A Generalized Frequency Response for Nonlinear Systems

The principal aim of this paper is to present the idea of ‘generalizedfrequency response’ of a nonlinear input-output map S_xo.It is defined as · S_xo · ^-1 where is the canonicalisomorphsim from the weighted space L²_w[0, ] to ² with respectto a given basis for L²_w(0 , ) and the standard basis for ².Realization results are presented pertainig to the linear, bilinear,and nonlinear cases.     

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.     

A generalization of the concept of similarity of infinite-dimensional linear systems with applications to stability, stabilizability, and detectability

In the present paper a generalization of the concept of similaritycalled D similarity for infinite-demensional linear systemsis introduced. We show that this D similarity preserves thespectrum in the standard sense, and the exponential growth bound,stability, stabilizability, and detectability in the extensionsense. For the case where one of the semigroups is stronglydifferentiable at some t₀ 0, it is shown that D similaritypreserves the exponential growth bound and weak and exponentialstability in the standard sense. For the case where one of thesemigroups is analytic, it is shown that D similarity preservesstabilizability and detectability in the standard sense.     

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.     

A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following thermoelasticplate equation with Dirichlet boundary conditions w_tt + ²w + = a₁(x)u₁ + ... + a_m(x)u_m, t 0, x , _t – ß – w_t = d₁(x)u₁ + ... + d_m(x)u_m,t 0, x , = w = w = 0, t 0, x , where 0, ß > 0, is a sufficiently regular boundeddomain in R^N, a_i, d_i, L² (; R), the control functions u_i L²(0, t₁; R); i = 1, 2, ..., m. This condition is easy to checkand is given by Rank [P_jBA_jP_jBA²_jP_jB ... A^3_j–1_jP_jB] = 3_j,BU=b₁U₁+...+b_mU_m,b_i=[0, a_i, d_i], A_j=[0, –²_j, 0, 1, 0, –_j, 0, _j, –ß_j]P_j, j1, where _j, S are the eigenvalues of – with Dirichlet boundarycondition and P_j, S are the projections on the correspondingeigenspace.     

Approximation of exact controllability problem involving parabolic differential equations

In this paper we examine computation of optimal control u^* ofthe exact controllability problem (referred to as the constraintproblem) governed by the following type of linear parabolicdifferential equations: (y/t) + Ay = u in Q y = 0 on y(0) = y₀ on where A is the second-order elliptic differential operator, is a bounded domain in ^k with smooth boundary , Q = (0, T)x , = (0, T) x and T > 0. This is achieved by approximatingu^* through a sequence {u_n} of controls corresponding to unconstrainedproblems involving a penalty function arising from the controllabilityconstraint.     

A Twisted Sampling Theorem

A generalized sampling theorem is proved for a set of functionsf that are well behaved in the frequency domain. Specifically,let W be a positive bounded integrable function, let be a reconstructionfunction, and let A denote the operation of multiplication inthe frequency domain by a bounded function A(s). Then coefficientsh_n (–<n<) are determined to minimize the maximalL²-error in an approximation scheme for functions f ranging over the set is the Fourier of f. Formulae are given for the minimal obtainablemaximal error, and the optimal choice (choices) of h_n (–<n<)is characterized through its discrete Fourier transform .     

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.     

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.     

Smooth Approximation of Sobolev Functions on Planar Domains

We examine two related problems concerning a planar domain .The first is whether Sobolev functions on can be approximatedby global C functions, and the second is whether approximationcan be done by functions in C() which, together with all derivatives,are bounded on . We find necessary and sufficient conditionsfor certain types of domains, such as starshaped domains, andwe construct several examples which show that the general problemis quite difficult, even in the simply connected case.     

Closed trajectories and global controllability in the plane

In the plane, results are given about the structure of closedtrajectories which may occur as simple closed curves or generalclosed curves with self-intersections. Necessary conditionsfor the global controllability of nonlinear systems that arein the so-called linear-analytic form = f(x)+ug(x), where x R² and u 1, are given. It is provedthat, if there exists a closed trajectory of the system, then either contains a point where f and g are linearly dependent,or encloses some zeros of f + ug for all u [-1,1]. Then thisresult is used to prove that, if the linear-analytic systemis controllable and the vector field g is never zero in W R²,then W contains some zeros of f + ug for some u [-1,1]. A topologicalapproach is taken. Remarks are made about the size of the regionwhere a closed trajectory can lie, and about the shape of theclosed trajectories. Further implications are discussed.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Lp Lq Estimates for Parabolic Systems in Non-Divergence Form with VMO Coefficients

Consider a parabolic NxN-system of order m on ⁿ with top-ordercoefficients a VMOL. Let 1 < p, q < and let be a Muckenhouptweight. It is proved that systems of this kind possess a uniquesolution u satisfying whereAu = _||m a Du and J = [0,). In particular, choosing = 1, therealization of A in L^p(ⁿ)^N has maximal L^p – L^q regularity.     

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 Note on a System Theoretic Approach to a Conjecture by Peller-Khrushchev: The General Case

Based on the construction of infinite-dimensional balanced realizations,an alternative solution to the following inverse spectral problemis presented. Given a decreasing sequence of positive numbers(_n)_n>1 (i.e. ₁₂₃...0), does there exist a Hankel operatorwhose sequance of singular values is (_n)_n>1? This paper isan extension of a previously published paper in which the sameapproach was taken in the case of a monotonically decreasingsequence(_n)_n>1.     

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.     

Controllability of a system of parabolic equations with non-diagonal diffusion matrix

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following systemof parabolic equations with Dirichlet boundary condition: {z_t = D z + b₁(x)u₁ + ··· + b_m(x)u_m, t 0, z ⁿ, z = 0, on where is a sufficiently smooth bounded domain in ^N, b_i L²(;ⁿ), the control functions u_i L²(0, t₁; ); i = 1, 2, ..., mand D is an n x n non-diagonal matrix whose eigenvalues aresemi-simple with positive real part. This algebraic conditionis checkable since it is given in terms of the n_j x m matricesDP_j and P_jB, i.e. Rank [P_jBDP_jBD²P_jB··· D^n_j–1 P_jB]= n_j, where P_jBu = P_jb₁u₁ + ··· + P_jb_mu_m. Finally,this result can be applied to those systems of partial differentialequations that can be rewritten as a diffusion system (see deOliveira, 1998).     

Contractions et Hyperdistributions a Spectre de Carleson

Let =(_n)_n1 be a log concave sequence such that lim inf_n+n/n^c>0for some c>0 and ((log _n)/n)_n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||T^–n||=O(_n)as n tends to + with _n11/(n log _n)=+, then T is unitary. Onthe other hand, if _n11/(n log _n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||T^–n||=O(_n) as n tends to +, andlim sup_n+||T^–n||=+.     

Hankel operators of systems with feedthrough delays

We study the singular values of and the essential spectrumof * when is the (non compact) Hankel operator of a systemwith feedthrough delays.