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Topology of classical molecular optimal control landscapes in phase space
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10.1063/1.4797498
/content/aip/journal/jcp/138/12/10.1063/1.4797498
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/12/10.1063/1.4797498

Figures

Image of FIG. 1.
FIG. 1.

Initial field and final optimal control for the single oscillator with a strong field. The initial field yields the phase space point . The target state was , and the optimal field achieved .

Image of FIG. 2.
FIG. 2.

Phase plane trajectories with the initial field and final optimal control fields for the single oscillator. The point denoted with × marks the final state of the trajectory with the optimal field, while that denoted with + marks the final state with the initial field. The ■ marks the initial point .

Image of FIG. 3.
FIG. 3.

Optimal control field (a) and phase plane plot (b) for the single oscillator with weak control field. The “×” marks the final point with the optimal field, and the “+” marks the final point with the initial field; the “□” marks the initial point . With the initial field, the final state is and with the optimal field it is . The target state is .

Image of FIG. 4.
FIG. 4.

Evolution of the objective functional J versus s for the single oscillator with a weak control field. The optimization was stopped when J = −8.46 × 10−6.

Image of FIG. 5.
FIG. 5.

Presentation of δz(T)/δε(t) for the single oscillator with weak control field. The singular values are 3.97 8 and 0.23 6, thereby showing clear linear independence of the two functions.

Image of FIG. 6.
FIG. 6.

The evolution of the objective functional J versus s for sixteen different simulations using the two coupled oscillators. The objective functional value approached zero in all cases, and the optimization was stopped when the values reached those shown in Table I .

Image of FIG. 7.
FIG. 7.

Initial and final optimal control fields for simulation case o with the two coupled oscillators. The initial field yields the state coordinates = (10.61, −0.40, 0.52, 0.23). The target state was , and the optimal field achieved = (2.02, 0.49, −0.013, −0.22).

Image of FIG. 8.
FIG. 8.

Position (a) and momentum (b) trajectories for case o of the two coupled oscillators with initial and final control fields shown in Fig. 7 .

Image of FIG. 9.
FIG. 9.

The singular values of δz(T)/δε(s, t) as a function of s over the control field evolution to climb the landscape for case o of the two coupled oscillators. The finite, nonzero singular values are consistent with the surjectivity assumption in the analysis of Sec. II .

Image of FIG. 10.
FIG. 10.

Phase-space illustrations (a) for (q 1, p 1) and (b) for (q 2, p 2) of the chaotic system's response to perturbations in the initial state values with no control field (nominal initial state: (q 1(0), q 2(0), p 1(0), p 2(0)) = (−0.4, 0.6, 0.2, 0.5); perturbed initial state: (q 1(0), q 2(0), p 1(0), p 2(0)) = (−0.414, 0.617, 0.197, 0.515)). A square marks the (approximate) location of the initial states in the phase plots; a “+” marks the final state of the nominal trajectories, and a “×” marks the final state of the perturbed trajectory.

Image of FIG. 11.
FIG. 11.

Phase-space illustrations (a) for (q 1, p 1) and (b) for (q 2, p 2) of the chaotic system's response to perturbations in the initial state values with the initial control field present (nominal initial state: (q 1(0), q 2(0), p 1(0), p 2(0)) = (−0.4, 0.6, 0.2, 0.5); perturbed initial state: (q 1(0), q 2(0), p 1(0), p 2(0)) = (−0.3985, 0.6014, 0.2014, 0.5007)). A square marks the (approximate) location of the initial states in the phase plots; a “+” marks the final state of the nominal trajectories, and a “×” marks the final state of the perturbed trajectory.

Image of FIG. 12.
FIG. 12.

Initial and final optimal control fields. The initial field yields the phase space point (q 1(T), q 2(T), p 1(T), p 2(T)) = (−0.43, 0.31, −0.02, 0.54). The target state was (q 1,tar, q 2,tar, p 1,tar, p 2,tar) = (0.5, −0.5, 0.3, 0), and the optimal field achieved (q 1(T), q 2(T), p 1(T), p 2(T)) = (0.480, −0.593, 0.277, 0.013).

Image of FIG. 13.
FIG. 13.

The evolution of the objective functional J versus s. The objective continued to increase monotonically and the process was stopped when J = −0.0097 was achieved at s = 14.8.

Image of FIG. 14.
FIG. 14.

Phase plane plots of the position and momentum, (q 1, p 1) in (a) and (q 2, p 2) in (b), trajectories with initial and optimal control fields. The final points are marked with a “+” and a “×” for the initial and final fields, respectively; the initial point at t = 0 is marked with a square.

Tables

Generic image for table
Table I.

Initial state conditions and final state coordinates with the optimal fields of the simulations shown in Fig. 6 .

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/content/aip/journal/jcp/138/12/10.1063/1.4797498
2013-03-29
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Topology of classical molecular optimal control landscapes in phase space
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/12/10.1063/1.4797498
10.1063/1.4797498
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