^{1,a)}, O. Cots

^{2}, N. Shcherbakova

^{1}and D. Sugny

^{3,b)}

### Abstract

In this article, we study the energy minimization problem of dissipative two-level quantum systems whose dynamics is governed by the Kossakowski–Lindblad equations. In the first part, we classify the extremal curve solutions of the Pontryagin maximum principle. The optimality properties are analyzed using the concept of conjugate points and the Hamilton–Jacobi–Bellman equation. This analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed. In the final section, an application in nuclear magnetic resonance is presented.

I. INTRODUCTION

II. GEOMETRIC ANALYSIS OF THE EXTREMAL CURVES

A. Maximum principle

B. Geometric computations of the extremals

1. Normal extremals in spherical coordinates

2. Abnormal extremals in spherical coordinates

C. The analysis in the normal case

D. Normal extremals in meridian planes

1. The integrable case

2. The general case

E. Normal extremals in nonmeridian planes

1. The integrable case

2. Analysis in the case

III. THE OPTIMALITY PROBLEM

A. Existence theorem

B. Optimality concepts

C. Symmetries and optimality

1. The integrable case

2. The general case

D. The geometric properties of the variational equation and estimation of conjugate points

1. Preliminaries

2. The integrable case

3. Computation of the conjugate locus for short periodic orbits in the meridian case

E. The value function and Hamilton–Jacobi–Bellman equation

1. The abnormal case

2. The Hamilton–Jacobi–Bellman theory in the normal case

3. The global Hamilton–Jacobi–Bellman equation

IV. GEOMETRIC ALGORITHMS AND NUMERICAL SIMULATIONS

A. The COTCOT code

B. The smooth continuation method

C. Robustness issues

D. Numerical simulations

1. Application in nuclear magnetic resonance

2. Extremals and conjugate points in the integrable case

3. Conjugate loci, spheres, and wave fronts

4. Extremals and conjugate points in the nonintegrable case

5. Continuation results

### Key Topics

- Lagrangian mechanics
- 20.0
- Vector fields
- 15.0
- Polynomials
- 12.0
- Nuclear magnetic resonance
- 11.0
- Singularity theory
- 11.0

## Figures

Phase portraits in the case for (left) and (right). In the left panel, numbers 1, 2, 3, 4, and 5 are, respectively, associated with , , , , and . In the right panel, they correspond, respectively, to , , , , and .

Phase portraits in the case for (left) and (right). In the left panel, numbers 1, 2, 3, 4, and 5 are, respectively, associated with , , , , and . In the right panel, they correspond, respectively, to , , , , and .

Phase portraits in the case for (top), (middle), and for (bottom).

Phase portraits in the case for (top), (middle), and for (bottom).

Behavior near the origin of a family of orbits (see the text).

Behavior near the origin of a family of orbits (see the text).

Foliation of (a) and level sets of the value function (b). (Bottom panel) Conjugate point analysis. Numerical values are taken to be , from top to bottom.

Foliation of (a) and level sets of the value function (b). (Bottom panel) Conjugate point analysis. Numerical values are taken to be , from top to bottom.

(Top) Generic microlocal case. (Bottom) The Grushin case.

(Top) Generic microlocal case. (Bottom) The Grushin case.

(Top) Plot of the different optimal trajectories (top) and of the corresponding control fields (bottom) for different dissipation parameters. Numerical values are, respectively, taken to be , 1.24, and 2.02 for the red (dark gray), green (light gray), and blue (black) trajectories.

(Top) Plot of the different optimal trajectories (top) and of the corresponding control fields (bottom) for different dissipation parameters. Numerical values are, respectively, taken to be , 1.24, and 2.02 for the red (dark gray), green (light gray), and blue (black) trajectories.

