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

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

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].
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]

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

[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]

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

[4]

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

[5]

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

[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]

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

[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]

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

[10]

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

[11]

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

[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]

Cédric Villani. Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 559-571. doi: 10.3934/dcds.2011.30.559

[14]

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

[15]

Jean-Pierre de la Croix, Magnus Egerstedt. Analyzing human-swarm interactions using control Lyapunov functions and optimal control. Networks & Heterogeneous Media, 2015, 10 (3) : 609-630. doi: 10.3934/nhm.2015.10.609

[16]

Thomas Y. Hou, Pengfei Liu. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4451-4476. doi: 10.3934/dcds.2016.36.4451

[17]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129

[18]

David González-Sánchez, Onésimo Hernández-Lerma. On the Euler equation approach to discrete--time nonstationary optimal control problems. Journal of Dynamics & Games, 2014, 1 (1) : 57-78. doi: 10.3934/jdg.2014.1.57

[19]

Max Gunzburger, Sung-Dae Yang, Wenxiang Zhu. Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 569-587. doi: 10.3934/dcdsb.2007.8.569

[20]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

 Impact Factor: 

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (0)

[Back to Top]