Sliding mode control of the forced generalized Burgers equation

^** Email: smaoui{at}mcs.sci.kuniv.edu.kw This paper deals with the sliding mode control (SMC) of theforced generalized Burgers equation via the Karhunen-Loève(K-L) Galerkin method. The decomposition procedure of the K-Lmethod is presented to illustrate the use of this method inanalysing the numerical simulations data which represent thesolutions of the forced generalized Burgers equation for viscosityranging from 1 to 100. The K-L Galerkin projection is used asa model reduction technique for non-linear systems to derivea system of ordinary differential equations (ODEs) that mimicsthe dynamics of the forced generalized Burgers equation. Thedata coefficients derived from the ODE system are then usedto approximate the solutions of the forced Burgers equation.Finally, static and dynamic SMC schemes with the objective ofenhancing the stability of the forced generalized Burgers equationare proposed. Simulations of the controlled system are givento illustrate the developed theory.     

Non-Gaussian non-stationary models for natural hazard modeling

This paper addresses the construction of probabilistic models for time or space dependent natural hazards. The proposed method uses Karhunen-Loève expansion in order to construct an empirical model matching the non-stationarity and the randomness of natural phenomena such as earthquakes or other complex environmental processes. The terms of the Karhunen-Loève expansion are identified directly from measured data. The approach is illustrated and its performance assessed through two academic examples. It is then applied to seismic ground motion modeling using recorded data.     

Kalman filtering on approximate state-space models

Several state-space models for estimating a second-order stochastic process are proposed in this paper on the basis of the approximate Karhunen-Loève expansion. Properties of these models are studied and then the Kalman filtering method is applied. The accuracy of the models on the basis of two different situations, deterministic or random inputs, is studied by means of a simulation of a Brownian motion.This work was supported in part by DGICYT, Project No. PS93-0201.     

Nontrivial solutions for Neumann problems with an indefinite linear part

In this paper, we consider a semilinear Neumann problem with an indefinite linear part and a Carathéodory nonlinearity which is superlinear near infinity and near zero, but does not satisfy the Ambrosetti-Rabinowitz condition. Using an abstract existence theorem for C¹-functions having a local linking at the origin, we establish the existence of at least one nontrivial smooth solution.     

A MAPLE package of new ADM-Padé approximate solution for nonlinear problems

Based on the new definition of four classes of Adomian polynomials proposed by Rach in 2008, a MAPLE package of new Adomian-Padé approximate solution for solving nonlinear problems is presented. This package combines the merits of the Adomian decomposition method and the diagonal Padé technique, and may give more accurate solutions of nonlinear problems with strong nonlinearity. Besides, the package is user-friendly and efficient, one only needs to input the initial conditions, governing equation and four optional parameters, then our package will output the analytic approximate solution within a few seconds, where the equation is decomposed into three parts, they are the linear term R, nonlinear term NN and source function g, which are all in functional form. Meanwhile, several graphs generated from the above solutions are displayed and demonstrate a favorable comparison. In this paper, several different types of examples are given to illustrate the validity and promising flexibility of the package. This package provides us with a convenient and useful tool for dealing with nonlinear problems, as well as its electronic version is free to download via the journal website.     

Asymptotic stability in the pth moment for stochastic differential equations with Lévy noise

This paper is devoted to study a class of stochastic differential equations with Lévy noise. In comparison to the standard Gaussian noise, Lévy noise is more versatile and interesting with a wider range of applications. However, Lévy noise makes the analysis more difficult owing to the discontinuity of its sample paths. In this paper, we attempt to overcome this difficulty. We propose several sufficient conditions under which we investigate the long-time behavior of the solution including the asymptotic stability in the pth moment and almost sure stability. Also, we discuss two types of continuity of the solution: continuous in probability and continuous in the pth moment. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.     

On some aspects of Bäcklund transformations

The Painlevé differential equations (P₂-P₆) possess Bäcklund transformations which relate one solution to another solution either of the same equation, with different values of the parameters, or another such equation. We review a method for deriving difference equations, the discrete Painlevé equations in particular, from Bäcklund transformations of the continuous Painlevé equations. Then, we prove the existence of an algebraic formula relating three inconsecutive solutions of the same Bäcklund hierarchy for P₃ and P₄.     

