The optimal control of the path to a specified 6nal 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 is established by proving the existence of an optimal solution. A I.agrange-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 6nite di8'erences is used to reduce the in6nite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for 6nal-state control of a diatomic molecule represented by a Morse potential. %ithia the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wavepacket control and 6nally to achieve a speci6ed dissociative wave packet at a given time. The optimal external optical Aelds 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.
Current experimental and theoretical progress toward the goal of controlling quantum dynamics is summarized. Two key developments have now revitalized the field. First, appropriate ultrafast laser pulse shaping capabilities have only recently become practical. Second, the introduction of engineering control concepts has put the required theoretical framework on a rigorous foundation. Extrapolations to determine what is realistically possible are presented.
The purpose of this brief note is to show that the set of states reachable from a given initial state for a finite dimensional quantum system is equal to the orbit under the Lie group generated by the Lie algebra generated by the internal and external
We investigate energy ampli cation in parallel channel ows, where background noise is modeled as stochastic excitation of the linearized Navier-Stokes equations. We show analytically that the energy of three-dimensional streamwise-constant disturbances achieves O(R 3) ampli cation. Our basic technical tools are explicit analytical calculations of the traces of solutions of operator Lyapunov equations, which yield the covariance operators of the forced random velocity elds. The dependence of these quantities on both the Reynolds number and the spanwise wave-number are explicitly computed. We show how the ampli cation mechanism is due to a coupling between normal velocity and vorticity disturbances, which in turn is due to non-zero mean shear and disturbance spanwise variation. This mechanism is viewed as a consequence of the non-normality of the dynamical operator, and not necessarily due to the existence of near resonances or modes with algebraic growth.
Learning control is an iterative approach to the problem of improving transient behavior for processes that are repetitive in nature. In this article, we present some results on iterative learning control. A complete review of the literature is given first. Then, a general formulation of the problem is given. Next, we present a complete analysis of the learning control problem for the case of linear, time-invariant plants and controllers. This analysis offers: (1) insight into the nature of the solution of the learning control problem by deriving sufficient convergence conditions; (2) an approach to learning control for linear systems based on parameter estimation; and (3) an analysis that shows that for finite-horizon problems it is possible to design a learning control algorithm that converges, with memory, in one step. Finally, a time-varying learning controller is given for controlling the trajectory of a nonlinear robot manipulator. A brief simulation example is presented to illustrate the effectiveness of this scheme.
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