the controllable flows, and the positionsof the boundaries of any partition can only be approximatelyspecified. In this paper, conditions are given under which thedynamics of a high-level HHC system M, and its associated setof control laws, are robust (that is to say, insensitive) withrespect to small perturbations of the partitions defining Mand small perturbations of the controllable flow. This analysisemploys two notions of the deformation of a partition specifiedrespectively by (i) maps of the boundaries and (ii) maps ofthe blocks of . Further, the robustness properties of the partitionmachine M are investigated for deformations of the controllableflow and, finally, for certain mechanical systems.     

On an Operator Theory Approach to the Corona Problem

This paper deals with an operator theory approach to the coronaconjecture for H(Dⁿ). Treil gave a counter-example to this conjecturein the case where n = 1 for operator-valued functions; thusone might hope to use this to disprove the corona conjecturefor H(Dⁿ) (for n 2). This paper shows that this natural approachtowards a negative answer fails. On the other hand, the secondresult here shows that ‘commutant lifting’ cannotbe true for more than two contractions for any constant. Thisobstructs a natural attempted proof of the corona conjecturefor H(Dⁿ) (for n 2) by our previous result.     

An LMI-based approach for robust stabilization of time delay systems containing saturating actuators

Email: elhaous_fati{at}yahoo.fr Corresponding author. Email: elh_tissir{at}yahoo.fr Received on September 8, 2005; Accepted on July 24, 2006 This paper deals with the problem of robust stabilization foruncertain systems with input saturation and time delay in thestate. The parameter uncertainties are time-varying and unknownbut are norm bounded. Sufficient conditions obtained via a linearmatrix inequality formulation are stated to guarantee the localstabilization. The method of synthesis consists in determiningsimultaneously a state feedback control law and an associateddomain of safe admissible states for which the stability ofthe closed-loop system is guaranteed when control saturationseffectively occur. Numerical examples are used to demonstratethe effectiveness of the proposed design technique.     

A posteriori estimates for approximations of time-dependent Stokes equations

Charalambos Makridakis ^{In this paper, we derive a posteriori error estimates for space-discreteapproximations of the time-dependent Stokes equations. By usingan appropriate Stokes reconstruction operator, we are able towrite an auxiliary error equation, in pointwise form, that satisfiesthe exact divergence-free condition. Thus, standard energy estimatesfrom partial differential equation theory can be applied directly,and yield a posteriori estimates that rely on available correspondingestimates for the stationary Stokes equation. Estimates of optimalorder in L(L2) and L(H1) for the velocity are derived for finite-elementand finite-volume approximations.}     

New conditions on absolute stability of uncertain Lur'e systems and the maximum admissible perturbed bound

Email: fhao{at}buaa.edu.cn Received on April 5, 2006; Accepted on September 10, 2006 The robust absolute stability problem for structured uncertainLur'e systems is considered in this paper by using Popov criterionand extended strictly positive real lemma. The conditions onrobust absolute stability for Lur'e systems with structureduncertainties are established in terms of multilinear matrixinequalities. An estimate of the maximum bound of all admissibleperturbations is given by a generalized eigenvalue problem.Finally, a numerical example is worked out to illustrate theefficiency of the main results.