Observability of the 1-D Wave Equation with Mixed Boundary Conditions in a Non-cylindrical Domain

We study a wave equation with mixed boundary conditions in a bounded interval with a moving endpoint. We derive precise estimates for the energy of the solution and show observability results at each endpoint. The observability constants are explicitly given and the obtained time of observability is sharp.     

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.     

Exact Boundary Controllability and Exact Boundary Observability for a Coupled System of Quasilinear Wave Equations

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local boundary(weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.     

On the Exact Controllability of the Wave Equation with Interior and Boundary Controls

The exact controllability of the wave equation in a bounded domain of Rⁿ, with controls on a part of the boundary and in the interior, is studied. Feedback laws are established.     

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.     

Sharp Observability Inequalities for the 1-D Plate Equation with a Potential

This paper deals with the problem of sharp observability inequality for the 1-D plate equation wtt + wxxxx + q(t,x)w = 0 with two types of boundary conditions w = wxx = 0 or w = wx = 0,and q(t,x) being a suitable potential.The author shows that the sharp observability constant is of order exp(C q ∞27) for q ∞≥ 1.The main tools to derive the desired observability inequalities are the global Carleman inequalities,based on a new point wise inequality for the fourth order plate operator.     

Boundary Controllability and Observability of a One-Dimensional Nonuniform SCOLE System

A hybrid system, composed of an elastic beam governed by an Euler-Bernoulli beam equation with variable coefficients and a linked rigid body governed by an ordinary differential equation, is considered. Various controllability/observability properties of the system under bounday control/observation are studied. It is shown that an open-loop smooth/singular controller of either torque control or force control is sufficient to make the system exactly controllable in arbitrarily short time duration. The work was carried out with the support of the National Natural Science Foundation of China and the Russian Foundation for Fundamental Researches, Grant 02-01-00554.     

Null Controllability of Three-dimensional Heat Equation in Singular Domains

We study in this paper a class of parabolic equations in singular domains. These equations are defined in a singular cylindrical domain whose cross-section contains one reentrant corner or one straight emerging crack. We assume that the diffusion coefficients are non-smooth in the normal direction. We will show some spectral inequality thanks to Carleman type estimates and the construction of a suitable weight function satisfying some properties. As in Benabdallah et al. (C. R. Acad. Sci. Paris 344(6):357–362, 2007), we deduce the null-controllability of these equations with the help of the Lebeau-Robbiano method.     

Approximate Controllability of a Stochastic Wave Equation

In this paper we discuss the controllability of a wave equation with random noise. Our main tools are the Ito representation theorem and an adaptation of the Hilbert uniqueness method for the exact controllability of deterministic equations.     

Approximate Controllability of a Stochastic Wave Equation

In this paper we discuss the controllability of a wave equation with random noise. Our main tools are the Ito representation theorem and an adaptation of the Hilbert uniqueness method for the exact controllability of deterministic equations.     

二维波动方程部分Dirichlet边界控制正则性的数值分析

通过数值方法,研究边界充分(逐段)光滑区域上的二维波动方程在部分Dirichlet边界控制下的正则性问题.数值结果表明:在所选条件下,系统是Salamon-Weiss意义下正则的.     

Exact Controllability of a Damped Wave Equation with Distributed Controls

We study the controllability problem of the one-dimensional damped wave equation $$\rho {\text{(}}x{\text{)}}u_{tt} - \frac{d}{{dx}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {p(x)u_x } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2d(x)\rho (x)u_t + q(x)\rho (x)u = h(x,t),{\text{ }}x \in {\text{(0,1)}}$$ This equation describes the forced motion of a nonhomogeneous string subject to a viscous damping. It is proved that the solution can be exactly controlled in finite time by means of distributed control forces h which vanish outside of any fixed nonempty subinterval of (0, 1). Moreover the optimal time of controllability is given.     

Null Controllability of Some Reaction-Diffusion Systems with Only One Control Force in Moving Domains

In this article, the authors establish the local null controllability property for semilinear parabolic systems in a domain whose boundary moves in time by a single control force acting on a prescribed subdomain. The proof is based on Kakutani's fixed point theorem combined with observability estimates for the associated linearized system.     

The Boltzmann Equation with Specular Boundary Condition in Convex Domains

We establish the global well‐posedness and stability of the Boltzmann equation with the specular reflection boundary condition in general smooth convex domains when an initial datum is close to the Maxwellian with or without a small external potential. In particular, we have completely solved the longstanding open problem after an announcement by Shizuta and Asano in 1977. © 2017 Wiley Periodicals, Inc.     

Exact Boundary Controllability for a Coupled System of Wave Equations with Neumann Boundary Controls

This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls.In order to establish the corresponding observability inequality,the authors introduce a compact perturbation method which does not depend on the Riesz basis property,but depends only on the continuity of projection with respect to a weaker norm,which is obviously true in many cases of application.Next,in the case of fewer Neumann boundary controls,the non-exact boundary controllability for the initial data with the same level of energy is shown.     

Null Controllability in Unbounded Domains for the Semilinear Heat Equation with Nonlinearities Involving Gradient Terms

We consider the null controllability problem for the semilinear heat equation with nonlinearities involving gradient terms in an unbounded domain of ^N with Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain such that the uncontrolled region \ is bounded. Using Carleman inequalities, we prove first the null controllability of the linearized equation. Then, by a fixed-point method, we obtain the main result for the semilinear case. This result asserts that, when the nonlinearity is ^C1 and globally Lipschitz, the system is null controllable.     

Boundary Controllability of a Dynamical System Governed by the Wave Equation on a Class of Graphs (Trees)

The boundary control problem for the wave equation on a planar graph consisting of strings of variable densities with Dirichlet control at boundary vertices is considered. The exact controllability in L₂-classes of controls and states is established in the case where the graph is a tree; a sharp estimate of the time of controllability is given. Bibliography: 7 titles. Dedicated to J. E. Lagnese on the occasion of his 65th birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 23–47.