Exact boundary observability for 1‐D quasilinear wave equations

By means of a direct and constructive method based on the theory of semi‐global C² solution, the local exact boundary observability and an implicit duality between the exact boundary controllability and the exact boundary observability are shown for 1‐D quasilinear wave equations with various boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi‐global C² solution, we establish the local exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the first‐order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.     

Exact boundary observability for a kind of second order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C² solution, we establish the local exact boundary observability for a kind of second order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary observability for a kind of first order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled.     

Exact boundary observability for nonautonomous first-order quasilinear hyperbolic systems

By means of the theory on the semiglobal C¹ solution to the mixed initial-boundary value problem for first-order quasilinear hyperbolic systems, we establish the local exact boundary observability for general nonautonomous first-order quasilinear hyperbolic systems without zero eigenvalues and reveal the essential difference between nonautonomous hyperbolic systems and autonomous hyperbolic systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings

Based on the theory of semi‐global piecewise C² solutions to 1D quasilinear wave equations, the local exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings with general topology is obtained by a constructive method. The principles of providing nodal profiles and of choosing and transferring boundary controls are presented, respectively. Copyright © 2013 John Wiley & Sons, Ltd.     

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System — the HUM Method*

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish ...     

Exact boundary controllability of nodal profile for 1-D quasilinear wave equations

Based on the theory of semi-global C ² solution for 1-D quasilinear wave equations, the local exact boundary controllability of nodal profile for 1-D quasilinear wave equations is obtained by a constructive method, and the corresponding global exact boundary controllability of nodal profile is also obtained under certain additional hypotheses.     

Stabilization of elasticity–viscoporosity system by linear boundary feedback

The initial boundary value problem for linear elastodynamic system for viscoporous materials is considered. Exponential decay of solutions via the linear boundary feedback is established. Existence of solutions is obtained through the method of c₀‐semigroups. Exponential stabilization is derived via a proper collection of ideas of observability inequality, energy identity and c₀‐semigroup of contractions. Copyright © 2008 John Wiley & Sons, Ltd.     

一维线性波动方程耦合组的精确边界同步能观性

作者介绍了多种精确同步能观性,并对一维波动方程耦合组在多种边界条件下分别实现了精确边界同步能观性,分组精确边界同步能观性以及分组精确边界零能观性与同步能观性.     

Exponential stability for a one‐dimensional compressible viscous micropolar fluid

In this paper, we consider one‐dimensional compressible viscous and heat‐conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H¹(0,1))⁴, and then we establish the global existence and exponential stability of solutions in (H²(0,1))⁴ under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.     

Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω₀ = ℝ^{n –1} × (–1, 1), n ≥ 2, in L^q ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω⁺₀ = ℝ^{n –1} × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–₀ = ℝ^{n –1} × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ²‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.     

Paralinearization of the Dirichlet to Neumann Operator,and Regularity of Three-Dimensional Water Waves

This paper is concerned with a priori C ^∞ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a Diophantine condition are automatically C ^∞. In particular, we prove that the solutions defined by Iooss and Plotnikov are C ^∞. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.     

Exact observability and controllability for a nonsimple elastic rod

In this paper, we prove exact observability, boundary and internal exact controllability results for a nonsimple elastic rod. By the spectral analysis method, it is shown that the considered problem has a sequence of eigenfunctions, which forms a Riesz basis for the state space. The exact observability is proved by application of a modified version of Ingham’s Theorem. Then, via the Hilbert Uniqueness method, we prove a boundary controllability result with only one controller and an internal controllability result.     

Well‐posedness of the upper convected Maxwell fluid in the limit of infinite Weissenberg number

An iteration scheme is used to show the well‐posedness of the initial‐boundary value problem for incompressible hypoelastic materials, which arise as a high Weissenberg number limit of viscoelastic fluids. We first assume that the stress is a rank‐one matrix T=qq^T, q∈?ⁿ, and develop energy estimates to show that the problem is locally well‐posed. This problem is related to incompressible ideal magnetohydrodynamics (MHD). We show that the general case T=CC^T, C∈?^n×n can be handled by a generalization of the method we developed. Copyright © 2010 John Wiley & Sons, Ltd.     

Approximation of the bifurcation function for elliptic boundary value problems

The bifurcation function for an elliptic boundary value problem is a vector field B(ω) on R ^d whose zeros are in a one‐to‐one correspondence with the solutions of the boundary value problem. Finite element approximations of the boundary value problem are shown to give rise to an approximate bifurcation function B^h(ω), which is also a vector field on R ^d. Estimates of the difference B(ω) − B^h(ω) are derived, and methods for computing B^h(ω) are discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 194–213, 2000     

Exact controllability for stochastic Schrödinger equations

This paper is addressed to studying the exact controllability of stochastic Schrödinger equations by two controls. One is a boundary control and the other is an internal control in the diffusion term. By means of the duality argument, the control problem is converted into an observability problem for backward stochastic Schrödinger equations, and the desired observability estimate is obtained by a global Carleman estimate. At last, we give a result about the lack of exact controllability, which shows that the action of two controls is necessary.     

An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

Exact boundary observability for a kind of second-order quasilinear hyperbolic system

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global classical solution, we establish the local exact boundary observability for a kind of second-order quasilinear hyperbolic system in which the number of positive eigenvalues and the number of negative ones are not equal. As an application, we obtain the one-sided local exact boundary observability and two-sided local exact boundary observability with fewer observed values for first-order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative ones are decoupled.