50 research outputs found
Small time reachable set of bilinear quantum systems
This note presents an example of bilinear conservative system in an infinite
dimensional Hilbert space for which approximate controllability in the Hilbert
unit sphere holds for arbitrary small times. This situation is in contrast with
the finite dimensional case and is due to the unboundedness of the drift
operator
Approximate controllability of the Schr\"{o}dinger Equation with a polarizability term in higher Sobolev norms
This analysis is concerned with the controllability of quantum systems in the
case where the standard dipolar approximation, involving the permanent dipole
moment of the system, is corrected with a polarizability term, involving the
field induced dipole moment. Sufficient conditions for approximate
controllability are given. For transfers between eigenstates of the free
Hamiltonian, the control laws are explicitly given. The results apply also for
unbounded or non-regular potentials
Regular propagators of bilinear quantum systems
The present analysis deals with the regularity of solutions of bilinear
control systems of the type where the state belongs to some
complex infinite dimensional Hilbert space, the (possibly unbounded) linear
operators and are skew-adjoint and the control is a real valued
function. Such systems arise, for instance, in quantum control with the
bilinear Schr\"{o}dinger equation. For the sake of the regularity analysis, we
consider a more general framework where and are generators of
contraction semi-groups.Under some hypotheses on the commutator of the
operators and , it is possible to extend the definition of solution for
controls in the set of Radon measures to obtain precise a priori energy
estimates on the solutions, leading to a natural extension of the celebrated
noncontrollability result of Ball, Marsden, and Slemrod in 1982. Complementary
material to this analysis can be found in [hal-01537743v1
Which notion of energy for bilinear quantum systems?
In this note we investigate what is the best L^p-norm in order to describe
the relation between the evolution of the state of a bilinear quantum system
with the L^p-norm of the external field. Although L^2 has a structure more easy
to handle, the L^1 norm is more suitable for this purpose. Indeed for every
p>1, it is possible to steer, with arbitrary precision, a generic bilinear
quantum system from any eigenstate of the free Hamiltonian to any other with a
control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm
are computed on an example
Controllability of the bilinear Schr\"odinger equation with several controls and application to a 3D molecule
We show the approximate rotational controllability of a polar linear molecule
by means of three nonresonant linear polarized laser fields. The result is
based on a general approximate controllability result for the bilinear
Schr\"odinger equation, with wavefunction varying in the unit sphere of an
infinite-dimensional Hilbert space and with several control potentials, under
the assumption that the internal Hamiltonian has discrete spectrum
Efficient finite dimensional approximations for the bilinear Schrodinger equation with bounded variation controls
This the text of a proceeding accepted for the 21st International Symposium
on Mathematical Theory of Networks and Systems (MTNS 2014). We present some
results of an ongoing research on the controllability problem of an abstract
bilinear Schrodinger equation. We are interested by approximation of this
equation by finite dimensional systems. Assuming that the uncontrolled term
has a pure discrete spectrum and the control potential is in some sense
regular with respect to we show that such an approximation is possible.
More precisely the solutions are approximated by their projections on finite
dimensional subspaces spanned by the eigenvectors of . This approximation is
uniform in time and in the control, if this control has bounded variation with
a priori bounded total variation. Hence if these finite dimensional systems are
controllable with a fixed bound on the total variation of the control then the
system is approximatively controllable. The main outcome of our analysis is
that we can build solutions for low regular controls such as bounded variation
ones and even Radon measures
Sparse Stabilization and Control of Alignment Models
From a mathematical point of view self-organization can be described as
patterns to which certain dynamical systems modeling social dynamics tend
spontaneously to be attracted. In this paper we explore situations beyond
self-organization, in particular how to externally control such dynamical
systems in order to eventually enforce pattern formation also in those
situations where this wished phenomenon does not result from spontaneous
convergence. Our focus is on dynamical systems of Cucker-Smale type, modeling
consensus emergence, and we question the existence of stabilization and optimal
control strategies which require the minimal amount of external intervention
for nevertheless inducing consensus in a group of interacting agents. We
provide a variational criterion to explicitly design feedback controls that are
componentwise sparse, i.e. with at most one nonzero component at every instant
of time. Controls sharing this sparsity feature are very realistic and
convenient for practical issues. Moreover, the maximally sparse ones are
instantaneously optimal in terms of the decay rate of a suitably designed
Lyapunov functional, measuring the distance from consensus. As a consequence we
provide a mathematical justification to the general principle according to
which "sparse is better" in the sense that a policy maker, who is not allowed
to predict future developments, should always consider more favorable to
intervene with stronger action on the fewest possible instantaneous optimal
leaders rather than trying to control more agents with minor strength in order
to achieve group consensus. We then establish local and global sparse
controllability properties to consensus and, finally, we analyze the sparsity
of solutions of the finite time optimal control problem where the minimization
criterion is a combination of the distance from consensus and of the l1-norm of
the control.Comment: 33 pages, 5 figure
Mean-field sparse Jurdjevic-Quinn control
International audienceWe consider nonlinear transport equations with non-local velocity, describing the time-evolution of a measure, which in practice may represent the density of a crowd. Such equations often appear by taking the mean-field limit of finite-dimensional systems modelling collective dynamics. We first give a sense to dissipativity of these mean-field equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, we address the problem of controlling such equations by means of a time-varying bounded control action localized on a time-varying control subset with bounded Lebesgue measure (sparsity space constraint). Finite-dimensional versions are given by control-affine systems, which can be stabilized by the well known Jurdjevic–Quinn procedure. In this paper, assuming that the uncontrolled dynamics are dissipative, we develop an approach in the spirit of the classical Jurdjevic–Quinn theorem, showing how to steer the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function, and enjoy sparsity properties in the sense that the control support is small. Finally, we show that our result applies to a large class of kinetic equations modelling multi-agent dynamics
Implementation of logical gates on infinite dimensional quantum oscillators
6 pagesInternational audienceIn this paper we study the error in the approximate simultaneous controllability of the bilinear Schrodinger equation. We provide estimates based on a tracking algorithm for general bilinear quantum systems and on the study of the finite dimensional Galerkin approximations for a particular class of quantum systems, weakly-coupled systems. We then present two physical examples: the perturbed quantum harmonic oscillator and the infinite potential well