Controllability of the Ornstein-Uhlenbeck equation

^** Email: Leiva{at}ula.ve In this paper we study the controllability of the followingcontrolled Ornstein–Uhlenbeck equation [graphic: see PDF] then the system is approximately controllable on [0, t₁]. Moreover,the system can never be exactly controllable.     

Space-time discretization of the heat equation

An algorithm for a stable parallelizable space-time Petrov-Galerkin discretization for linear parabolic evolution equations is given. Emphasis is on the reusability of spatial finite element codes.     

Controllability problems for the string equation

For the string equation controlled by boundary conditions, we establish necessary and sufficient conditions for 0-and ε-controllability. The controls that solve such problems are found in explicit form. Moreover, using the Markov trigonometric moment problem, we construct bangbang controls that solve the problem of ε-controllability. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 939–952, July, 2007.     

Controllability and asymptotic stabilization of the Camassa-Holm equation

We investigate the problems of exact controllability and asymptotic stabilization of the Camassa-Holm equation on the circle, by means of a distributed control. The results are global, and in particular the control prevents the solution from blowing up.     

Improved upwind discretization of the advection equation

The simplest upwind discretization of the advection equation is only first‐order accurate in time and space and very diffusive. In this article, the first‐order upwind method is improved by changing its basis functions. The resulting scheme, called exponentially fitted, proves to be more accurate in both space and time. In addition, it inherits some qualitative behaviors of the advection equation. The proposed approach is able to be generalized for more complicated problems provided that appropriate relations between the fitting parameters of the method are imposed. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 773–787, 2014     

Finite difference discretization of the Kuramoto-Sivashinsky equation

Summary We analyze a Crank-Nicolson-type finite difference scheme for the Kuramoto-Sivashinsky equation in one space dimension with periodic boundary conditions. We discuss linearizations of the scheme and derive second-order error estimates.Work supported by the Institute of Applied and Computational Mathematics of the Research Center of Crete-FORTH     

Effective discretization of the two-dimensional wave equation

The wave equation and associated spherical means are a widespread model in modern imaging modalities like photoacoustic tomography. We consider a discretization for the Cauchy problem of the two dimensional wave equation by plane waves. The considered frequencies lie on a Cartesian or on a polar grid which gives rise to efficient algorithms for the computation of the spherical means. The theoretical findings are illustrated by a some numerical experiments. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite difference discretization of the Rosenau-RLW equation

In this paper, numerical solutions of the Rosenau-RLW equation are considered using Crank–Nicolson type finite difference method. Existence of the numerical solutions is derived by Brouwer fixed point theorem. A priori bound and the error estimates as well as conservation of discrete mass and discrete energy for the finite difference solutions are discussed. Unconditional stability, second-order convergence and uniqueness of the scheme are proved using discrete energy method. Some numerical experiments have been conducted in order to verify the theoretical results.     

Controllability of the Ginzburg–Landau equation

This Note investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Null-controllability results are derived from a Carleman estimate and an analysis based upon the theory of sectorial operators. To cite this article: L. Rosier, B.-Y. Zhang, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Finite difference discretization of the cubic Schrdinger equation

We analyze the discretization of an initial-boundary value problemfor the cubic Schrdinger equation in one space dimension bya Crank-Nicolson-type finite difference scheme. We then linearizethe corresponding equations at each time level by Newton's methodand discuss an iterative modification of the linearized schemewhich requires solving linear systems with the same tridiagonalmatrix. We prove second-order error estimates. Work supported by the Institute of Applied and ComputationalMathemati of the Research Center of Crete-FORTH.     

On the discretization of a delay differential equation

The dynamics of the delay difference equation μ[δx_n+αδx_n-N]=-X_n+1+f(x_n-N)asn→:∞is studied for small positive μ. The equation is shown to possess stable periodic solutions that correspond to hyperbolic attracting cycles of the one–dimensional map ?.     

Monotonic dynamical systems under spatial discretization

We estimate the probability of replicating the asymptotic behaviour of a dynamical system generated by a monotonic mapping for randomly centered roundoff lattices.

