Application of optimal control theory for selective vibrational excitation in molecules modeled as harmonic physical systems

The design of optimal electromagnetic fields producing selective vibrational excitation in molecules modeled as harmonic physical systems is shown to be equivalent to minimizing a quadratic cost functional balancing the energy distribution in the molecule and the fluence of the input. In the control problem, two approaches are employed to insure that the final excitation is attained. One method uses a control strategy that employs a terminal constraint and in the other approach the cost functional is augmented with a terminal cost. The asymptotic form of the state and costate is investigated for both strategies in the limit that the final time approaches infinity, and some mathematical results on the form of the Lagrange parameter are presented for the first type of controller. These two results allow for a detailed discussion on the appropriate choice of practical design constants. For the example of a linear chain molecule, and approximation for the eigenvalues of the Hamiltonian matrix is derived for the limiting case where the weighting on the fluence of the optical field in the cost functional increases to infinity. Also, for the linear chain it is shown that the eigenvalues are bounded and that this bound does not depend on the length of the chain.