Hamiltonian and Quantum Mechanical Control Systems

Abstract

After a brief introduction to the notions of controllability and observability for classical control systems, these notions are elaborated for Hamiltonian control systems. It is shown that to every Hamiltonian control system there corresponds a Lie algebra of functions under the Poisson bracket, which completely characterizes the Hamiltonian system from an input-output point of view. Special attention is paid to the case that this Lie algebra is finite-dimensional, leading to an application of the theory of coadjoint representations. It is shown how Hamiltonian control systems can be quantized to quantum mechanical control systems, which are themselves Hamiltonian con trol systems on a Hilbert space. Some relations between the classical and quantized control system are being discussed. Finally the question is addressed which control systems are actually Hamiltonian, and so can be quantized. This leads to a generalization of the Helmholtz conditions for the inverse problem in classical mechanics.