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.     

A global Carleman inequality and exact controllability of parabolic equations with mixed boundary conditions

This paper establishes a global Carleman inequality of parabolic equations with mixed boundary conditions and an estimate of the solution. Further, we prove exact controllability of the equation by controls acting on an arbitrarily given subdomain or subboundary.     

One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws

In this paper, the one-sided exact boundary null controllability of entropy solutions is studied for a class of general strictly hyperbolic systems of conservation laws, whose negative (or positive) characteristic families are all linearly degenerate. The authors first prove the well-posedness of semi-global solutions constructed as the limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions and they establish various properties of both the ε-approximate front tracking solutions and such solutions. By means of essential modifications of the strategy suggested by the first author in [17] originally for the local exact boundary controllability in the framework of classical solutions, the one-sided local exact boundary null controllability of entropy solutions can then be realized via boundary controls acting on one side of the boundary, where the incoming characteristics are all linearly degenerate.     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.     

Interior local null controllability for multi-dimensional compressible flow near a constant state

We consider compressible flow with periodic boundary conditions. In a neighborhood of a state of constant density and nonzero velocity, we prove exact null controllability of the system with a control localized in space and acting only on the momentum equation.     

Observabilité, contrôlabilité et stabilisation frontière du système d’élasticité linéaire

We study homogeneous but not necessarily isotropic linear elastodynamic systems. Applying the Hilbert Uniqueness Method (HUM), we establish their boundary observability and their exact boundary controllability. Then we construct boundary feedbacks leading to arbitrarily large energy decay rates. Finally, under some geometrical conditions, we prove that a “natural” boundary feedback leads also to exponentiel energy decay.     

A new relation between the condensation index of complex sequences and the null controllability of parabolic systems

In this note, we present a new result that relates the condensation index of a sequence of complex numbers with the null controllability of parabolic systems. We show that a minimal time is required for controllability. The results are used to prove the boundary controllability of some coupled parabolic equations.     

Approximate controllability for linearized compressible barotropic Navier–Stokes system in one and two dimensions

In this paper we consider compressible barotropic Navier–Stokes equations in one and two dimensions linearized around a constant steady state with Dirichlet boundary conditions. We explore the controllability of this linearized system using a control only for the velocity equation. We prove that the system with homogeneous Dirichlet boundary conditions, is approximately controllable by a localized interior control when time is sufficiently large.     

一类非局部非线性扩散方程解的全局爆破

主要研究在Dirichlet边界条件或Neumann边界条件下的一类非局部非线性的扩散方程问题.在适当的假设下,证明解的存在性、唯一性、比较原则、以及解对初边值条件的连续依赖性,并就给定的初边值条件,证明解在有限时刻全局爆破.     

Boundary controllability for the quasi-linear wave equations coupled in parallel

We study the boundary exact controllability for a system of two quasi-linear wave equations coupled in parallel with springs and viscous terms. We prove the locally exact controllability around superposition equilibria under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the coupled quasi-linear system moves from a superposition equilibrium in one location to a superposition equilibrium in another location. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and superposition equilibria of the system.     

On the controllability of the fifth-order Korteweg–de Vries equation

In this paper, we consider the fifth-order Korteweg–de Vries equation in a bounded interval. We prove that this equation is locally well-posed when endowed with suitable boundary conditions, and establish a result of local controllability to the trajectories.     

Remarks on global controllability for the Burgers equation with two control forces

In this paper we deal with the viscous Burgers equation. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. In a first result, we prove that the global exact null controllability does not hold for small time. In a second one, we prove that the exact controllability result does not hold even for large time.     

Superconvergence and a posteriori error estimation for triangular mixed finite elements

Summary. In this paper,we prove superconvergence results for the vector variable when lowest order triangular mixed finite elements of Raviart-Thomas type [17] on uniform triangulations are used, i.e., that the -distance between the approximate solution and a suitable projection of the real solution is of higher order than the -error. We prove results for both Dirichlet and Neumann boundary conditions. Recently, Duran [9] proved similar results for rectangular mixed finite elements, and superconvergence along the Gauss-lines for rectangular mixed finite elements was considered by Douglas, Ewing, Lazarov and Wang in [11], [8] and [18]. The triangular case however needs some extra effort. Using the superconvergence results, a simple postprocessing of the approximate solution will give an asymptotically exact a posteriori error estimator for the -error in the approximation of the vector variable. Received December 6, 1992 / Revised version received October 2, 1993     

Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

In this paper, we prove the controllability of a quasi-linear heat equation involving gradient terms with Fourier boundary conditions in a bounded domain of ? ^N . The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.     

Boundary stabilization for a coupled system

We investigate in this work the global existence of weak solutions for a nonlinear coupled system with mixed type boundary conditions. More precisely, Dirichlet and feedback boundary conditions. Further, we also prove the exponential decay of the energy associated with these solutions.     

Uniform null controllability of a linear KdV equation using two controls

In this paper we consider a linear KdV equation posed on a bounded interval. We study the behavior of the cost of null controllability when two boundary controls are employed. By means of suitable Carleman inequalities and a new exponential dissipation estimate, we prove that uniform null controllability with respect to the dispersion coefficient holds, contrary to the case when one control is used at the left end-point of the interval.     

Symmetry and monotonicity of positive solutions of elliptic equations with mixed boundary conditions in a super-spherical cone

In this paper, we study symmetry and monotonicity properties for positive solutions of semilinear elliptic equations with mixed boundary conditions. We establish a version of the maximum principle for mixed boundary conditions in a narrow domain, and prove the positive solution in super-spherical cone is axially symmetric and monotone in some direction by the methods of moving planes.     

Radial symmetry and partially overdetermined problems in a convex cone

We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.     

Boundary Controllability for the Quasilinear Wave Equation

We study the boundary exact controllability for the quasilinear wave equation in high dimensions. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical conditions. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.     

On the Theoretical and Numerical Control of a One-Dimensional Nonlinear Parabolic Partial Differential Equation

This paper deals with the analysis of the internal and boundary control of a one-dimensional parabolic partial differential equation with nonlinear diffusion. First, we prove a local null controllability result with distributed controls, locally supported in space. The proof relies on local inversion (more precisely, we use Liusternik’s Inverse Function Theorem), together with some appropriate specific estimates. We also establish a similar result with controls on one side of the boundary. Then, we consider an iterative algorithm for the computation of null controls, we prove the convergence of the iterates, and we perform some numerical experiments.