Optimal Bilinear Control of Gross--Pitaevskii Equations

Hintermüller, Michael; Marahrens, Daniel; Markowich, Peter A.; Sparber, Christof

doi:10.1137/120866233

Cited by 33 publications

(55 citation statements)

References 35 publications

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“…Remark 2.8. Optimal bilinear control is also studied in [15] and [13] for linear and nonlinear deterministic Schrödinger equations respectively. In both papers, some compactness conditions of initial data or controls are needed for the existence of the optimal control.…”

Section: )mentioning

confidence: 99%

Optimal bilinear control of nonlinear stochastic Schrödinger equations driven by linear multiplicative noise

Barbu¹,

Röckner²,

Deng³

2018

Ann. Probab.

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Here is investigated the bilinear optimal control problem of quantum mechanical systems with final observation governed by a stochastic nonlinear Schrödinger equation perturbed by a linear multiplicative Wiener process. The existence of an open loop optimal control and first order Lagrange optimality conditions are derived, via Skorohod's representation theorem, Ekeland's variational principle and the existence for the linearized dual backward stochastic equation. Moreover, our approach in particular applies to the deterministic case.

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Section: )mentioning

confidence: 99%

Optimal bilinear control of nonlinear stochastic Schrödinger equations driven by linear multiplicative noise

Barbu¹,

Röckner²,

Deng³

2018

Ann. Probab.

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“…This equation describes dynamics of Bose-Einstein condensate, which is important both from a theoretical point of view and for creation of new technologies, e.g., atomic chips [109]. Modelling of controlled dynamics of a Bose-Einstein condensate is done using the Gross-Pitaevskii equation with control [23,39,65,95,96,[108][109][110][111][112][113].…”

Section: Gross-pitaevskii Equation and Optimal Control Of Bose-einstementioning

confidence: 99%

“…Meaning ψ(·, t), we will use ψ(t) for shortness. Potentials of various form are used [23,39,65,95,96,[108][109][110][111][112][113]. In the article [65] (S.E.…”

Section: Gross-pitaevskii Equation and Optimal Control Of Bose-einstementioning

confidence: 99%

“…where F is defined, for example, using (2.5); λ u , λ du ≥ 0; S is a shape function. The rigorous formulation of the optimal control problem for the Gross-Pitaevskii equation for the potential V (x, u) = U (x) + u(t)Ṽ (x), where U andṼ are given and u(t) is the control, is provided in [111]. In this formulation the potential V ∈ W 1,∞ (R d ) (the space of Lipschitz functions), the potential U satisfies…”

Section: Gross-pitaevskii Equation and Optimal Control Of Bose-einstementioning

confidence: 99%

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Krotov method for optimal control of closed quantum systems

Моржин¹,

Pechen²

2019

Russ. Math. Surv.

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Mathematical problems of optimal control in quantum systems attract high interest in connection with fundamental questions and existing and prospective applications. An important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method initially proposed beyond quantum control in the articles by V.F. I.N. Feldman (1978, 1983). The method was used to develop a novel approach for finding optimal controls for quantum systems in [D.J. Tannor, V. Kazakov, V. Orlov, In: Time-Dependent Quantum Molecular Dynamics, Boston, Springer, 347-360 (1992)] and [J. Somlói, V.A. Kazakov, D.J. Tannor, Chem. Phys., 172:1, 85-98 (1993)], and in many works of various scientists, as described in details in this review. The review discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with respect to defining the improvement function (which in most cases is linear or linear-quadratic), constraints on control spectrum and on the states of a quantum system, regularizers, etc. The review describes applications of the Krotov method to control of molecular dynamics, manipulation of Bose-Einstein condensate, quantum gate generation. We also discuss comparison with GRAPE (GRadient Ascent Pulse Engineering), CRAB (Chopped Random-Basis), the Zhu Rabitz and the Maday Turinici methods.Bibliography: 154 titles.3 quantum tomography [53,54]. Quantum machine learning is considered [55]. One of the commonly used methods for constructing program controls for quantum systems is the Krotov 5 method. This method was initially proposed beyond quantum control by V.F. I.N. Feldman [56, 57] (1978, 1983) based on the Krotov optimality principle [58,59] and further developed by A.I. Konnov and V.F. Krotov [60] (1999). An example with control of an open (i.e. interacting with the environment) quantum system was analyzed by V.A. Kazakov and V.F. Krotov in 1987 in the article [61] (also in [62]). Crucial step in development to quantum systems was done in 1992-1993, when D.J. Tannor and coauthors used the 1st order Krotov method to develop a general approach for finding optimal controls for quantum systems [63,64]. In 2002, the 2nd order Krotov method [57,60] was adapted by S.E. Sklarz and D.J. Tannor for modeling of optimal control for Bose-Einstein condensate, whose dynamics is defined in terms of a controlled Gross-Pitaevskii equation [65]. In 2008, J.P. Palao, R. Kosloff, and C.P. Koch developed the method to optimal control for the problem of obtaining an objective in a subspace of the Hilbert space while avoiding population transfer to other subspaces [70]. The Krotov method was applied with various modifications and taking into account specific details of quantum optimal control problems, for atomic and molecular dynamics [9,19,42,[66][67][68][69][70][71][72][73][74][75][76][77]; qubits, quantum gates, quantum networks [78-94]; manipulation of Bose-Einstein condensate [39,65,95,96]; nuclear magnetic reson...

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“…Optimal control problem and exact controllability of Schrödinger equations have been extensively studied in the deterministic case. See, e.g., [11,35,38,46]. In the stochastic situation, there are many results on optimal control problems of dissipative equations, see, e.g., [30,31].…”

Section: Introductionmentioning

confidence: 99%

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

Deng

2020

Probab. Theory Relat. Fields

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We study optimal bilinear control problems for stochastic nonlinear Schrödinger equations in both the mass subcritical and critical case. For general initial data of the minimal L 2 regularity, we prove the existence and first order Lagrange condition of an open loop control. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type U 2 and V 2 , adapted to evolution operators related to linear Schrödinger equations with lower order perturbations. In particular, we obtain a new temporal regularity of rescaled (backward) stochastic solutions, which is the key ingredient in the proof of tightness of approximating controls induced by Ekeland's variational principle.

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Optimal Bilinear Control of Gross--Pitaevskii Equations

Cited by 33 publications

References 35 publications

Optimal bilinear control of nonlinear stochastic Schrödinger equations driven by linear multiplicative noise

Optimal bilinear control of nonlinear stochastic Schrödinger equations driven by linear multiplicative noise

Krotov method for optimal control of closed quantum systems

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

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