Timeoptimal torus theorem and control of spin systems
Abstract
Given a compact, connected Lie group G with Lie algebra mathfrak{g}. We discuss timeoptimal control of bilinear systems of the form dot U(t) = left( {H_d + sumlimits_{j = 1}^m {v_j (t)H_j } } right)U(t), where H _{ d }, H _{ j } ∈ mathfrak{g}, U ∈ G, and the v _{ j } act as control variables. The case G = SU(2^{ n }) has found interesting applications to questions of timeoptimal control of spin systems. In this context Eq. (I) describes the dynamics of an nparticle system with fixed drift Hamiltonian H _{ d }, which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v _{ j }. The question of interest here is to transfer the system from a given initial state U _{0} to a prescribed final state U _{1} in least possible time. Denote by [Figure not available: see fulltext.] the Lie algebra spanned by H _{1}, ..., H _{ m }, and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space G/K we discuss in detail timeoptimal control strategies for system (I) in the case where G/K carries the structure of a Riemannian symmetric space.
 Publication:

Optics and Spectroscopy
 Pub Date:
 September 2007
 DOI:
 10.1134/S0030400X07090019
 Bibcode:
 2007OptSp.103..343S
 Keywords:

 03.67.a;
 32.80.Qk