Complete observability of differential-algebraic systems with delays

We study the statement and solvability of complete observability problems for linear stationary differential-algebraic dynamical systems with delays (DAD systems. Since in the general case, the state space of such systems is infinite-dimensional and is not necessarily “minimal,” we consider various statements of problems depending on what states are observed. Our attention is focused on the simplest DAD system in symmetric form. We obtain efficient parametric criteria and analyze relationships between various notions of complete observability for DAD systems. In the case of DAD systems with scalar coefficients, we obtain a complete classification of notions of complete observability in the class of continuous initial functions with the continuous matching condition. We analyze the problem of computing the minimum number of outputs of a spectrally observable DAD system.     

A Perturbation Result for Dynamical Contact Problems

This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.     

CONSERVATION OF THREE—POINT COMPACT SCHEMES ON SINGLE AND MULTIBLOCK PATCHED GRIDS FOR HYPERBOLIC PROBLEMS

For nonlinear hyperbloic problems,Conservation of the numerical scheme is important for convergence to the correct weak solutions.In this paper the the conservation of the well-known compact scheme up to fourth order of accuracy on a single and uniform grid is studied,and a conservative interface treatment is derived for compact schemes on patched grids .For a pure initial value problem,the compact scheme is shown to be equivalent to a scheme in the usual conservative form .For the case of a mixed initial boundary value problem,the compact scheme is conservative only if the rounding errors are small enough.For a pactched grid interface,a conservative interface condition useful for mesh fefiement and for parallel computation is derived and its order of local accuracy is analyzed.     

On uniqueness conditions for optimal curve fitting

We show that the standard uniqueness condition for an important class of optimal curve-fitting problems is equivalent to an observability condition on an associated dynamical model. This fact allows uniqueness to be checked by testing the rank of a matrix.     

An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.     

约束力学系统的联络及其运动方程的测地性质

用现代整体微分几何方法研究非定常约束力学系统运动方程的测地性质，得到非定常力学系统的动力学流关于１＿射丛上的联络具有测地性质的充分必要条件·非定常情形下的动力学流关于无挠率的联络总具有测地性质，因此任何非定常约束力学系统在外力作用下的运动总可以表示为关于１＿射丛上无挠率的动力学联络的测地运动，这与定常力学的情形有所区别·     

Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface

In one space dimension we address the homogenization of the spectral problem for a singularly perturbed diffusion equation in a periodic medium. Denoting by ε the period, the diffusion coefficient is scaled as ε². The domain is made of two purely periodic media separated by an interface. Depending on the connection between the two cell spectral equations, three different situations arise when ε goes to zero. First, there is a global homogenized problem as in the case without an interface. Second, the limit is made of two homogenized problems with a Dirichlet boundary condition on the interface. Third, there is an exponential localization near the interface of the first eigenfunction. Received: January 10, 2001; in final form: July 9, 2001?Published online: June 11, 2002     

Inertial Manifolds for von Karman Plate Equations

Inertial manifolds associated with nonlinear plate models governed by dynamical von Karman equations are considered. Three different dissipative mechanisms are discussed: viscous, structural and thermal damping. Though the systems considered are subject to some dissipation, the overall dynamics may not be dissipative. This means that the energy may not be decreasing. The main result of the paper establishes the existence of an inertial manifold subject to the spectral gap condition for linearized problems. The validity of the spectral gap condition depends on the geometry of the domain and the type of damping. It is shown that the spectral gap condition holds for plates of rectangular shape. In the case of viscous damping, which is associated with hyperbolic-like dynamics, it is also required that the damping parameter be sufficiently large. This last requirement is not needed for other types of dissipation considered in the paper.     

Sensitivity of the slip rate coefficient in fluid flow – poroelastic coupling conditions

To model flow-induced structural vibrations, an interface to couple fluid flow and poroelastic material in a finite element formulation has been developed. One parameter of this interface condition is the slip rate coefficient, resulting from the so-called Beavers-Joseph-Saffman condition. This condition states that the jump in tangential velocity at a fluid flow – porous interface is proportional to the shear stress. Up to now no a priori determination of this parameter exists, and the known parameter range has been deducted from measurements, i. e., in our case from the results of the pore-resolving simulations. When modeling realistic problems assuming incompressible fluids, vectorial flow velocity and scalar pressure interact with the poroelastic material. As the slip rate coefficient only influences the tangential contributions, its overall influence is not clear. In this work, the sensitivity of the slip rate coefficient regarding the interface's coupling conditions is evaluated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

DAE-formulation for optimal solutions of a multirate model

A dynamical system including frequency modulated signals can be transformed into multirate partial differential algebraic equations. Optimal solutions are determined by a necessary condition. A method of lines yields a semi-discretisation in the case of initial-boundary value problems. We show that the resulting system can be written in a standard formulation of differential algebraic equations. Hence appropriate time integration schemes are available for a numerical solution. We present results for a test example modelling the electric circuit of a ring oscillator. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-scale homogenization in transmission problems of elasticity with interface jumps

The elasticity problem in a periodic structure with prescribed interface jumps in displacements and tractions and oscillating Neumann condition on a part of the external boundary is considered. This work is just a generalization of inhomogeneous Dirichlet and Neumann conditions on the oscillating interface. Such interface jumps arise, e.g. in contact problems with known periodic contact interface. Two-scale approach was applied to the problem and the two-scale convergence was proven. This article also provides a detailed auxiliary analysis for Sobolev functions with interface jumps.     

