An application of minimax robust optimal control theory for selective vibrational excitation in molecules

In recent investigations control theory was applied to design electromagnetic fields capable of producing vibrational excitation in molecular systems. This approach has been applied to linear or non-linear classical approximations of molecular systems or to quantal systems using distributed cost functionals. Practical computations of molecular optimal control theory for large molecules especially with anharmonic potentials become difficult due to the increased dimensionality and the mixed nature of the boundary conditions. This paper proposes to approach the control design for such systems by treating a portion of the molecule containing the target and dipole bonds in full detail while the effect of the remainder of the system is modelled as a disturbance of limited energy. The optimal field minimizes the cost functional which is simultaneously maximized with respect to the disturbance. Such assumptions give rise to a robust controller akin to the H∞ theory of robust estimation. We investigate the various field designs for truncated harmonic systems associated with different disturbance energies and demonstrate that the existence of the solution to the associated Ricatti equation ensures the existence of the equilibrium game point. In addition, in the range of physically reasonable disturbance energy the optimal field could be accurately predicted from an asymptotic expansion involving only the undisturbed reference case. As an application we show the optimal field design for a 20 atom truncated molecular chain containing both the target bond (the 5th bond) and the dipole bonds (1st and 9th) where the disturbance only affects the end bond of the system attached to the remainder of the chain. In an effort to improve on the efficiency of the bond energy deposition we investigate shortened target times and also a 40 atom truncated chain. This approach presents very conservative estimates of possible disturbances but provides insight into the sensitivity of different configurations with respect to external disturbances. The minimax approach can be generalized to non-linear molecular systems by modelling the original system as a linear system plus an energy constrained disturbance.