In this letter, we apply optimal control theory to design minimum-energy π2 and π pulses for the Bloch system in the presence of relaxation with constrained control amplitude. We consider a commonly encountered case in which the transverse relaxation rate is much larger than the longitudinal one so that the latter can be neglected. Using Pontryagin's maximum principle, we derive optimal feedback laws which are characterized by the number of switches, depending on the control bound and the coordinates of the desired final state.

Original languageEnglish
Pages (from-to)601-607
Number of pages7
JournalSystems and Control Letters
Issue number10
StatePublished - Oct 2010


  • Bloch equations
  • Maximum principle


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