Time-optimal torus theorem and control of spin systems

Given a compact, connected Lie group G with Lie algebra . We discuss time-optimal control of bilinear systems of the form

((I))

where H _d , H _j ∈ , U ∈ G, and the v _j act as control variables. The case G = SU(2 ⁿ ) has found interesting applications to questions of time-optimal control of spin systems. In this context Eq. (I) describes the dynamics of an n-particle 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 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 time-optimal control strategies for system (I) in the case where G/K carries the structure of a Riemannian symmetric space. The text submitted by the author in English.