Molecular Control

 


One of the primary goals of chemists is to stimulate chemical reactions to form desired products. Recently chemists have sought more selective ways to stimulate chemical reactions by means of laser fields. This has led to the problem of designing optical fields to control structure and dynamics at the molecular scale. We applied optimal control theory to the design of laser fields for controlling molecular motion and subsequently extended this methodology to the design of controllers for uncertain quantum systems. This paradigm has been adopted by many subsequent researchers and has been the theme of a number of workshops (Telluride, 1991 and the Fields Institute 1992, which I co-organized). The possible applications of this work include the manufacture of new drugs, the preparation of surfaces, and the design of experiments to probe molecular potentials. We were invited by the IEEE Proceedings to write a review paper as a recognition of our contribution to this area.n this paper we analyze the controllability of quantum systems arising in molecular dynamics. We model these systems as systems with finite numbers of levels, and examine their controllability. To do this we pass to their unitary generators and use results on the controllability of invariant systems on Lie groups. Examples of molecular systems, modeled as finite-dimensional control systems, are provided. A simple algorithm to detect the controllability of a molecular system is provided. Finally, we apply this algorithm to a five-level system.

6. Control of Molecular Motion
by M. Dahleh, A. Peirce, H. Rabitz, and V. Ramakrishna
Proceedings of the IEEE, Vol. 84, No. 1, pp. 7-15, 1996.
 

 

 

Prolog to
Control of Molecular Motion
An introduction to the paper by Dahleh, Peirce, Rabitz and Ramakrishna

Traditionally, the way to stimulate chemical reactions so as to form desired end products has involved altering global thermodynamic variables like temperature and pressure and combining reagents. But reagents also form undesirable byproducts; moreover, some products cannot be formed in traditional ways. These shortcomings drive research into alternative methods. One way to better control molecular motion or reactions is to utilize radiative coupling-to prepare the molecule in certain quantum states, which will then evolve into the desired end product. Radiative coupling relies on determining the correct field to apply-a field robust enough for modeling and the rigors of the laboratory. Previous intuitive usage of electrical and optical fields to provide molecular control have been unsuccessful. At this time, there is a need to exploit the dynamics and quantum interference structure of the molecule and to apply it to field design.

The application of control theory to molecular motion has led to major advances..Control theory shows us how to design an input to force the evolution of a system to a desired end. Techniques of optimal control span many analytical and computational methods. With respect to molecular control, a: particularly useful approach is the variational method: the solution of two coupled boundary-value problems derived from the quantum description of molecular dynamics. This solution provides gradient information about the desired end-information used to search for a suitable optimal control input. Such an approach is outlined in this paper by means of a case study describing the control of a diatomic molecule.
Applied control theory has led to significant developments in field design for molecular systems, including the excitation of vibrational modes of polyatomic molecules, rotational excitations, and the dissociation of molecular structures. Nonlinear control methods also play an important role. One of the most important control objectives that the paper analyzes is the problem of constructive controllability: How does one, given an initial state, prepare a desired final state? Optimal control is not without its drawbacks; while it is very flexible and useful for field design, it is computationally demanding. Other techniques may in some cases prove helpful, namely "inverse control" and "exact tracking." There are also limitations to these techniques when applied to practical lab situations. Since dynamic molecular events happen so quickly, traditional real-time feedback is currently impractical. Also, the complexity of the control equations and variability of the fields may make implementation difficult However, experiments could be performed sequentially in an adaptive feedback learning fashion at the molecular scale. Such an adaptive feedback mechanism is described in the paper. One enticing application of molecular control would be to attain molecular "inversion," assuming a guiding algorithm can be developed. We may see "smart," adaptively controlled spectrometers designed especially for inversion. The effort toward molecular control is making new demands from control theory itself. Problematic factors include the complexity of quantum mechanical models, the very small time scale of molecular systems, and the incomplete knowledge of molecular Hamiltonians. As for nonlinear control, there are the problems of "motion planning," the need for a controllability framework for laboratory observables, and techniques for the exact tracking of multiple objectives.

-Jim Esch


5. Controllability of Molecular Systems
by V. Ramakrishna, M. Salapaka, M. Dahleh, H. Rabitz, and A. Peirce
Physical Review A, Vol. 51, No. 2, pp. 960-966, 1995.


