Controllability of linear systems,differential geometry curves in grassmannians and generalized grassmannians,and riccati equations

We study the linear system =Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators e^t B. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ⁿ. We then view (t)=Im(e^t B) as a curve in appropriate Grassmannian and see that, in local coordinates, is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on that are equivalent to the controllability of the system. To get quantitiative results, we lift to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ⁿ). Explicit and simple expressions concerning the geometry of are computed in terms of the Lie algebra of GL(ⁿ), and these are related to the controllability of the system.James Wolper was a visiting professor in the Department of Mathematics at Texas Tech University while much of this research was conducted. He would like to express appreciation for the hospitality he received during his visit.     

P-harmonic obstacle problems

We prove an extension theorem for Sobolev functions which plays an important role in the partial regularity theory of vector valued functions which minimize degenerate variational integrals subject to some nonlinear side conditions in the image space.     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem

Summary. This paper is concerned with a high order convergent discretization for the semilinear reaction-diffusion problem: , for , subject to , where . We assume that on , which guarantees uniqueness of a solution to the problem. Asymptotic properties of this solution are discussed. We consider a polynomial-based three-point difference scheme on a simple piecewise equidistant mesh of Shishkin type. Existence and local uniqueness of a solution to the scheme are analysed. We prove that the scheme is almost fourth order accurate in the discrete maximum norm, uniformly in the perturbation parameter . We present numerical results in support of this result. Received February 25, 1994     

Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points

We consider first the initial-boundary value problem for the parabolic equation

Optimal inspection and perfect repair

Models are developed for decision making about monitoring andmaintenance of systems whose performance through time is describedby a general stochastic process. The system is monitored andpreventive and corrective maintenance actions are carried outin response to the observed system state. The decision processis simplified by using the maximum process as a decision variable.The models developed generalize age replacement models and othersimple maintenance strategies. The approach can deal with failuresthat prevent the system functioning further, and also failuresthat are defined by regulation or economic considerations. Attentionis restricted to perfect repair and inspection, but the structureprovides the framework for further developments.     

Equivalence between mixed finite element and multi-point finite volume methods

We consider the lowest-order Raviart–Thomas mixed finite element method for elliptic problems on simplicial meshes in two or three space dimensions. This method produces saddle-point type problems for scalar and flux unknowns. We show how to easily eliminate the flux unknowns, which implies an equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. We describe the stencil of the final matrix and give sufficient conditions for its symmetry and positive definiteness. We present a numerical example illustrating the performance of the proposed method. To cite this article: M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Corrigendum: The complexity of counting graph homomorphisms

We close a gap in the proof of Theorem 4.1 in our paper “The complexity of counting graph homomorphisms” [Random Structures Algorithms 17 (2000), 260–289]. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004