Optimal control of quantum objects (Q1084659)

The control of quantum objects is considered by examining the system \[ \partial \psi /\partial t-iH\psi +iu(x)\psi =0, \] \[ \psi |_ s=0 \] (s the boundary of the region considered for all time), \[ \psi |_{t=0}=v(x)\;(\| v\| =1), \] and H is a particular form of partial differential equation. The controls considered are using u(x) (i.e. control by the exterior field) or v(x) (i.e. control using the initial conditions). The control problems considered include finding u(x) so that the system should be in a general state with maximal probability or that the probability of being in a region is minimal. Other problems using v(x) to exercise control (where \(u=0)\) include finding the optimal v so that \(\psi\) is close to a particular \(y\in s.\) The theorems that the various functions can be optimized to give the optimal controls are stated in the main part of the paper and proved in the appendices.