Abstract: In this paper we analyze the controllability of quantum systems arising in molecular dynamics. We model these systems as systems with finite numbers of levels, and examine their controllability. To do this we pass to their unitary generators and use results on the controllability of invariant systems on Lie groups. Examples of molecular systems, modeled as finite-dimensional control systems, are provided. A simple algorithm to detect the controllability of a molecular system is provided. Finally, we apply this algorithm to a five-level system.

4. Design Challenges for Control of Molecular Dynamics
by M. Dahleh, A. P. Peirce, and H. Rabitz
IEEE Control Systems Magazine, Vol. 12, No. 2, pp. 93-94, April, 1992.


Abstract: Material modification at the molecular scale is considered. Optimal control is explored as a theoretical framework for exploiting molecular dynamics. The possibility of using optical fields to break bonds selectively is discussed

3. Numerical Solution of a Class of Parabolic Partial Differential Equations Arising in Optimal Control Problems with Uncertainty
by M. Dahleh and A. P. Peirce
Numerical Methods for Partial Differential Equations, Vol. 8, No. 1, pp. 77-95, Jan., 1992.


Abstract: In this paper the optimal control of uncertain parabolic systems of partial differential equations is investigated. In order to search for controllers that are insensitive to uncertainties in these systems, an iterative optimization procedure is proposed. This procedure involves the solution of a set of operator valued parabolic partial differential equations. The existence and uniqueness of solutions to these operator equations is proved, and a stable numerical algorithm to approximate the uncertain optimal control problem is proposed. The viability of the proposed algorithm is demonstrated by applying it to the control of parabolic systems having two different types of uncertainty.

2. Optimal Control of Uncertain Quantum Systems
by M. Dahleh, A. Peirce, and H. Rabitz
Physical Review A, Vol. 42, No. 3, pp. 1065-1079, August, 1990.


Abstract: The design of optimal final-state controllers of quantum-mechanical systems that are insensitive to errors in the molecular Hamiltonian or to errors in the initial state of the system is considered. Control arises through the interaction of the system with an external field; the goal is optimal design of these latter fields for various physical objectives in the presence of system uncertainty. Sensitivity to modeling errors and other uncertainties in the molecular Hamiltonian is minimized by considering averaged costs for a family of Hamiltonian functions $H(\alpha)$ indexed by the random variable $\alpha$ taking values on a compact set in Euclidean space. Similarly, sensitivity of the optimal control to the initial state is minimized by viewing the initial condition as a Hilbert-space-valued random variable and considering an optimization problem with a cost functional that is averaged over the class of initial conditions. A precise formulation of the control problem is given, and its well-posedness is established. Cost propagators are defined to display the dependence of the performance index on the initial conditions explicitly, which allows analytic averaging of initial conditions. The constrained optimization problem is reduced to an unconstrained optimization problem by the i introduction of Lagrange-multiplier operators. Necessary conditions for the unconstrained problem provide the basis for a gradient search for an optimal solution. Finite-difference schemes are utilized to provide a numerical approximation of the optimal control problem. Numerical examples are given for final-state control of a diatomic molecule represented by a Morse potential illustrating design for systems with initial-phase uncertainty and parametric uncertainty. The resultant insensitive controllers execute different strategies depending on the design requirements. The controller designed to be insensitive to errors in the initial phases adopts a strategy of phase imprinting during the initial stages of the control interval to compensate for a lack of knowledge of the initial phases. It is also shown that it is not possible to coerce a system from a state with completely random initial phases to a correlated state using the class of averaged controllers considered here. The controller designed to be insensitive to parametric Hamiltonian errors adopts a strategy of amplitude restraint to prevent the wave packet from taking significant excursions into the regions where the potential is uncertain. The interesting structure exhibited by the controllers in response to the different design requirements, and the superior performance of the insensitive controllers when compared with controllers designed at nominal phases and parameters, illustrate the usefulness of the cost-averaging technique for design in the presence of uncertainties.

1. Optimal Control of Quantum Mechanical Systems: Existence, Numerical Approximations and Applications
by A. P. Peirce, M. Dahleh and H. Rabitz
Physical Review A, Vol. 37, No. 12, pp. 4950-4964, 15 June, 1988.

Abstract: The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wavepacket control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time, period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives.