# Application of Optimal Control Theory to Fourier Transform Ion Cyclotron Resonance

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## Abstract

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## 1. Introduction

## 2. Formulation of the Control Problem

#### 2.1. The Model System

#### 2.2. The Rotating Wave Approximation

## 3. Optimal Control Theory

#### 3.1. A Short Introduction to Optimal Control Theory

#### 3.2. Optimal Gradient-Based Algorithm

- Choose guess fields ${u}_{x}\left(t\right)$ and ${u}_{y}\left(t\right)$.
- Propagate forward the state of every ion k and compute $({v}_{x}^{\left(k\right)}\left({t}_{f}\right),{v}_{y}^{\left(k\right)}\left({t}_{f}\right))$.
- Propagate backward the adjoint state of the system from Equation (8).
- Compute the corrections $\delta {u}_{x}\left(t\right)$ and $\delta {u}_{y}\left(t\right)$ to the control fields, $\delta {u}_{x}\left(t\right)=\u03f5{\sum}_{k}{p}_{x}^{\left(k\right)}$, $\delta {u}_{y}\left(t\right)=\u03f5{\sum}_{k}{p}_{y}^{\left(k\right)}$ where $\u03f5$ is a small positive constant.
- Define the new control fields ${u}_{x}\left(t\right)\mapsto {u}_{x}\left(t\right)+\delta {u}_{x}\left(t\right)$${u}_{y}\left(t\right)\mapsto {u}_{y}\left(t\right)+\delta {u}_{y}\left(t\right)$.
- Truncate the new control fields ${u}_{x}\left(t\right)$ and ${u}_{y}\left(t\right)$ to satisfy the constraint $\sqrt{{u}_{x}{\left(t\right)}^{2}+{u}_{y}{\left(t\right)}^{2}}\le {u}_{max}$:$${u}_{x}\left(t\right)\mapsto \frac{{u}_{x}\left(t\right){u}_{max}}{\sqrt{{u}_{x}{\left(t\right)}^{2}+{u}_{y}{\left(t\right)}^{2}}},{u}_{y}\left(t\right)\mapsto \frac{{u}_{y}\left(t\right){u}_{max}}{\sqrt{{u}_{x}{\left(t\right)}^{2}+{u}_{y}{\left(t\right)}^{2}}}.$$
- Go to Step 2 until a given accuracy is reached.

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

OCT | Optimal Control Theory |

LQOCT | Linear Quadratic Optimal Control Theory |

ICR | Ion Cyclotron Resonance |

PMP | Pontryagin Maximum Principle |

NMR | Nuclear Magnetic Resonance |

RWA | Rotating Wave Approximation |

## Appendix A. The Rotating Wave Approximation

## Appendix B. Adiabatic Excitation of ICR Process

**Figure A1.**Excitation of an ensemble of ions by an adiabatic pulse: Evolution of the final radius as a function of the frequency. The parameters are set to ${t}_{f}=10$ ms, ${E}_{0}$ = 3.2 V/m and ${B}_{0}$ = 7 T. The red (light gray) solid line represents the stationary phase approximation. The vertical blue (dark gray) solid lines indicate the range of frequency of the pulse.

## Appendix C. Excitation by the SWIFT Approach

## Appendix D. Application of LQOCT to ICR

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**Figure 1.**Comparison between the optimal and the SWIFT approaches for the robust control of an ensemble of ions in the frequency range $[400,600]$ kHz. The small insert is a zoom of the profile around the frequency $f=600$ kHz. (

**a**,

**e**) The evolution of the final radius and phase as a function of f. Note that an arbitrary constant has been added to the phase in order to superimpose the curves (the three lines are practically indistinguishable in (

**e**)). The black, blue (dark gray) and red (light gray) solid lines depict, respectively, the optimal solutions computed without and with the RWA and the SWIFT pulse. The SWIFT and optimal control laws are plotted in (

**b**,

**c**) (optimal without RWA) and (

**d**) (optimal with RWA). The number of discretized frequency points is set to 601 in the optimization process in the range $[350,650]$ kHz.

**Figure 2.**Same as Figure 1 but for different slopes of the excitation profile. The parameter a is fixed, respectively, to 0.25 and 0.75 in (

**a**,

**b**). The optimal pulses without RWA are represented.

**Figure 3.**(

**a**) The final radius r as a function of the frequency f. The black, blue (dark gray) and red (light gray) curves represent, respectively, the ideal profile, the one obtained with the optimization algorithm and the one corresponding to the optimal pulse of Figure 1 whose amplitude has been abruptly limited (see the text for details). Note that the black and blue lines in (

**a**) are almost superimposed. The amplitude ${E}_{0}$ of the optimal fields in the rotating frame with (black curve) and without (red or light gray curve) constraints are depicted in (

**b**).

**Figure 4.**Plot of the optimal amplitudes ${E}_{0x}$ (blue or dark gray) and ${E}_{0y}$ (red or light gray) for a maximum amplitude of 100 V·m${}^{-1}$ (

**a**)) and 50 V·m${}^{-1}$ (

**b**). (

**c**) The corresponding total amplitude ${E}_{0}=\sqrt{{E}_{0x}^{2}+{E}_{0y}^{2}}$.

**Figure 5.**Evolution of the logarithm of the cost functional $\mathcal{J}$ as a function of the control time for different maximum amplitudes (black, 30 V·m${}^{-1}$; blue or dark gray, 50 V·m${}^{-1}$; red or light gray, 70 V·m${}^{-1}$).

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**MDPI and ACS Style**

Martikyan, V.; Beluffi, C.; Glaser, S.J.; Delsuc, M.-A.; Sugny, D.
Application of Optimal Control Theory to Fourier Transform Ion Cyclotron Resonance. *Molecules* **2021**, *26*, 2860.
https://doi.org/10.3390/molecules26102860

**AMA Style**

Martikyan V, Beluffi C, Glaser SJ, Delsuc M-A, Sugny D.
Application of Optimal Control Theory to Fourier Transform Ion Cyclotron Resonance. *Molecules*. 2021; 26(10):2860.
https://doi.org/10.3390/molecules26102860

**Chicago/Turabian Style**

Martikyan, Vardan, Camille Beluffi, Steffen J. Glaser, Marc-André Delsuc, and Dominique Sugny.
2021. "Application of Optimal Control Theory to Fourier Transform Ion Cyclotron Resonance" *Molecules* 26, no. 10: 2860.
https://doi.org/10.3390/molecules26102860