7,101 research outputs found
Controllability for a Wave Equation with Moving Boundary
We investigate the controllability for a one-dimensional wave equation in domains with moving boundary. This model characterizes small vibrations of a stretched elastic string when one of the two endpoints varies. When the speed of the moving endpoint is less than 1 − 1/√ , by Hilbert uniqueness method, sidewise energy estimates method, and multiplier method, we get partial Dirichlet boundary controllability. Moreover, we will give a sharper estimate on controllability time that only depends on the speed of the moving endpoint
Controllability for a Wave Equation with Moving Boundary
We investigate the controllability for a one-dimensional wave equation in domains with moving boundary. This model characterizes small vibrations of a stretched elastic string when one of the two endpoints varies. When the speed of the moving endpoint is less than 1-1/e, by Hilbert uniqueness method, sidewise energy estimates method, and multiplier method, we get partial Dirichlet boundary controllability. Moreover, we will give a sharper estimate on controllability time that only depends on the speed of the moving endpoint
Variants of global Carleman weights in one-measurement inverse problems and fluid-structure controllability problems
We review some recent results on variants of global Carleman weights and Carleman inequalities applied to singular controllability and inverse problems partially developed in collaboration with the authors in a series of papers. First of all, we explain how we can modify weights to study one measurement inverse problems for the heat and wave equations with discontinuous coefficients in the principal part, in a case of locally supported boundary observations for recovering coefficients in the wave equation and we mention also some recent results for the Sch¨odinger equation. As another important application, we show how time-variable global Carleman weights are applied to study the null- controllability for a Navier-Stokes-rigid solid problem in moving domains
Geometric control condition for the wave equation with a time-dependent observation domain
We characterize the observability property (and, by duality, the
controllability and the stabilization) of the wave equation on a Riemannian
manifold with or without boundary, where the observation (or control)
domain is time-varying. We provide a condition ensuring observability, in terms
of propagating bicharacteristics. This condition extends the well-known
geometric control condition established for fixed observation domains. As one
of the consequences, we prove that it is always possible to find a
time-dependent observation domain of arbitrarily small measure for which the
observability property holds. From a practical point of view, this means that
it is possible to reconstruct the solutions of the wave equation with only few
sensors (in the Lebesgue measure sense), at the price of moving the sensors in
the domain in an adequate way.We provide several illustrating examples, in
which the observationdomain is the rigid displacement in of a fixed
domain, withspeed showing that the observability property depends both on
and on the wave speed. Despite the apparent simplicity of some of
ourexamples, the observability property can depend on nontrivial
arithmeticconsiderations
Generation of two-dimensional water waves by moving bottom disturbances
We investigate the potential and limitations of the wave generation by
disturbances moving at the bottom. More precisely, we assume that the wavemaker
is composed of an underwater object of a given shape which can be displaced
according to a prescribed trajectory. We address the practical question of
computing the wavemaker shape and trajectory generating a wave with prescribed
characteristics. For the sake of simplicity we model the hydrodynamics by a
generalized forced Benjamin-Bona-Mahony (BBM) equation. This practical problem
is reformulated as a constrained nonlinear optimization problem. Additional
constraints are imposed in order to fulfill various practical design
requirements. Finally, we present some numerical results in order to
demonstrate the feasibility and performance of the proposed methodology.Comment: 21 pages, 7 figures, 1 table, 69 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Null Controllability for Wave Equations with Memory
We study the memory-type null controllability property for wave equations
involving memory terms. The goal is not only to drive the displacement and the
velocity (of the considered wave) to rest at some time-instant but also to
require the memory term to vanish at the same time, ensuring that the whole
process reaches the equilibrium. This memory-type null controllability problem
can be reduced to the classical null controllability property for a coupled
PDE-ODE system. The later is viewed as a degenerate system of wave equations,
the velocity of propagation for the ODE component vanishing. This fact requires
the support of the control to move to ensure the memory-type null
controllability to hold, under the so-called Moving Geometric Control
Condition. The control result is proved by duality by means of an observability
inequality which employs measurements that are done on a moving observation
open subset of the domain where the waves propagate
Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus
We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional
torus T = R/(2{\pi}Z). We prove a Unique Continuation Property (UCP) for small
data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain
BBM-like equations, including the equal width wave equation and the KdV-BBM
equation. Applications to the stabilization of the above equations are given.
In particular, we show that when an internal control acting on a moving
interval is applied in BBM equation, then a semiglobal exponential
stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove
that the BBM equation with a moving control is also locally exactly
controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H
s (T) for any s \geq 1
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