90 research outputs found
Lipschitz stability in an inverse problem for the wave equation
We are interested in the inverse problem of the determination of the
potential from the measurement of the
normal derivative on a suitable part of the
boundary of , where is the solution of the wave equation
set in and
given Dirichlet boundary data. More precisely, we will prove local uniqueness
and stability for this inverse problem and the main tool will be a global
Carleman estimate, result also interesting by itself
Robust control of a bimorph mirror for adaptive optics system
We apply robust control technics to an adaptive optics system including a
dynamic model of the deformable mirror. The dynamic model of the mirror is a
modification of the usual plate equation. We propose also a state-space
approach to model the turbulent phase. A continuous time control of our model
is suggested taking into account the frequential behavior of the turbulent
phase. An H_\infty controller is designed in an infinite dimensional setting.
Due to the multivariable nature of the control problem involved in adaptive
optics systems, a significant improvement is obtained with respect to
traditional single input single output methods
Convergence of an inverse problem for discrete wave equations
International audienceIt is by now well-known that one can recover a potential in the wave equation from the knowledge of the initial waves, the boundary data and the flux on a part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are interested in proving that trying to fit the discrete fluxes, given by discrete approximations of the wave equation, with the continuous one, one recovers, at the limit, the potential of the continuous model. In order to do that, we shall develop a Lax-type argument, usually used for convergence results of numerical schemes, which states that consistency and uniform stability imply convergence. In our case, the most difficult part of the analysis is the one corresponding to the uniform stability, that we shall prove using new uniform discrete Carleman estimates, where uniform means with respect to the discretization parameter. We shall then deduce a convergence result for the discrete inverse problems. Our analysis will be restricted to the 1-d case for space semi-discrete wave equations discretized on a uniform mesh using a finite differences approach
An inverse problem for Schr\"odinger equations with discontinuous main coefficient
This paper concerns the inverse problem of retrieving a stationary potential
for the Schr\"odinger evolution equation in a bounded domain of RN with
Dirichlet data and discontinuous principal coefficient a(x) from a single
time-dependent Neumann boundary measurement. We consider that the discontinuity
of a is located on a simple closed hyper-surface called the interface, and a is
constant in each one of the interior and exterior domains with respect to this
interface. We prove uniqueness and lipschitz stability for this inverse problem
under certain convexity hypothesis on the geometry of the interior domain and
on the sign of the jump of a at the interface. The proof is based on a global
Carleman inequality for the Schr\"odinger equation with discontinuous
coefficients, result also interesting by itself
Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation.
International audienceThis article develops the numerical and theoretical study of the reconstruction algorithm of a potential in a wave equation from boundary measurements, using a cost functional built on weighted energy terms coming from a Carleman estimate. More precisely, this inverse problem for the wave equation consists in the determination of an unknown time-independent potential from a single measurement of the Neumann derivative of the solution on a part of the boundary. While its uniqueness and stability properties are already well known and studied, a constructive and globally convergent algorithm based on Carleman estimates for the wave operator was recently proposed in [BdBE13]. However, the numerical implementation of this strategy still presents several challenges, that we propose to address here
Global Carleman estimates for waves and applications
In this article, we extensively develop Carleman estimates for the wave
equation and give some applications. We focus on the case of an observation of
the flux on a part of the boundary satisfying the Gamma conditions of Lions. We
will then consider two applications. The first one deals with the exact
controllability problem for the wave equation with potential. Following the
duality method proposed by Fursikov and Imanuvilov in the context of parabolic
equations, we propose a constructive method to derive controls that weakly
depend on the potentials. The second application concerns an inverse problem
for the waves that consists in recovering an unknown time-independent potential
from a single measurement of the flux. In that context, our approach does not
yield any new stability result, but proposes a constructive algorithm to
rebuild the potential. In both cases, the main idea is to introduce weighted
functionals that contain the Carleman weights and then to take advantage of the
freedom on the Carleman parameters to limit the influences of the potentials.Comment: 31 page
Robust Measurement Feedback Control of an Inclined Cable
International audienceConsidering the partial differential equation model of the vibrations of an inclined cable, we are interested in applying robust control technics to stabilize the system with measurement feedback when it is submitted to external disturbances. This paper focuses indeed on the construction of a standard linear infinite dimensional state space system and an H_infinity feedback control of vibrations with partial observation of the state. The control and observation are performed using an active tendon
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