Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups

Abstract

The controllability of bilinear systems is well understood for finite dimensional isolated systems where the control can be implemented exactly. However when perturbations are present some interesting theoretical questions are raised. We consider in this paper a control system whose control cannot be implemented exactly but is shifted by a time independent constant in a discrete list of possibilities. We prove under general hypothesis that the collection of possible systems (one for each possible perturbation) is simultaneously controllable with a common control. The result is extended to the situations where the perturbations are constant over a common, long enough, time frame. We apply the result to the controllability of quantum systems. Furthermore, some examples and a convergence result are presented for the situation where an infinite number of perturbations occur. In addition, the techniques invoked in the proof allow to obtain generic necessary and sufficient conditions for ensemble controllability.

Quantum Hamiltonian and dipole moment identification in presence of large control perturbations

“…The proof is similar with the exception that the simultaneous controllability result to be used is the corollary 5 page 25 in [3].…”

Robust controllability of two-qubit Hamiltonian dynamics

“…By a similar procedure of the first example, we can see [18] for arbitrary polynomial functions f i (ω 2 ). Assume there are robustly controllable functions f 1 , f 3 satisfying f 1 (ω 2 )−ωf 3 (ω 2 ) = θ 1 and f 3 (ω 2 ) + ωf 1 (ω 2 ) = θ 2 for any θ 1 , θ 2 ∈ R. Then…”

“…In the following section, we will numerically investigate the robust controllability by using a method called discretization [18,23,24] for a given region of the unknown ω to provide a robust control pulse and investigate the robust controllability of System E for the region. In addition, System E is a good candidate to see the difference between the robust controllability for continuous and discretized unknown parameters as mentioned in Sec.…”

“…We also consider the robust controllability of the two-qubit systems where the values of the unknown parameters are given as finite sets, or a discretized subset of a given continuous set. The robust controllability of the systems with unknown parameters given in finite (discretized) sets has been well studied [18,23,24], but the differences between continuous and discretized cases has not been clarified, i.e., whether or not the robust controllability of the systems with unknown parameters in all discretized subsets of a given continuous set implies the robust controllability of the continuous one. We investigate this problem by combining analytical and numerical approaches.…”

Stochastic learning control of inhomogeneous quantum ensembles

“…A first natural question is whether this is at all possible, i.e., whether a single field can drive several distinct molecules to a common target; the answer is given by the theory of ensemble control controllability, see [26,5,17,4,6] and is in general positive. However the theory does not explain how to find the control (except under specific regimes, see [2]).…”