OPTIMALLY CONTROLLED QUANTUM MOLECULAR-DYNAMICS - A PERTURBATION FORMULATION AND THE EXISTENCE OF MULTIPLE SOLUTIONS


DEMIRALP M., RABITZ H.

PHYSICAL REVIEW A, vol.47, no.2, pp.809-816, 1993 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 47 Issue: 2
  • Publication Date: 1993
  • Doi Number: 10.1103/physreva.47.809
  • Title of Journal : PHYSICAL REVIEW A
  • Page Numbers: pp.809-816

Abstract

This work considers optimal control of quantum-mechanical systems within the framework of perturbation theory with respect to the controlling optical electric field. The control problem is expressed in terms of a cost functional including the physical objective, the penalties, and constraints. The resultant nonlinear variational equations are linearized by considering the lowest-order term in an expansion in powers of the optical-field strength. The optical field is found to satisfy a linear integral equation, and the solution may be expressed in terms of a generalized eigenvalue problem associated with the corresponding kernel. A full determination of the field is specified through the solution to the integral equation and the roots of an accompanying linearized spectral equation for a characteristic multiplier parameter. Each discrete value of the latter parameter corresponds to a particular solution to the variational equations. As a result, it is argued that under very general conditions there will be a denumerably infinite number of solutions to well-posed quantum-mechanical optimal-control problems.