Asymptotic Tracking Of Two-spin1/2Systems

This paper considers the differential equation modeling and the vector space characterizing fortwo-spin1/2systems as well as the asymptotic tracking problem of two-spin1/2systems. Recently,spin-1/2system have become an important quantum system to work in and promises a broad range oftechnological applications. Motivated by some recent advances in the areas which have rendered theintroduction of a series of constructive mathematical techniques, the primary results obtained in thispaper include the following three parts.In the first part, the problem of the differential equation modeling for two-spin1/2systems hasbeen investigated. Based on a scaled Pauli basis of operator space, the density operator equation(Liouville-von Neumann Equation) of two-spin1/2systems can be transformed into the coordinatedifferential equation by using matrices of adjoint operators of two-spin1/2systems. Using the coor-dinate differential equation obtained, the problems of two-spin1/2systems can be investigated in theframework of theory of nonlinear control systems.The second part focuses on characterizing the vector space for two-spin1/2systems. Because ofthe nature of the density matrix itself, there does not exist a one-to-one correspondence between thedensity matrix and the coordinates of the density operator on the basis of {λj1j2}j1,j2∈I4. Using themathod of characterizing the Bloch-vector space for arbitrary N-level systems as well as the algorithmfor computing trace of cubic density operator, we restrict the vector space to be a proper subset of aball, which makes the density operator equation of two-spin1/2systems can be completely presentedby the coordinate differential equation obtained.The final part is concerned with the asymptotic tracking problem of two-spin1/2systems. Norminvariance property of trajectory of the transformed coordinate differential system is established whichforms a foundation to formulate tracking problem for two-spin1/2systems. The tracking problem isto design a state feedback controller as well as to determine the set of initial points such that trajectoryof the two-spin1/2system is asymptotically tracking the reference trajectory of a given two-spin1/2system. A Lyapunov function based control design approach is proposed to solve the trackingproblem. Trajectory convergence result of the controlled systems is established by using LaSalleinvariance principle. Using the trajectory convergence result, the asymptotic tracking problem oftwo-spin1/2systems is solved through further constraint on the initial points of the spin systems.