Decentralized dynamic pole assignment with low-order compensators

N. Karcanias Control Engineering Centre, School of Engineering and Mathematical Sciences, City University, Northampton Square London EC1V OHB, UK Email: n.karcanias{at}city.ac.uk Received on June 14, 2006; Accepted on October 2, 2006 The problem of arbitrary pole placement via dynamic decentralizedoutput feedback is studied for minimal systems described bya proper transfer function matrix P(s) R^{m x p}(s) (m = m_i andp = p_i), with McMillan degree n. The family of controllersto be used includes those decentralized controllers with channelswhose ith channel has maximum observability index at most d_i.The method presented here is based on asymptotic linearizationaround a decentralized degenerate compensator of the pole placementmap related to the problem. It is shown that the method worksgenerically when m⁺p > n, where m⁺ = min{d_i(p_i + m_i –1) + m_i}, i = 1, ..., , and the smallest d_i of the compensatorof the ith channel is the integral part of (n – pm_i)/p(p_i+ m_i – 1).     

On spurious asymptotic numerical solutions of explicit Runge-Kutta methods

The bifurcation diagram associated with the logistic equationⁿ⁺¹ = aⁿ(1 – ⁿ) is by now well known, as is its equivalenceto solving the ordinary differential equation (ODE) u' = u(1– u) by the explicit Euler difference scheme. It has alsobeen noted by Iserles that other popular difference schemesmay not only exhibit period doubling and chaotic phenomena butalso possess spurious fixed points. We investigate, both analyticallyand computationally, Runge-Kutta schemes applied to the equationu'=f(u), for f(u) = u{1 – u) and f(u) = au(1 – u)(b– u), contrasting their behaviour with the explicit Eulerscheme. We determine and provide a local analysis of bifurcationsto spurious fixed points and periodic orbits. In particularwe show that these may appear below the linearised stabilitylimit of the scheme, and may consequently lead to erroneouscomputational results. Major part of the material was published as an internal report-NASATechnical Memorandum 102919, April 1990, also as Universityof Reading Numerical Analysis Report 3/90, March 1990. This work was performed whilst a visiting scientist at NASAAmes Research Center, Moffett Field. CA 94035 USA. Staff Scientist, Fluid Dynamics Division.     

Maximin investment problems for discounted and total wealth

Email: nikolaidokuchaev{at}trentu.ca Received on 27 March 2006. Accepted on 14 September 2007. We study an optimal investment problem for a continuous-timeincomplete market model such that the risk-free rate, the appreciationrates and the volatility of the stocks are all random; theyare not necessarily adapted to the driving Brownian motion,and their distributions are unknown, but they are supposed tobe currently observable. The optimal investment problem is statedin ‘maximin’ setting which leads to maximizationof the minimum of expected utility over all distributions ofparameters. We found that the presence of the non-discountedwealth in the performance criterion (in addition to the discountedwealth) implies an additional condition for the saddle pointof the maximin problem: the saddle point must include the minimumof the possible risk-free return. This is different from thecase when the utility depends on the discounted wealth only.Using this result, the maximin problem is reduced to a linearparabolic equation and minimization over two scalar parameters.It is an important development of the results obtained in Dokuchaev(2002, Dynamic Portfolio Strategies: Quantitative Methods andEmpirical Rules for Incomplete Information. Boston: Kluwer;2006, IMA J. Manage. Math., 17, 257–276).     

Optimality conditions for n x n infinite-order parabolic coupled systems with control constraints and general performance index

W. Kotarski Institute of Informatics, Silesian University, Bedzinska 60, 41-200 Sosnowiec, Poland Email: bahaa_gm{at}hotmail.com Email: kotarski{at}gate.math.us.edu.pl Received on March 14, 2006; Accepted on December 20, 2006 A distributed control problem for n x n parabolic coupled systemsinvolving operators with infinite order is considered. The performanceindex is more general than the quadratic one and has an integralform. Constraints on controls are imposed. Making use of theDubovitskii–Milyutin theorem, the necessary and sufficientconditions of optimality are derived for the Dirichlet problem.Yet, the problem considered here is more general than the problemsin El-Saify & Bahaa (2002, Optimal control for n x n hyperbolicsystems involving operators of infinite order. Math. Slovaca,52, 409–424), El-Zahaby (2002, Optimal control of systemsgoverned by infinite order operators. Proceeding (Abstracts)of the International Conference of Mathematics (Trends and Developments)of the Egyptian Mathematical Society, Cairo, Egypt, 28–31December 2002. J. Egypt. Math. Soc. (submitted)), Gali &El-Saify (1983, Control of system governed by infinite orderequation of hyperbolic type. Proceeding of the InternationalConference on Functional-Differential Systems and Related Topics,vol. III. Poland, pp. 99–103), Gali et al. (1983, Distributedcontrol of a system governed by Dirichlet and Neumann problemsfor elliptic equations of infinite order. Proceeding of theInternational Conference on Functional-Differential Systemsand Related Topics, vol. III. Poland, pp. 83–87) and Kotarskiet al. (200b, Optimal control problem for a hyperbolic systemwith mixed control-state constraints involving operator of infiniteorder. Int. J. Pure Appl. Math., 1, 241–254).     

