An application of minimax analysis to robust optimal control of molecular dynamics

Recent theoretical research has shown that a molecule can be manipulated and controlled by an optical field designed through optimal control theory. In practice, a number of disturbances/uncertainties arise, e.g., in the laboratory generated control field, in the molecular Hamiltonian, or in the interaction term (i.e., the dipole moment). It is, therefore, important to design an optimal control field which is robust to such disturbances. In this paper, we apply minimax analysis to design optimal control fields in the presence of disturbances having finite magnitude. The minimax analysis seeks the best control field in the presence of the worst disturbance of bounded magnitude, thus, the minimax solution gives a conservative estimate of the extent to which a physical objective can be achieved under the possed disturbances. As an example, we consider a diatomic molecule modeled as a Morse oscillator. The control objective is to achieve either selective excitation of a particular energy state or to transform an initial wave packet to an arbitrary nonstationary wave packet, in the presence of the three aforementioned disturbances. For the former objective, the results for the bounded worst disturbances and robust optimal control fields are similar to that of previous perturbation calculations. For the second objective, however, the results significantly differ from analogous perturbation calculations. Under certain severe conditions useful minimax solutions may be difficult to find or simply do not exist.