Molecular Control
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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. |
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Prolog to 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. -Jim Esch |
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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. |
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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 |
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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. |
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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. |
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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. |