Feedback operator for a pseudoparabolic optimal control problem

The feedback operator of a linear pseudoparabolic problem with quadratic criterion is obtained by decoupling of the optimality condition. The feedback operator is shown to be related to the solution of a Riccati equation formulated in theB*-algebra of bounded linear operators onL ²(). This approach shows that the linear feedback operator may be considered as a bounded operator fromL ²() intoH ₀ ¹ (). Finally, we give a theorem establishing the convergence behavior for the feedback operators for these problems as they formally approach an analogous problem of parabolic type.This work was supported in part by the National Science Foundation, Grant No. MCS-7902037.     

Entropy and Convergence on Compact Groups

We investigate the behaviour of the entropy of convolutions of independent random variables on compact groups. We provide an explicit exponential bound on the rate of convergence of entropy to its maximum. Equivalently, this proves convergence of the density to uniformity, in the sense of Kullback–Leibler. We prove that this convergence lies strictly between uniform convergence of densities (as investigated by Shlosman and Major), and weak convergence (the sense of the classical Ito–Kawada theorem). In fact it lies between convergence in L ¹⁺ and convergence in L ¹.     

A Locking-Free Nonconforming Finite Element Method for Planar Linear Elasticity

We propose a locking-free nonconforming finite element method based on quadrilaterals to solve for the displacement variable in the pure displacement boundary value problem of planar linear elasticity. The method proposed in this paper is optimal and robust in the sense that the convergence estimates in the energy and L ²-norms are independent of the Lamé parameter .     

Calculation of linear bestL p -approximations

In this paper we are concerned with finding theL _p -solution (i.e. minimizing theL _p -norm of the residual vector) to a linear approximation problem or, equivalenty, to an overdetermined system of linear equations. An embedding method is described in which the damped Newton iteration is applied to a series of perturbed problems in order to guarantee convergence and also increase the convergence rate.     

The regulator problem for parabolic equations with Dirichlet boundary control

This paper considers the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain , with control functionu acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes theL ₂(0, ; L₂())-norm of the solutiony and theL ₂(0, ; L₂())-norm of the Dirichlet controlu. The paper is divided in two parts. Part I derives, in a constructive way, the algebraic Riccati equation satisfied by the candidate Riccati operator solution (unique in our case) and, moreover, studies the regularity properties of the optimal pairu ⁰, y⁰. Part II studies a Galerkin approximation of the regulator problem. It shows first the uniform analyticity and the uniform exponential stability of the underlying discrete (approximating) semigroups. Then it establishes the desired convergence properties, in particular, pointwise Riccati operators convergence and, as a final goal, convergence of the original dynamics acted upon by the discrete feedbacks.Research partially supported by the National Science Foundation under Grant DMS-8301668.     

Limit theorems for the minimum interpoint distance between any pair of i.i.d. random points in R d

The limit theorem for the minimum interpoint distance between any pair of i.i.d. random points in R ^d with common density fL ² was studied by a method which makes use of the convergence of point processes. Some one-dimensional examples with fL ² (including the cases Beta and Gamma distributions) were also considered.     

The Central Approximation Theorems for the Method of Left Gamma Quasi-Interpolants in L p spaces

The optimal degree of approximation of the method of Gammaoperators G _n in L _p spaces is O(n ^-1). In order to obtain much faster convergence, quasi-interpolants G _n ^(k) of G _n in the sense of Sablonnière are considered. We show that for fixed k the operator-norms G _n ^(k) _p are uniformly bounded in n. In addition to this, for the first time in the theory of quasi-interpolants, all central problems for approximation methods (direct theorem, inverse theorem, equivalence theorem) could be solved completely for the L _p metric. Left Gamma quasi-interpolants turn out to be as powerful as linear combinations of Gammaoperators [6].     

On Two-Scale Convergence

In this paper we present the foundations of the two-scale convergence theory. We discuss the class of admissible functions in the mean-value formula and in the definition of two-scale convergence. We discuss the relation between strong and weak convergence and prove general theorems on the semicontinuity from below for convex functionals, and we also discuss the relation between two-scale convergence and monotonicity. The explanations are conducted on the level of L ^p -convergence; we do not deal with convergence in Sobolev spaces.     

