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.     

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).     

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.     

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.     

Theta-characteristics on singular curves

On a smooth curve a theta-characteristic is a line bundle L,the square of which is the canonical line bundle . The equivalentcondition om(L, ) L generalizes well to singular curves, asapplications show. More precisely, a theta-characteristic isa torsion-free sheaf of rank 1 with om(, ) . If the curvehas non-ADE singularities, then there are infinitely many theta-characteristics.Therefore, theta-characteristics are distinguished by theirlocal type. The main purpose of this article is to compute thenumber of even and odd theta-characteristics (that is withh⁰(C, ) 0 and h⁰(C, ) 1 modulo 2, respectively) in terms ofthe geometric genus of the curve and certain discrete invariantsof a fixed local type.     

On the Optimum Criterion of Polynomial Stability

The purpose of this note is to answer the question raised byNie & Xie (1987). Let f(x)=a₀xⁿ+a₁x^n–1+...+a_n be apositive-coefficient polynomial. The numbers ₁=a_i-1a_i+2/a_ia_i+1(i=1, ..., n–2) are called determining coefficients. Theoptimum criterion problem was posed as follows: for n3, findthe maximal number (n) such that the polynomial f(x) is stableif _i < (n) (1in–2). For n6, we show that (n)=ß,where ß is the unique real root of the equation x(x+1)²=1.     

Root-Clustering Criteria (I): The Composite-Matrix Approach

In a series of articles we investigate different approachesto root clustering criteria. In particular, given A C^nxn andan algebraic region A C, we are looking for a criterion (asingle set of inequalities) ensuring the inclusion (A) A. Here,in the first article, we collect previous ideas on transformableregions and composite matrices, state new results, and correctand refine others.     

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 .     

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.     

ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN

In this paper, we consider the following semilinear ellipticproblem: where f(x, t) tends to p(x) and q(x) L(^N), respectively, ast 0 and t +. We prove that there exist two numbers l and Lwith L < l such that problem (P) has at least one positivesolution for (– l, –L) and has no positive solutionfor all [–l,–L]. The existence and non-existenceof positive solutions for problem (P) at = –l and =–L are also 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.     

Approximate parametric optimization of infinite-dimensional systems

We consider the problem of finding g M_n such that where M_n is the n-dimensional subspace of the complexHilbert space L²(0, ) spanned by an n-tuple of normalized eigenvectoesof the operator , corresponding to eigenvalues. The solution is g = P_nf and P_n denotesthe orthoprojector onto M_n. From Grabowski (1991) we know thatP_n can be expressed in terms of the Malmquist functions. Wegive an alternative approach, more convenient for applicationof the standard mathematical software. The problem of convergenceas n is discussed from both theoretical and numerical viewpoint.The reslts are illustrated by the problems of finding the optimaladjustment of the proportional controller stabilizing a distributedplant. Email: pgrab{at}ia.agh.edu.pl     

Sparse Systems of Functions Closed on Large Sets in RN

Let E(Z) = {e^inx}_nZ denote the trigonometrical exponential system.It is well known that E(Z) forms an orthogonal basis in thespace L²(0, 2). In 1964, H. Landau discovered that the trigonometricalsystem has the following property: certain small perturbationsof E(Z) yield exponential systems which are complete in L² onany finite union of 2-periodic translations of any interval(, 2–), 0 < < .     

A Class of Infinite Dimensional Simple Lie Algebras

Let A be an abelian group, F be a field of characteristic 0,and , ß be linearly independent additive maps fromA to F, and let ker()\{0}. Then there is a Lie algebra L = L(A,, ß, ) = _xA Fe_x under the product [e_x, e_y]]=(x–y)e_x+y+(ß) (x, y) e_x+y–. If, further, ß() = 1, and ß(A) = Z, thereis a subalgebra L⁺:=L(A⁺, , ß, ) = _xA⁺ Fe_x, whereA⁺ = {xA|ß(x)0}. The necessary and sufficient conditionsare given for L' = [L, L] and L⁺ to be simple, and all semi-simpleelements in L' and L⁺ are determined. It is shown that L' andL⁺ cannot be isomorphic to any other known Lie algebras andL' is not isomorphic to any L⁺, and all isomorphisms betweentwo L' and all isomorphisms between two L⁺ are explicitly described.     

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.     

The gradient iteration for approximation in reproducing kernel Hilbert spaces

Consider the bounded linear operator, L: F Z, where Z R^N andF are Hilbert spaces defined on a common field X. L is madeup of a series of N bounded linear evaluation functionals, L_i:F R. By the Riesz representation theorem, there exist functionsk(x_i, ^·) F : L_if = f, k(x_i, ^·)_F. The functions,k(x_i, ^·), are known as reproducing kernels and F is areproducing kernel Hilbert space (RKHS). This is a natural frameworkfor approximating functions given a discrete set of observations.In this paper the computational aspects of characterizing suchapproximations are described and a gradient method presentedfor iterative solution. Such iterative solutions are desirablewhen N is large and the matrix computations involved in thebasic solution become infeasible. This is also exactly the casewhere the problem becomes ill-conditioned. An iterative approachto Tikhonov regularization is therefore also introduced. Unlikeiterative solutions for the more general Hilbert space setting,the proofs presented make use of the spectral representationof the kernel.     

Sub-Laplacians of Holomorphic Lp-Type on Exponential Solvable Groups

Let L denote a right-invariant sub-Laplacian on an exponential,hence solvable Lie group G, endowed with a left-invariant Haarmeasure. Depending on the structure of G, and possibly alsothat of L, L may admit differentiable L^p-functional calculi,or may be of holomorphic L^p-type for a given p 2. ‘HolomorphicL^p-type’ means that every L^p-spectral multiplier for Lis necessarily holomorphic in a complex neighbourhood of somenon-isolated point of the L²-spectrum of L. This can in factonly arise if the group algebra L¹(G) is non-symmetric. Assume that p 2. For a point in the dual g^* of the Lie algebrag of G, denote by ()=Ad^*(G) the corresponding coadjoint orbit.It is proved that every sub-Laplacian on G is of holomorphicL^p-type, provided that there exists a point g^* satisfying Boidol'scondition (which is equivalent to the non-symmetry of L¹(G)),such that the restriction of () to the nilradical of g is closed.This work improves on results in previous work by Christ andMüller and Ludwig and Müller in twofold ways: on theone hand, no restriction is imposed on the structure of theexponential group G, and on the other hand, for the case p>1,the conditions need to hold for a single coadjoint orbit only,and not for an open set of orbits. It seems likely that the condition that the restriction of ()to the nilradical of g is closed could be replaced by the weakercondition that the orbit () itself is closed. This would thenprove one implication of a conjecture by Ludwig and Müller,according to which there exists a sub-Laplacian of holomorphicL¹ (or, more generally, L^p) type on G if and only if there existsa point g^* whose orbit is closed and which satisfies Boidol'scondition.     

Finite Element Approximation of a Rigid Punch Indenting a Membrane

Optimal order H¹ and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, _c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R², given ,g R⁺ and an obstacle defined over E we consider the minimization of |v|²_1,+gµover (v, µ) H¹₀() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (_c,^h_c)Ch ln 1/h, where ^h_c is the finite element approximation to_c. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.