110 research outputs found
Controllability of the 1D Schrodinger equation by the flatness approach
We derive in a straightforward way the exact controllability of the 1-D
Schrodinger equation with a Dirichlet boundary control. We use the so-called
flatness approach, which consists in parameterizing the solution and the
control by the derivatives of a "flat output". This provides an explicit
control input achieving the exact controllability in the energy space. As an
application, we derive an explicit pair of control inputs achieving the exact
steering to zero for a simply-supported beam
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
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
An obstruction to small time local null controllability for a viscous Burgers' equation
In this work, we are interested in the small time local null controllability
for the viscous Burgers' equation on the line
segment , with null boundary conditions. The second-hand side is a
scalar control playing a role similar to that of a pressure. In this setting,
the classical Lie bracket necessary condition introduced by
Sussmann fails to conclude. However, using a quadratic expansion of our system,
we exhibit a second order obstruction to small time local null controllability.
This obstruction holds although the information propagation speed is infinite
for the Burgers equation. Our obstruction involves the weak norm of
the control . The proof requires the careful derivation of an integral
kernel operator and the estimation of residues by means of weakly singular
integral operator estimates
Local exact controllability of a 1D Bose-Einstein condensate in a time-varying box
We consider a one dimensional Bose-Einstein condensate in a in finite square-well (box) potential. This is a nonlinear control system in which the state is the wave function of the Bose Einstein condensate and the control is the length of the box. We prove that local exact controllability around the ground state (associated with a fi xed length of the box) holds generically with respect to the chemical potential ; i.e. up to an at most countable set of values. The proof relies on the linearization principle and the inverse mapping theorem, as well as ideas from analytic perturbation theory
Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations
International audienceWe consider a system of an arbitrary number of \textsc{1d} linear Schrödinger equations on a bounded interval with bilinear control. We prove global exact controllability in large time of these equations with a single control. This result is valid for an arbitrary potential with generic assumptions on the dipole moment of the considered particle. Thus, even in the case of a single particle, this result extends the available literature. The proof combines local exact controllability around finite sums of eigenstates, proved with Coron's return method, a global approximate controllability property, proved with Lyapunov strategy, and a compactness argument
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 Schrödinger 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 (Beauchard et al., 2015) of the local exact controllability of NLS
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