     

Controllability of the wave equation with moving point control

This paper deals with approximate and exact controllability of the wave equation in finite time with interior point control acting along a curve specified in advance in the system's spatial domain. The structure of the control input is dual to the structure of the observations which describe the measurements of velocity and gradient of the solution of the dual system, obtained from the moving point sensor. A relevant formalization of such a control problem is discussed, based on transposition. For any given timeinterval [0,T] the existence of the curves providing approximate controllability inH _D ^–[n/2]–1 ()×H _D ^–[n/2]–1 () (wheren stands for the space dimension) is established with controls fromL ²(0,T; R ⁿ +1). The same curves ensure exact controllability inL ²() × H^–1() if controls are allowed to be selected in [L (0,T; R ⁿ+1)]. Required curves can be constructed to be continuous on [0,T).This work was supported in part by NSF Grant ECS 89-13773 and NASA Grant NAG-1-1081.     

On the exact discretization of the classical harmonic oscillator equation

We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical applications.     

Implicit time discretization for the mean curvature flow equation

In this paper we apply the method of implicit time discretization to the mean curvature flow equation including outer forces. In the framework ofBV-functions we construct discrete solutions iteratively by minimizing a suitable energy-functional in each time step. Employing geometric and variational arguments we show an energy estimate which assures compactness of the discrete solutions. An additional convergence condition excludes a loss of area in the limit. Thus existence of solutions to the continuous problem can be derived. We append a brief discussion of the related Mullins-Sekerka equation.This work was supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 256, Bonn     

Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations

The multidimensional quasi-gasdynamic system written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization of these equations on a nonuniform rectangular grid is constructed (with the basic unknown functions—density, velocity, and temperature—defined on a common grid and with fluxes and viscous stresses defined on staggered grids). Primary attention is given to the analysis of entropy behavior: the discretization is specially constructed so that the total entropy does not decrease. This is achieved via a substantial revision of the standard discretization and applying numerous original features. A simplification of the constructed discretization serves as a conservative discretization with nondecreasing total entropy for the simpler quasi-hydrodynamic system of equations. In the absence of regularizing terms, the results also hold for the Navier–Stokes equations of a viscous compressible heat-conducting gas.     

Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes

We present and analyze several ways of discretizing first-order Hamilton-Jacobi equations on unstructured meshes. We first discuss two Godunov-type Hamiltonians: the first one is an extension of a result by Bardi and Osher, where a particular decomposition of the initial condition is assumed, and we point out its practical limits; the other one arises from a particular decomposition of the Hamiltonian. Despite its complexity this decomposition enables us to construct a Lax-Friedrichs Hamiltonian. These schemes all share common properties: They are consistent, monotonic, and independent of the geometrical interpretation of the piecewise linear initial condition. Under these assumptions and classical ones on the mesh, we show these schemes are convergent. We describe their high-order extensions using the ENO technique and provide numerical illustrations. © 1996 John Wiley & Sons, Inc.     

Error estimates for mixed finite element discretization of the Richards equation

We present a numerical scheme based on the mixed finite element method (MFEM) for the Richards equation, a nonlinear, degenerate parabolic equation. Due to the degeneracy, the solution of the equation has low regularity and therefore only lower order finite elements are recommended. We review the main posibilities for proving the convergence of the scheme. Especially for the case without using the Kirchhoff transformation a new result is given. We also briefly discuss how to solve the nonlinear fully discrete problems appearing at each time step and refer to papers where the convergence of these methods is rigurously studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time discretization of an evolution equation via Laplace transforms

Following earlier work by Sheen, Sloan, and Thomée concerningparabolic equations we study the discretization in time of aVolterra type integro-differential equation in which the integraloperator is a convolution of a weakly singular function andan elliptic differential operator in space. The time discretizationis accomplished by using a modified Laplace transform in timeto represent the solution as an integral along a smooth curveextending into the left half of the complex plane, which isthen evaluated by quadrature. This reduces the problem to afinite set of elliptic equations with complex coefficients,which may be solved in parallel. Stability and error boundsof high order are derived for two different choices of the quadraturerule. The method is combined with finite-element discretizationin the spatial variables.