4,732 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
Local controllability of 1D linear and nonlinear Schr\"odinger equations with bilinear control
We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control, that represents a quantum particle in an electric field (the
control). We prove the controllability of this system, in any positive time,
locally around the ground state. Similar results were proved for particular
models (by the first author and with J.M. Coron), in non optimal spaces, in
long time and the proof relied on the Nash-Moser implicit function theorem in
order to deal with an a priori loss of regularity. In this article, the model
is more general, the spaces are optimal, there is no restriction on the time
and the proof relies on the classical inverse mapping theorem. A hidden
regularizing effect is emphasized, showing there is actually no loss of
regularity. Then, the same strategy is applied to nonlinear Schr\"odinger
equations and nonlinear wave equations, showing that the method works for a
wide range of bilinear control systems
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
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