35 research outputs found
Simultaneous local exact controllability of 1D bilinear Schr\"odinger equations
We consider N independent quantum particles, in an infinite square potential
well coupled to an external laser field. These particles are modelled by a
system of linear Schr\"odinger equations on a bounded interval. This is a
bilinear control system in which the state is the N-tuple of wave functions.
The control is the real amplitude of the laser field. For N=1, Beauchard and
Laurent proved local exact controllability around the ground state in arbitrary
time. We prove, under an extra generic assumption, that their result does not
hold in small time if N is greater or equal than 2. Still, for N=2, we prove
using Coron's return method that local controllability holds either in
arbitrary time up to a global phase or exactly up to a global delay. We also
prove that for N greater or equal than 3, local controllability does not hold
in small time even up to a global phase. Finally, for N=3, we prove that local
controllability holds up to a global phase and a global delay
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
Approximate controllability for a 2D Grushin equation with potential having an internal singularity
This paper is dedicated to approximate controllability for Grushin equation
on the rectangle with an inverse square
potential. This model corresponds to the heat equation for the Laplace-Beltrami
operator associated to the Grushin metric on , studied by Boscain
and Laurent. The operator is both degenerate and singular on the line .
The approximate controllability is studied through unique continuation of the
adjoint system. For the range of singularity under study, approximate
controllability is proved to hold whatever the degeneracy is.
Due to the internal inverse square singularity, a key point in this work is
the study of well-posedness. An extension of the singular operator is designed
imposing suitable transmission conditions through the singularity.
Then, unique continuation relies on the Fourier decomposition of the 2D
solution in one variable and Carleman estimates for the 1D heat equation solved
by the Fourier components. The Carleman estimate uses a suitable Hardy
inequality
Minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field
In this work, we study the minimal time to steer a given crowd to a desired
configuration. The control is a vector field, representing a perturbation of
the crowd velocity, localized on a fixed control set. We will assume that there
is no interaction between the agents. We give a characterization of the minimal
time both for microscopic and macroscopic descriptions of a crowd. We show that
the minimal time to steer one initial configuration to another is related to
the condition of having enough mass in the control region to feed the desired
final configuration. The construction of the control is explicit, providing a
numerical algorithm for computing it. We finally give some numerical
simulations
Minimal time problem for discrete crowd models with a localized vector field
In this work, we study the minimal time to steer a given crowd to a desired
configuration. The control is a vector field, representing a perturbation of
the crowd velocity, localized on a fixed control set. We characterize the
minimal time for a discrete crowd model, both for exact and approximate
controllability. This leads to an algorithm that computes the control and the
minimal time. We finally present a numerical simulation
Controllability and optimal control of the transport equation with a localized vector field
We study controllability of a Partial Differential Equation of transport
type, that arises in crowd models. We are interested in controlling such system
with a control being a Lipschitz vector field on a fixed control set .
We prove that, for each initial and final configuration, one can steer one to
another with such class of controls only if the uncontrolled dynamics allows to
cross the control set . We also prove a minimal time result for such
systems. We show that the minimal time to steer one initial configuration to
another is related to the condition of having enough mass in to feed
the desired final configuration
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