Time-optimal torus theorem and control of spin systems

Given a compact, connected Lie group G with Lie algebra g. We discuss time-optimal control of bilinear systems of the form [GRAPHICS] where H-d, H-j is an element of, U is an element of 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 Hd, which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters vj. 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 f 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.