Optimal control experiments

Many of the initial theoretical models used to vahdate the concept of coherent control and optimal control have been based on the interaction of the electric field of the laser light with a molecular dipole moment [43, 60, 105]. This represents just the first, or lowest, term in the expression for the interaction of an electric field with a molecule. Many of the successful optimal control experiments have used electric fields that are capable of ionizing the molecules and involve the use of electric field strengths that lead to major distortions of the molecular electronic structure. With this in mind, there has been discussion in the... 

In the second experiment, we carried out an optimal control experiment on the production of highly charged ions [11]. The optimal control technique [12] consists of varying a large number of parameters that characterize the laser pulse shape in an automated way in order to... 

The pulse shapes from the optimal control experiment provide strong support to the nanoplasma model. The optimal result can easily be interpreted in terms of a first pulse that creates an overdense plasma which subsequently undergoes expansion and a second pulse, which interacts with the plasma when the plasma frequency equals the laser frequency. 

It is very likely that incomplete or missing records would prevent the verification of data integrity. Source records should be complete to facilitate an understanding of actual study conduct for critical phases of method development, method validation, and subject sample analysis. The records should confirm whether the testing was conducted in an appropriate manner, with well-designed and optimally controlled experiments. The documentation of actual laboratory events should demonstrate that the quantitative measures are suitable to achieve the objectives of the clinical or nonclinical protocol. The records should confirm that the reported results accurately reflect the actual concentration of the analyte in the biological matrix. It should be noted that the failure to adequately document critical details of study conduct has resulted in rejection of bioanalytical data for regulatory purposes. 

Mode selective excitation can therefore be achieved by adjusting the pulse intervals in a pulse train, which supports the interpretation of the optimal control experiment by Laarmann et al. [9], though the applied pulses used in the experiment were of A = 800 nm. More direct comparison with our theoretical results can be achieved by using longer wavelengths in the optimal control experiments of Ceo. 

This work prompted optimal control experiments on a large cyanine [88, 89], which confirmed that the branching ratio could be controlled by a pulse leading to excitation of the same skeletal deformations in the initial wavepacket (see Fig. 7.11). This control strategy was further validated in a quantum dynamics context with Gaussian-based direct dynamics simulations [90]. 

The results from the model were compared semi-quantitatively with experimental measurements to validate the model. Because of both the extremely long time constants in the system and the variations in the ambient conditions in the laboratory where the extruder was placed, it was not possible to rigorously test the model against experimental data. To conserve demands of time and materials for experiments on the extruder, numerical experiments were used to provide data for developing an optimal control system. The goal of the numerical experiments was to develop a reduced-order model suitable for optimizing the control system. 

The original optimal control problem can also be simplified (by reducing its dimensionality) by partitioning the manipulated variables u(t) into two groups U and u2. One group U could be kept constant throughout the experiment and hence, the optimal inputs for subgroup u2(t) are only determined. 

The type of associated anion has only a minor effect on the properties of a cationic retarder. General practical experience [39,41] suggests that optimal control is achieved if the... 

It is clear that pulse sequences may not only be designed by analytical means, they may also be designed numerically (see, e.g., reviews on numerical aspects of solid-state NMR in [54, 65, 66]) using standard nonlinear optimization to well-defined analytical expressions [67, 68], by optimizing pulse sequences directly on the spectrometer [69], or by optimal control procedures [70-72] to name but a few of the possibilities. We will in this review restrict ourselves to optimal control design procedures that recently in analytical and numerical form have formed a new basis for efficient NMR experiment design. 

While all methods described above have been derived by analytical means, we will in this section describe the design of recoupling experiments by numerical means in an optimal control setting. The idea behind the use of alternative design strategies... 

The first example, also being the example introducing optimal control to solid-state NMR [40] and further elaborated on later [161], is optimal control versions of the DCP experiment. This experiment was a natural choice for numerical improvements as it is widely used and it is well known that this experiment is sensitive to offsets, rf mismatch relative to the MAS-modified Hartmann-Hahn condition, and rf inhomogeneity. In particular the two latter effects may reduce significantly the performance of 15N to 13C transfers, severely complicate setup of such experiments, and render these critically sensitive to altered tuning/rf conditions in the course of potentially long experiments for biological samples. 

Fig. 11 (a) 2D NCO experiment with optimal control element inserted for 15N — 13C transfer. Transfer efficiencies for the ocNCO experiment optimized for 12 kHz spinning speed as function of (b) resonance offsets for 13C and 1SN and (c) rf inhomogeneity/adjustment in terms of scaling factors on the nominal rf field strengths for 13C and 15N. (d) Experimental ocNCO 2D spectrum of uniformly 13C,15N-labeled ubiquitin with the projections to the left comparing ocNCO experiment most intense) and DCP (less intense) based NCO experiments [reproduced with permission from [161] (a, d) and [164] (c)]... 

The final optimal control example addresses the possibility to design experiments characterized by a specific Hamiltonian. We consider the well-established... 

We note that optimal control is a universal tool for experiment design and has also, in solid-state NMR spectroscopy, found additional applications in the design of homonu-clear dipolar recoupling [41], broadband rf pulses and quantum gates [71], building blocks of symmetry-based recoupling experiments [129], quadrupolar multiple-quantum MAS experiments [165], and improved pulses for quadrupolar nuclei [166]. Numerous references to further applications with regard to liquid-state NMR can be found in [72]. 

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