Symmetry Based Modelling And Control Of Infinite Dimensional Quantum Mechanical Systems

For many emerging high technologies, it is important to solve the control problems in quantum mechanical systems from a systematic point of view, which has become an international prosperous research area. Quantum mechanical control systems on infinite-dimensional manifolds is a ubiquitous and important class of systems, but have not been studied so far. In the spirit of symmetries, in this thesis a class of quantum mechanical control systems are modelled including such type of systems, and firstly applies infinite dimensional differential geometry methods to the controllability problems. The main results are as follows:( 1 ) A class of quantum mechanical control system models that possess good algebraic structures are constructed with ideas of symmetries in modern physics. Based on the Lie algebra that describes the symmetries, it is proved in this dissertation that a sufficiently large class of control Hamiltonians can be provided by the universal enveloping algebra of the Lie algebra, on which the quantum mechanical control systems can be modelled. The approach largely simplifies the system analysis, and build a necessary basis for the application of differential geometric methods.(2) For quantum mechanical control systems with structures of universal enveloping algebras, the thesis studies the enlargability of the universal enveloping algebras and the domain problems. First, it is shown from a new viewpoint that the universal enveloping algebra generates a Lie subgroup of a known classical infinite dimensional Lie group, therefore the enlargability is ensured. Next, for the key domain problems caused by unbounded operators, the smooth domain of the symmetry algebra is proved to be a well-defined domain for the quantum control systems, and can be constructed from the ILH-chain. These results make it possible to analyze the control systems on infinite dimensional manifolds.( 3 ) The definition and criterion of smooth controllability are presented, which develop the theory of analytic controllability. With smooth controllability defined on the smooth domain, a generalized HTC theorem is proposed and proved. These results enlarge the domain of analytic controllability, and more essentially, they are applicable to more general quantum mechanical systems with infinite dimensional control Lie algebras. Especially, the smooth controllability may be conveniently applied to thecontrol problems of complex quantum mechanical systems which possess continuous spectra.( 4 ) By the intuitive idea of approximate an infinite dimensional manifold by finite dimensional manifolds, concepts of projected subsystems and project limit controllability are proposed. Based on the filtered Lie algebra of the universal enveloping algebra, the algebraic structure of projected subsystems are studied. It is proved that they have simple decompositions, and the construction of the representations is given as well as the corresponding criterion for project limit controllability.