光滑函数集列紧的一个判据

本文讨论紧区间上的C~1光滑函数,用6-网构造法证明了这种函数集合列紧的一个判据,从而使著名的Ascoli-Arzela引理在光滑情形下得到推广,这一结果对讨论光滑动力系统有重要意义。     

Exponential dichotomy of linear systems with almost periodic matrices

Using the Lyapunov direct method, we establish a condition of exponential dichotomy for the class of dynamical systems under weaker assumptions as compared to the case of an arbitrary continuous matrix on the derivative of the Lyapunov function along the trajectories. By way of application, we obtain a sufficient condition for the dichotomy of a second order almost periodic vector equation in terms of coefficients.     

Asymptotic analysis of the differences between the Stokes-Darcy system with different interface conditions and the Stokes-Brinkman system

We consider the coupling of the Stokes and Darcy systems with different choices for the interface conditions. We show that, comparing results with those for the Stokes-Brinkman equations, the solutions of Stokes-Darcy equations with the Beavers-Joseph interface condition in the one-dimensional and quasi-two-dimensional (periodic) cases are more accurate than are those obtained using the Beavers-Joseph-Saffman-Jones interface condition and that both of these are more accurate than solutions obtained using a zero tangential velocity interface condition. The zero tangential velocity interface condition is in turn more accurate than the free-slip interface boundary condition. We also prove that the summation of the quasi-two-dimensional solutions converge so that the conclusions are also valid for the two-dimensional case.     

Two-layer muti-parameterized Schwarz Alternating Method

The convergence rate of a numerical procedure based on Schwarz Alternating Method (SAM) for solving elliptic boundary value problems (BVP’s) depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that theRobin condition(mixed interface condition), controlled by a parameter, can optimize SAM’s convergence rate. Since the convergence rate is very sensitive to the parameter, Tang[17] suggested another interface condition calledover- determined interface condition. Based on the over-determined interface condition, we formulate thetwo-layer multi-parameterized SAM. For the SAM and the one-dimensional elliptic model BVP’s, we determine analytically the optimal values of the parameters. For the two-dimensional elliptic BVP’s, we also formulate the two-layer multiparameterized SAM and suggest a choice of multi-parameter to produce good convergence rate.     

On the theory of dynamic thermoelasticity : PMM, vol. 42, no. 6, 1978, pp. 1093–1098

Use of the causality principle as radiation condition in dynamical problems of thermoelasticity is proposed. It follows from an analysis of the fundamental mathematical models describing the thermoelastic behavior of a continuous medium and used in the solution of specific problems, that some will yield physically unrealizable solutions. To eliminate the ambiguity in the solution which occurs, an approach is possible which has an explicit physical meaning and is based on the causality principle [1, 2]; it is required that the time source not yield a response earlier than the time of starting up of the source. Different kinds of radiation conditions of the Sommerfeld type are known in thermoelasticity problems [3 – 6].

To extract the unique solution in dynamical thermoelasticity problems, it is proposed in this paper to use the causality principle, which is equivalent to the requirement of analyticity of the solution in the upper half of the complex frequency plane; there are studied the analytic properties of the solutions of the fundamental boundary value problems for the models used most often for thermoelastic media, and there are made deductions about their physical realizability.     

New conditions for synchronization in complex networks with multiple time-varying delays

In this paper, two kinds of synchronization problems of complex dynamical networks with multiple time-varying delays are investigated, that is, the cases with fixed topology and with switching topology. For the former, different from the commonly used linear matrix inequality (LMI) method, we adopt the approach basing on the scramblingness property of the network’s weighted adjacency matrix. The obtained result implies that the network will achieve exponential synchronization for appropriate communication delays if the network’s weighted adjacency matrix is of scrambling property and the coupling strength is large enough. Note that, our synchronization condition is very new, which would be easy to check in comparison with those previously reported LMIs. Moreover, we extend the result to the case when the interaction topology is switching. The maximal allowable upper bounds of communication delays are obtained in each case. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.     

A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions

We present and analyze a new fictitious domain model for the Brinkman or Stokes/Brinkman problems in order to handle general jump embedded boundary conditions (J.E.B.C.) on an immersed interface. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface Σ separating two subdomains: they are well chosen to get the coercivity of the operator. It is issued from a generalization to vector elliptic problems of a previous model stated for scalar problems with jump boundary conditions (Angot (2003, 2005) [2], [3]). The proposed model is first proved to be well-posed in the whole fictitious domain and some sub-models are identified. A family of fictitious domain methods can be then derived within the same unified formulation which provides various interface or boundary conditions, e.g. a given stress of Neumann or Fourier type or a velocity Dirichlet condition. In particular, we prove the consistency of the given-traction E.B.C. method including the so-called do nothing outflow boundary condition.     

Conservation laws for invariant functionals containing compositions§

The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler–Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work, we prove a generalization of the necessary optimality condition of DuBois–Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic.