^{1,a)}, Tak-San Ho

^{2}, Ruixing Long

^{2}, Herschel Rabitz

^{2,b)}and Rebing Wu

^{3}

### Abstract

Optimal control of molecular dynamics is commonly expressed from a quantum mechanical perspective. However, in most contexts the preponderance of molecular dynamics studies utilize classical mechanical models. This paper treats laser-driven optimal control of molecular dynamics in a classical framework. We consider the objective of steering a molecular system from an initial point in phase space to a target point, subject to the dynamic constraint of Hamilton's equations. The classical control landscape corresponding to this objective is a functional of the control field, and the topology of the landscape is analyzed through its gradient and Hessian with respect to the control. Under specific assumptions on the regularity of the control fields, the classical control landscape is found to be free of traps that could hinder reaching the objective. The Hessian associated with an optimal control field is shown to have finite rank, indicating the presence of an inherent degree of robustness to control noise. Extensive numerical simulations are performed to illustrate the theoretical principles on (a) a model diatomic molecule, (b) two coupled Morse oscillators, and (c) a chaotic system with a coupled quartic oscillator, confirming the absence of traps in the classical control landscape. We compare the classical formulation with the mathematically analogous quantum state-to-state transition probability control landscape.

This work was supported in parts by the US Department of Energy (DOE) and the Princeton Plasma Physics Laboratory. C.J.-W. acknowledges the NDSEG Fellowship. R.W. acknowledges support from NSFC (Grant Nos. 60904034 and 61134008).

I. INTRODUCTION

II. OPTIMAL PHASE SPACE CONTROL WITHIN CLASSICAL MECHANICS

A. Critical points of the control landscape

B. Nature of the control landscape critical points: Hessian analysis

C. Robustness analysis and level sets at the control landscape global maximum

III. COMPARING THE CLASSICAL AND QUANTUM CONTROL LANDSCAPES

IV. NUMERICAL SIMULATIONS

A. Numerical techniques: Symplectic integration and D-MORPH optimization algorithms

B. Example: Optimal control of a diatomic molecule

C. Example: Optimal control of two coupled Morse oscillators

D. Example: Optimal control of a chaotic coupled quartic oscillator

V. CONCLUSION

### Key Topics

- Critical point phenomena
- 31.0
- Oscillators
- 17.0
- Coupled oscillators
- 14.0
- Eigenvalues
- 8.0
- Electric dipole moments
- 6.0

## Figures

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 .

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 .

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 .

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 .

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 .

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 .

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}.

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}.

Presentation of δ**z**(*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.

Presentation of δ**z**(*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.

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 .

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 .

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).

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).

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

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

The singular values of δ**z**(*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 .

The singular values of δ**z**(*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 .

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.

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.

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.

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.

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).

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).

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.

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.

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.

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

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

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

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