11 research outputs found
Exact boundary observability for nonautonomous quasilinear wave equations
By means of a direct and constructive method based on the theory of
semiglobal solution, the local exact boundary observability is shown for
nonautonomous 1-D quasilinear wave equations. The essential difference between
nonautonomous wave equations and autonomous ones is also revealed.Comment: 18 pages, 5 figure
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
Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials
We establish a new family of Carleman inequalities for wave operators on
cylindrical spacetime domains containing a potential that is critically
singular, diverging as an inverse square on all the boundary of the domain.
These estimates are sharp in the sense that they capture both the natural
boundary conditions and the natural -energy. The proof is based around
three key ingredients: the choice of a novel Carleman weight with rather
singular derivatives on the boundary, a generalization of the classical
Morawetz inequality that allows for inverse-square singularities, and the
systematic use of derivative operations adapted to the potential. As an
application of these estimates, we prove a boundary observability property for
the associated wave equations.Comment: 31 pages; accepted versio
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
Exact controllability for multidimensional semilinear hyperbolic equations
In this paper, we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients. For this purpose, we establish an observability estimate for the linear hyperbolic equation with an unbounded potential, in which the crucial observability constant is estimated explicitly by a function of the norm of the potential. Such an estimate is obtained by a combination of a pointwise estimate and a global Carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem