American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Non ordered lower and upper solutions to fourth order problems with functional boundary conditions
  • PROC Home
  • This Issue
  • Next Article
    Existence for the linearization of a steady state fluid/nonlinear elasticity interaction
2011, 2011(Special): 198-208. doi: 10.3934/proc.2011.2011.198

Energy minimization in two-level dissipative quantum control: Th e integrable case

Bernard Bonnard 1, , Jean-Baptiste Caillau 1, and  Olivier Cots 1,

1. 

Math. Institute, Bourgogne Univ. & CRNS, 9 avenue Savary, F-21078 Dijon, France, France, France

Received  August 2010 Revised  March 2011 Published  October 2011

The aim of this contribution is to re ne some of the computations of [6]. The Lindblad equation modelling a two-level dissipative quantum system is investigated. The control can be interpretated as the action of a laser to rotate a molecule in gas phase, or as the e ect of a magnetic eld on a spin 1=2 particle. For the energy cost, normal extremals of the maximum principle are solution to a three-dimensional Hamiltonian with parameters. The analysis is focussed on an integrable submodel which de nes outside singularities a pseudo-Riemannian metric in dimension ve. Complete quadratures are given for this subcase by means of Weierstra elliptic functions. Preliminary computations of cut and conjugate loci are also provided for a two-dimensional restriction using [9].
Keywords: conjugate and cut loci, Lorentzian metrics, elliptic functions, Optimal control, Lindblad equation.
Mathematics Subject Classification: Primary: 49K15, 81Q05.
Citation: Bernard Bonnard, Jean-Baptiste Caillau, Olivier Cots. Energy minimization in two-level dissipative quantum control: Th e integrable case. Conference Publications, 2011, 2011 (Special) : 198-208. doi: 10.3934/proc.2011.2011.198
[1]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011
[2]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
[3]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041
[4]

Robert M. Strain. Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinetic & Related Models, 2012, 5 (3) : 583-613. doi: 10.3934/krm.2012.5.583
[5]

Dag Lukkassen, Annette Meidell, Peter Wall. On the conjugate of periodic piecewise harmonic functions. Networks & Heterogeneous Media, 2008, 3 (3) : 633-646. doi: 10.3934/nhm.2008.3.633
[6]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[7]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73
[8]

B. Bonnard, J.-B. Caillau, E. Trélat. Geometric optimal control of elliptic Keplerian orbits. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 929-956. doi: 10.3934/dcdsb.2005.5.929
[9]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279
[10]

Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016
[11]

William Clark, Anthony Bloch, Leonardo Colombo, Patrick Rooney. Time-minimum control of quantum purity for 2-level Lindblad equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1061-1073. doi: 10.3934/dcdss.2020063
[12]

Enrique Fernández-Cara, Juan Límaco, Laurent Prouvée. Optimal control of a two-equation model of radiotherapy. Mathematical Control & Related Fields, 2018, 8 (1) : 117-133. doi: 10.3934/mcrf.2018005
[13]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[14]

Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations & Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35
[15]

Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783
[16]

Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016
[17]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
[18]

Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021
[19]

Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053
[20]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020179

 Impact Factor: 

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences