UPPER AND LOWER BOUNDS ON THE CONTROL FIELD AND THE QUALITY OF ACHIEVED OPTIMALLY CONTROLLED QUANTUM MOLECULAR-MOTION

A large class of problems in optimally controlled quantum or classical molecular dynamics has multiple solutions for the control field ampli tude. A denumerably infinite number of solutions may exist depending o n the structure of the design cost functional. This fact has been rece ntly proved with the aid of perturbation theory by considering the ele ctric field as the perturbating agent. In carrying out this analysis, an eigenvalue (i.e., a spectral parameter) appears which gives the deg ree of deviation of the control objective from its desired value. In t his work, we develop a scheme to construct upper and lower bounds for the field amplitude and spectral parameter for each member of the denu merably infinite set of control solutions. The bounds can be tightened if desired. The analysis here is primarily restricted to the weak fie ld regime, although the bounds for the strong field nonlinear case are also presented.