J. G. B. Beumee

Robust optimal control theory for selective vibrational excitation in molecules: A worst case analysis

Recent research has demonstrated that optimal electromagnetic fields capable of producing selective vibrational excitation in molecules can be designed employing linear quadratic control methods using a cost functional that balances the energy distribution in the molecule, the fluence of the optical field, and a final cost to insure the desired excitation. Practical computations of molecular control theory for large molecules especially with anharmonic potentials become difficult to obtain due to the increased dimensionality and the accompanying uncertainty in the Hamiltonian. In this paper we reduce the complexity of the problem by treating a portion of the molecule including the target and optical dipoles in full detail, while the remainder of the molecule is modeled as an external disturbance of bounded energy. The optimal control field now minimizes the cost functional which is simultaneously maximized with respect to the energy constrained external disturbance to assure robustness. This optimal desig...

Sensitivity analysis of experimental data

This paper examines how a number of sensitivity techniques may be adapted to deal with the analysis of real experiments. We show that the fundamental quantities in this context are the partial derivatives of model input parameters with respect to experimental observables. They appear as a consequence of the procedure used to fit the model to the experiment. These so-called experimental elementary sensitivities are then combined with the usual model elementary sensitivities to yield coefficients which relate experimental and model observables. The importance of such coefficients for the planning of experiments is, in particular, discussed. Based on a linear analysis, we also derive simple expressions (containing the experimental elementary sensitivities) for the degree of parameter deviation arising from uncertainties in and discrepancies between model and measured observables. Finally, an overview of the role of sensitivity theory in analyzing experimental data is given.

Optimal control of classical anharmonic molecules represented with piecewise harmonic potential surfaces: analytic solution and selective dissociation of triatomic systems

Abstract In this paper we apply optimal control theory to manipulate the dynamics of molecular systems with anharmonic potential surfaces modelled as a series of continuous piecewise harmonic sections within the classical approximation. In analogy to the simple harmonic case, an analytic solution is possible, yet bond dissociation is allowed. This methodology was applied to several model systems. It was found that the molecular objective of selective dissociation was achieved for a model linear triatomic system.

An application of filtering theory to parameter identification using stochastic mechanics

An estimation method for unknown parameters in the initial conditions and the potential of a quantal system using the stochastic interpretation of quantum mechanics and some results in system theory are presented. According to this interpretation the possible trajectories of a particle through coordinate space may be represented by the realization of a stochastic process that satisfies a stochastic differential equation. The drift term in this equation is derived from the wave function and consequently contains all unknown parameters in the initial conditions and the potential. The main assumption of the paper is that a continuous sequence of position measurements on the trajectory of the particle can be identified with a realization of this stochastic process over the corresponding period of time. An application of the stochastic filtering theorems subsequently provides a minimum variance estimate of the unknown parameters in the drift conditional on this continuous sequence of measurements. As simple il...

An application of filtering theory to parameter identification in quantum mechanics

A method for inverting observations on quantum mechanical systems to obtain estimates of unknown parameters residing in the Hamiltonian is presented. The quantal system is represented in matrix form with respect to a chosen basis, and it is assumed that the associated expansion coefficients are truncated to a finite dimension. The uncontrollable laboratory noise will be modeled by means of an inhomogeneous white noise process so that the experimental observations are represented as stochastic variables satisfying a stochastic differential equation. It will be assumed that measurements obtained from an experiment are now equivalent to a realization of these stochastic variables. It is known from filtering theory that the minimum variance estimate of the unknown parameters in the quantal model is now given by the expectation of the unknowns conditional on this realization. This estimator can be calculated analytically from the associated a posteriori probability density if the original quantal system does n...

Applications of filtering theory for the inversion of temporal chemical systems

This paper presents a method for inverting temporal experimental data from chemical models to obtain estimates of unknown parameters. Most of the models under consideration are deterministic and we assume that the measurements obtained from experimental observations are represented as the solution of a differential equation containing the variables of the model. To incorporate any extraneous laboratory effects that are not included in the model, we assume that these equations are perturbed by a white noise process so that the measurements become time‐dependent stochastic variables. A particular measurement is then equivalent to a realization of these variables and applying stochastic estimation theory this realization can be used to obtain estimates of the unknown parameters in the model. As an example of this estimation method, we consider chemical kinetics models with various observational equations and construct an estimator for the unknown reaction rate constants. We also show the estimators for the s...

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 fieldminimizes the cost functional which is simultaneouslymaximized with respect to the disturbance. Such assumptions give rise to a robust controller akin to theH∞ 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.

Optimal control theory for selective vibrational excitation in molecules

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 optical field. To ensure that a desired final excitation is attained, a terminal constraint is introduced for the state. A physically reasonable controller requires that both the weighting parameter on the fluence in the cost functional and the final time be large. The authors present the asymptotic form of the state and costate for large final time using familiar LQG (linear quadratic Gaussian) techniques and give an approximation of the modes of the linear chain molecule in the limit in which the weighting parameter becomes large. They conclude with a discussion of the choice of practical design parameters.<<ETX>>