Detecting Fourier Subspaces

Let G be a finite abelian group. We examine the discrepancy between subspaces of \(l^2(G)\) which are diagonalized in the standard basis and subspaces which are diagonalized in the dual Fourier basis. The general principle is that a Fourier subspace whose dimension is small compared to \(|G| = \mathrm{dim}\left( l^2(G)\right) \) tends to be far away from standard subspaces. In particular, the recent positive solution of the Kadison–Singer problem shows that from within any Fourier subspace whose dimension is small compared to |G| there is a standard subspace which is essentially indistinguishable from its orthogonal complement.     

Quasi-reflexive Fréchet spaces and mean ergodicity

We characterize quasi-reflexive Fréchet spaces with a basis in terms of the properties of this basis. As a consequence we prove that a Fréchet space with a basis is quasi-reflexive of order one if and only if for every power bounded operator T, either T or T^′ is mean ergodic.     

Constrained multi-degree reduction of triangular Bézier surfaces using dual Bernstein polynomials

We propose a novel approach to the problem of multi-degree reduction of Bézier triangular patches with prescribed boundary control points. We observe that the solution can be given in terms of bivariate dual discrete Bernstein polynomials. The algorithm is very efficient thanks to using the recursive properties of these polynomials. The complexity of the method is O(n²m²), n and m being the degrees of the input and output Bézier surfaces, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined triangular Bézier surfaces, the result is a composite surface of global C^r continuity with a prescribed order r. Some illustrative examples are given.     

Carathéodory interpolation on the non-commutative polydisk

The Carathéodory problem in the N-variable non-commutative Herglotz-Agler class and the Carathéodory-Fejér problem in the N-variable non-commutative Schur-Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory-Fejér) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set Λ) takes operator values with positive semidefinite real part (resp., contractive operator values) on N-tuples of Λ-jointly nilpotent contractive n×n matrices, for all n∈N.     

Benders's method revisited

The master problem in Benders's partitioning method is an integer program with a very large number of constraints, each of which is usually generated by solving the integer program with the constraints generated earlier. Computational experience shows that the subset B of those constraints of the master problem that are satisfied with equality at the linear programming optimum often play a crucial role in determining the integer optimum, in the sense that only a few of the remaining inequalities are needed. We characterize this subset B of inequalities. If an optimal basic solution to the linear program is nondegenerate in the continuous variables and has p integer-constrained basic variables, then the corresponding set B contains at most 2^p inequalities, none of which is implied by the others. We give an efficient procedure for generating an appropriate subset of the inequalities in B, which leads to a considerably improved version of Benders's method.     

Near-field and far-field approximations by the Adomian and asymptotic decomposition methods

The Adomian decomposition method and the asymptotic decomposition method give the near-field approximate solution and far-field approximate solution, respectively, for linear and nonlinear differential equations. The Padé approximants give solution continuation of series solutions, but the continuation is usually effective only on some finite domain, and it can not always give the asymptotic behavior as the independent variables approach infinity. We investigate the global approximate solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations. For several examples we find that there exists an overlap between the near-field approximation and the far-field approximation, so we can match them to obtain a global approximate solution. For other nonlinear examples where the series solution from the Adomian decomposition method has a finite convergent domain, we can match the Padé approximant of the near-field approximation with the far-field approximation to obtain a global approximate solution representing the true, entire solution over an infinite domain.     

Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs

In this paper we study mathematically and computationally optimal control problems for stochastic elliptic partial differential equations. The control objective is to minimize the expectation of a tracking cost functional, and the control is of the deterministic, distributed type. The main analytical tool is the Wiener-Itô chaos or the Karhunen-Loève expansion. Mathematically, we prove the existence of an optimal solution; we establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations; we represent the input data in their Wiener-Itô chaos expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the optimality system through the discretizations of the probability space and the spatial space by the finite element method; we also derive error estimates in terms of both types of discretizations.     

A posteriori error estimates and domain decomposition with nonmatching grids

Let F be a nonlinear mapping defined from a Hilbert space X into its dual X, and let x be in X the solution of F(x)=0. Assume that, a priori, the zone where the gradient of the function x has a large variation is known. The aim of this article is to prove a posteriori error estimates for the problem F(x)=0 when it is approximated with a Petrov–Galerkin finite element method combined with a domain decomposition method with nonmatching grids. A residual estimator for a model semi-linear problem is proposed. We prove that this estimator is asymptotically equivalent to a simplified one adapted to parallel computing. Some numerical results are presented, showing the practical efficiency of the estimator. AMS subject classification 65J10, 65N55, 65M60T. Sassi: Present address: Université de Caen, Laboratoire de Mathématiques Nicolas Oresme, UFR Sciences Campus II, Bd Maréchal Juin, 14032 Caen Cedex, France.     