A particle method for first-order symmetric systems

Summary We present and study a conservative particle method of approximation of linear hyperbolic and parabolic systems. This method is based on an extensive use of cut-off functions. We prove its convergence inL ² at the order as soon as the cut-off function belongs toW ^m+1.1.Dedicated to Professor Joachim Nitsche on the occasion of his 60th birthday     

Approximation of infinitely differentiable multivariate functions is intractable

We prove that L_∞-approximation of C^∞-functions defined on [0,1]^d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε(0,1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.     

Tanaka Formulae for (α, d, β)-Superprocesses

We establish Tanaka like formulae for the local time of (, d, )-superprocess in the dimensions where the local time exists. The result generalizes the result of Adler, Lewin who proved existence of Tanaka formulae for a class of super-processes with finite variance. The fact that we abandon the finite variance assumption, requires using an L ¹⁺ convergence argument (with 0<<1) rather than L ² convergence, for the derivation of the Tanaka formulae.     

Complete controllability of perturbed linear control systems,revisited

LetL ¹(I, ^n,n )×M(I, ^n,m ) be the space of all pairs (A, B), whereA andB are measurable functions from a compact intervalI to ^n,n and ^n,m , respectively, andA is Lebesgue integrable. Also, let this space be endowed with the topology of theL ¹-norm with respect toA and the topology of convergence in measure with respect toB. Then, the set of all pairs (A, B), for which the corresponding linear control system

The surprising almost everywhere convergence of Fourier-Neumann series

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L^p requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L² is the celebrated Carleson theorem, proved in 1966 (and extended to L^p by Hunt in 1967).In this paper, we take the system

     

On the Equivalence of Spectra of Linear Partial Differential Operators in Certain Function Spaces

For linear differential operators with coefficients of class C on ⁿ, we prove theorems on the simultaneous invertibility and equivalence of spectra in the Lebesgue space L ^p, Stepanov space M ^p, and in a particular Banach space V ^p L ^p, p 1     

Boundary control of parabolic systems: Regularity of optimal solutions

Boundary control problems for the linear, parabolic equations and a quadratic performance index are considered. The controls are allowed to be in the spaceL ²[OT;L²()], where is a boundary. Exploiting the semigroup approach, it is shown that optimal control belongs toL ²[OT;H^1/2()] and, as a consequence, optimal trajectory belongs toL ¹[OT;H¹()]. This result is obtained for two kinds of domains. The first are the domains withC -boundary and the second are the domains being the parallelepipeds.     

The fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L²‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order in H¹‐norm for the velocity and in L²‐norm for the pressure, where is the penalty parameter. The L²‐norm error estimate for the velocity is upgraded to . Moreover, we derive the a priori estimates depending on for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate in H¹‐norm for the velocity and in L²‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L²‐norm, the convergence rate is improved to . The theoretical results are verified by the numerical experiments.     

Zur Konvergenz derL n -Diskrepanz gegen die extreme Diskrepanz

This paper gives bounds for the rate of convergence of theL _n -discrepancy of a sequence of points of thes-dimensional space ^s (s. Further, a formula to calculate theL _n -discrepancy for even numbersn is given.     

Long steps in an O(n 3 L) algorithm for linear programming

We consider partial updating in Ye's affine potential reduction algorithm for linear programming. We show that using a Goldstein—Armijo rule to safeguard a linesearch of the potential function during primal steps is sufficient to control the number of updates. We also generalize the dual step construction to apply with partial updating. The result is the first O(n ³ L) algorithm for linear programming whose steps are not constrained by the need to remain approximately centered. The fact that the algorithm has a rigorous primal-only initialization actually reduces the complexity to less than O(m ^1.5 n ^1.5 L).     

A θ-stable approximations of abstract Cauchy problems

Summary We study the approximation of linear parabolic Cauchy problems by means of Galerkin methods in space andA -stable multistep schemes of arbitrary order in time. The error is evaluated in the norm ofL _t ² (H _x ¹ ) L _t (L _x ² ).     

Maximal L p –L q Regularity for Parabolic Partial Differential Equations on Manifolds with Cylindrical Ends

We give a short, simple proof of maximal L^p–L^q regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of -boundedness for the resolvent.