324 research outputs found
Rigorous numerics for NLS: bound states, spectra, and controllability
In this paper it is demonstrated how rigorous numerics may be applied to the
one-dimensional nonlinear Schr\"odinger equation (NLS); specifically, to
determining bound--state solutions and establishing certain spectral properties
of the linearization. Since the results are rigorous, they can be used to
complete a recent analytical proof [6] of the local exact controllability of
NLS.Comment: 30 pages, 2 figure
On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension
In this paper, we consider the cost of null controllability for a large class
of linear equations of parabolic or dispersive type in one space dimension in
small time. By extending the work of Tenenbaum and Tucsnak in "New blow-up
rates for fast controls of Schr\"odinger and heat equations`", we are able to
give precise upper bounds on the time-dependance of the cost of fast controls
when the time of control T tends to 0. We also give a lower bound of the cost
of fast controls for the same class of equations, which proves the optimality
of the power of T involved in the cost of the control. These general results
are then applied to treat notably the case of linear KdV equations and
fractional heat or Schr\"odinger equations
Moving and oblique observations of beams and plates
We study the observability of the one-dimensional Schr{\"o}dinger equation
and of the beam and plate equations by moving or oblique observations. Applying
different versions and adaptations of Ingham's theorem on nonharmonic Fourier
series, we obtain various observability and non-observability theorems. Several
open problems are also formulated at the end of the paper
Local controllability of 1D Schr\"odinger equations with bilinear control and minimal time
We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control.
Beauchard and Laurent proved that, under an appropriate non degeneracy
assumption, this system is controllable, locally around the ground state, in
arbitrary time. Coron proved that a positive minimal time is required for this
controllability, on a particular degenerate example.
In this article, we propose a general context for the local controllability
to hold in large time, but not in small time. The existence of a positive
minimal time is closely related to the behaviour of the second order term, in
the power series expansion of the solution
Unique continuation property with partial information for two-dimensional anisotropic elasticity systems
In this paper, we establish a novel unique continuation property for
two-dimensional anisotropic elasticity systems with partial information. More
precisely, given a homogeneous elasticity system in a domain, we investigate
the unique continuation by assuming only the vanishing of one component of the
solution in a subdomain. Using the corresponding Riemann function, we prove
that the solution vanishes in the whole domain provided that the other
component vanishes at one point up to its second derivatives. Further, we
construct several examples showing the possibility of further reducing the
additional information of the other component. This result possesses remarkable
significance in both theoretical and practical aspects because the required
data is almost halved for the unique determination of the whole solution.Comment: 14 pages, 1 figur
Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback
We use the backstepping method to study the stabilization of a 1-D linear
transport equation on the interval (0, L), by controlling the scalar amplitude
of a piecewise regular function of the space variable in the source term. We
prove that if the system is controllable in a periodic Sobolev space of order
greater than 1, then the system can be stabilized exponentially in that space
and, for any given decay rate, we give an explicit feedback law that achieves
that decay rate
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