A Simple Proof for a Spectral Factorization Theorem

It is shown using simple methods that the transform Z(z), z C, of the coefficient sequence of the Wold decomposition ofany full-rank wide-sense stationary purely non-deterministicstochastic process satisfies (i) Z(z) H² (D) and (ii) Z^–1(z) H(D). Further it is shown that all spectral factors satisfying(i) and (ii) are equal up to right multiplication by orthogonalmatrices, and that among these the normalized (Z(0) =I) spectralfactors are equal to the transform of the Wold decomposition.An elementary proof of Youla's Theorem is then given togetherwith a simple proof that the rows of a Cholesky factor of abanded block Toeplitz matrix converge to the coefficients ofa stable matrix polynomial. Computer and Automation Institute of the Hungarian Academyof Sciences, Budapes, Hungary.     

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.     

Traces and extensions of matrix-weighted Besov spaces

Let V be a matrix weight on ⁿ⁺¹ and let W be a matrix weighton ⁿ, satisfying, for example, the matrix A_p condition. Definethe trace, or restriction, operator Tr by Tr (f)(x^')=f(x^', 0),where x^'n and f is a function on ⁿ⁺¹. If –1/p>n (1/p–1)₊+(β–n)/p,where β is the doubling exponent of W, then the trace operatoris bounded from into (matrix-weighted Besov spaces) if and only ifthe weights V and W uniformly satisfy an estimate controllingthe average of on anydyadic cube I ⁿ by the average of on Q(I)=Ix[0, (I)], for all . If V and W satisfy the converse inequality, then there existsa continuous linear map .If both inequalities hold, then Tr Ext is the identity on .     

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.     

On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

Optimal control problems of parabolic equations with an infinite number of variables and with equality constraints

Email: bahaa_gm{at}hotmail.com Received on December 6, 2005; Accepted on December 7, 2006 Optimal control problems of systems governed by parabolic equationswith an infinite number of variables and with additional equalityconstraints are considered. The extremum principle, as wellas sufficient condition of optimality, is formulated for theNeumann problem by using certain extensions of Dubovitskii–Milyutinmethod.     

Generalized compound quadrature formulae for finite-part integrals

Received on 31 July 1995. Revised on 19 August 1996. We investigate the error term of the dth degree compound quadratureformulae for finite-part integrals of the form where and p 1.We are mainly interested in error bounds of the form with best possible constants c. Itis shown that, for and n uniformlydistributed nodes, the error behaves as O(n^p–s–1for , p–1 <s d+1.In a previous paper we have shown that this is not true for As an improvement, we consider the case of non-uniformly distributednodes. Here, we show that for all p I and , an O(n^–s) error estimate can be obtainedin theory by a suitable choice of the nodes. A set of nodeswith this property is staled explicitly. In practice, this gradedmesh causes stability problems which are computationally expensiveto overcome. E-mail address: diethelm{at}informatik.uni-hildesheim.de     

On the solvability for the mixed-type Lyapunov equation

^** Email: xsf{at}math.pku.edu.cn^*** Email: mscheng{at}math.pku.edu.cn In this paper, the linear matrix equation X = AXB^* + BXA^* +Q is considered, which is called the mixed-type Lyapunov equation.Some necessary and sufficient conditions for the existence ofa unique solution are presented. Since a Hermitian positivesemidefinite solution is important from the application pointof view, some sufficient conditions for the existence of a Hermitianpositive semidefinite solution are derived.     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

Asymptotic behavior of positive solutions of some quasilinear elliptic problems

We discuss the asymptotic behavior of positive solutions ofthe quasilinear elliptic problem –_pu = a u^p–1–b(x)u^q, u| = 0, as q p – 1 + 0 and as q , via a scale argument.Here _p is the p-Laplacian with 1 < p and q > p –1. If p = 2, such problems arise in population dynamics. Ourmain results generalize the results for p = 2, but some technicaldifficulties arising from the nonlinear degenerate operator–_p are successfully overcome. As a by-product, we cansolve a free boundary problem for a nonlinear p-Laplacian equation.     

Numerical stabilization of polynomial and matrix

Email: jdhan{at}sia.cn Email: zhjiang{at}sia.cn Corresponding author Email: nyy{at}sia.cn Received on March 6, 2006; Accepted on September 4, 2006 In this paper, the conception of numerical stabilization, whichis related to mantissa digits of computer and dimensions ofsystem, is described; and several strategies for the numericalstabilization of polynomial and matrix are presented.     

Total Flux Estimates for a Finite-Element Approximation of Elliptic Equations

An elliptic boundary-value problem on a domain with prescribedDirichlet data on _I is approximated using a finite-elementspace of approximation power h^K in the L² norm. It is shownthat the total flux across _I can be approximated with an errorof O(h^K) when is a curved domain in Rⁿ (n = 2 or 3) and isoparametricelements are used. When is a polyhedron, an O(h^2K–2)approximation is given. We use these results to study the finite-elementapproximation of elliptic equations when the prescribed boundarydata on _I is the total flux. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton, Sussex BN1 9QH.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.     

Behaviour of a non-local equation modelling linear friction welding

A non-local parabolic equation modelling linear friction weldingis studied. The equation applies on the half line and containsa non-linearity of the form . For f(u) = e^u, global existence and convergence to a steadystate are proved. Numerical calculations are also carried outfor this case and for f(u) = (– u)^1/a.