Hamilton–Jacobi–Bellman equations for quantum optimal feedback control

John E. Gough, V. P. Belavkin, O. G. Smolianov

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41 Citations (SciVal)


We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton–Jacobi–Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton–Pontryagin set-up. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback. A controlled qubit with a feedback is considered as example.
Original languageEnglish
Article numberS237
JournalJournal of Optics B: Quantum and Semiclassical Optics
Issue number10
Publication statusPublished - 14 Sept 2005


  • optimal quantum control
  • quantum diffusion filtering
  • quantum feedback control
  • quantum dynamical programming
  • quantum Bellman principle
  • quantum Hamilton-Jacobi equation


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