On the trend to equilibrium for the Vlasov-Poisson-Boltzmann equation

The dynamics of dilute electrons and plasma can be modeled by Vlasov-Poisson-Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is O(t−∞), by using a method developed for the Boltzmann equation without external force in [L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 (2005) 245-316]. In particular, the idea of this method is to show that the solution f cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro-micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

The ovals in the Excerpta Mathematica and the origins of Descartes’ method of normals

The only occurrence of Descartes’ method of normals before La Géométrie (1637) is to be found in the Excerpta Mathematica. These mathematical fragments, published posthumously among others works in 1701, and dated by Tannery before 1629, deal with curves used in dioptrics which Descartes called ovals. I study in detail two of the texts on ovals together with the related texts in La Géométrie in order to shed light on the geometrical origins of Descartes’ method of normals.     

On the theories used for the wave propagation in laminated composite thin elastic shells

In this paper, the problem of propagation of free harmonic waves in cross-ply laminated thin elastic shells is considered. For this problem, a theoretical unification of the most commonly used, in physical and engineering applications, thin shell theories which take into consideration transverse shear deformation effects is presented. In more detail, the problem is formulated in such a way that by using some tracers, which have the form of Kronecker's deltas, the obtained stress-strain relations, constitutive equations and equations of motion produce, as special cases, the corresponding relations and equations of the transverse shear deformable analogs of Donnell's, Love's and Sanders' theories. Using an eigenvalue form solution of the equations of motion, a comparison of corresponding numerical results obtained on the basis of all of the afore-mentioned theories is made. Comparisons with corresponding results obtained on the basis of the classical thin shell theories of Donnell, Love, Sanders and Flugge are also made.

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Resumé On étudie un problème de propagation d'ondes harmoniques libres dans une fine coque cylindrique, élastique, composée de lamelles croisées. On présente une unification des théories les plus courantes que les ingénieurs et physiciens appliquent aux problèmes de coques minces sous considération de la déformation due au cisaillement transverse. En détails, le problème est formulé de telle façon qu'en utilisant des opérateurs de trace (sous forme du symbole de Kronecker) les relations obtenues: contrainte-déformation, équations constitutives et équations de mouvement donnent comme cas spéciaux les relations correspondantes et les équations des problèmes analogues (déformation de coques minces par cisaillement transverse) des théories de Donnell, Love et Sanders. En utilisant une solution aux valeurs propres des équations de mouvement, on compare les résultats numériques obtenus grâce aux théories mentionées ci-dessus aux résultats correspondants sur la base des théories classiques de Donnell, Love, Sander et Flugge.

     

A method of asymptotic constructions with improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation

The Dirichlet problem on an interval for quasilinear singularly perturbed parabolic convection-diffusion equation is considered. The higher order derivative of the equation is multiplied by a parameter ε that takes any values from the half-open interval (0, 1]. For this type of linear problems, the order of the ε-uniform convergence (with respect to x and t) for the well-known schemes is not higher than unity (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge ε-uniformly at the rate of O(N ^?2ln² N + N ^?2 ₀), where N + 1 and N ₀ + 1 are the numbers of the mesh points with respect to x and t, respectively. On the x axis, piecewise uniform meshes that condense in the boundary layer are used. If the parameter value is small compared to the effective step of the spatial grid, the domain decomposition method is used, which is motivated by “asymptotic constructions.” Monotone approximations of “auxiliary” subproblems describing the main terms of the asymptotic expansion of the solution outside a neighborhood of the boundary layer neighborhood are used. In the neighborhood of the boundary layer (of the width O(ε ln N)) the first derivative with respect to x is approximated by the central difference derivative. These subproblems are successively solved in the subdomains on uniform grids. If the parameter values are not sufficiently small (compared to the effective step of the mesh with respect to x), the classical implicit difference schemes approximating the first derivative with respect to x by the central difference derivative are applied. To improve the accuracy in t, the defect correction technique is used. Notice that the calculation of the solution of the constructed difference scheme (the scheme based on the method of asymptotic constructions) can be considerably simplified for sufficiently small values of the parameter ε.