Plot of the different optimal trajectories (top) and of the corresponding control fields (bottom) for different control durations. Numerical values are, respectively, taken to be , 1.5, and 2 for the red (dark gray), green (light gray), and blue (black) trajectories. The value refers to the time-minimum solution for the same dissipation parameters and a maximum normalized amplitude of the control field of . The black crosses indicate the positions of the different conjugate points. The middle panel is a zoom of the top figure near the origin.

Plot of the different optimal trajectories (top) and of the corresponding control fields (bottom) for different control durations. Numerical values are, respectively, taken to be , 1.5, and 2 for the red (dark gray), green (light gray), and blue (black) trajectories. The value refers to the time-minimum solution for the same dissipation parameters and a maximum normalized amplitude of the control field of . The black crosses indicate the positions of the different conjugate points. The middle panel is a zoom of the top figure near the origin.

(a) Plot of the potential as a function of . (b) Plot of four extremals corresponding to , 0, 1, and . (c) Same as (b) but up to the first conjugate point. Other numerical values are taken to be , , , , and . The extremals associated with and 1 are short periodic orbits with an energy equal to 5.5, while extremals with initial adjoint states and 2 are long periodic orbits with an energy equal to 6.5. The horizontal dashed lines indicate the positions of the parallel of equation and the antipodal one of equation where short and long periodic orbits, respectively, intersect with the same time.

(a) Plot of the potential as a function of . (b) Plot of four extremals corresponding to , 0, 1, and . (c) Same as (b) but up to the first conjugate point. Other numerical values are taken to be , , , , and . The extremals associated with and 1 are short periodic orbits with an energy equal to 5.5, while extremals with initial adjoint states and 2 are long periodic orbits with an energy equal to 6.5. The horizontal dashed lines indicate the positions of the parallel of equation and the antipodal one of equation where short and long periodic orbits, respectively, intersect with the same time.

Conjugate loci, spheres, and wave fronts for the Grushin (top) and the non-Grushin case (bottom).

Conjugate loci, spheres, and wave fronts for the Grushin (top) and the non-Grushin case (bottom).

(a) Evolution of the angle as a function of the angle for , , and . Initial values are taken to be , , , and , 0, and 1. (b) Evolution of the radial coordinate as a function of time. [(c) and (d)] Plot of the two optimal control fields and as a function of time for .

(a) Evolution of the angle as a function of the angle for , , and . Initial values are taken to be , , , and , 0, and 1. (b) Evolution of the radial coordinate as a function of time. [(c) and (d)] Plot of the two optimal control fields and as a function of time for .

The same as Fig. 10 but for .

The same as Fig. 10 but for .

The same as Fig. 10 but the extremals are plotted up to the first conjugate point.

The same as Fig. 10 but the extremals are plotted up to the first conjugate point.

The same as Fig. 12 but for .

The same as Fig. 12 but for .

Evolution as a function of the parameter of the initial adjoint states , , and for the Newton-type continuation (solid line) and the smooth-type one (dashed line). The other dissipative parameters are and and the control duration is equal to 0.5. The coordinates of the initial point are equal to , , , , , and for . The target state is (, , ).

Evolution as a function of the parameter of the initial adjoint states , , and for the Newton-type continuation (solid line) and the smooth-type one (dashed line). The other dissipative parameters are and and the control duration is equal to 0.5. The coordinates of the initial point are equal to , , , , , and for . The target state is (, , ).

Three extremal trajectories solutions of the smooth continuation method for , 2.5, and 3. The black dot indicates the position of the target state.

Three extremal trajectories solutions of the smooth continuation method for , 2.5, and 3. The black dot indicates the position of the target state.

The same as Fig. 14 but for a continuation in . The other dissipative parameters are and and the control duration is equal to 0.5. The coordinates of the initial point are equal to , , , , , and for . The target state is (, , ).

The same as Fig. 14 but for a continuation in . The other dissipative parameters are and and the control duration is equal to 0.5. The coordinates of the initial point are equal to , , , , , and for . The target state is (, , ).

Three extremal trajectories solutions of the smooth continuation method for , 0, and 0.1. The black dot indicates the position of the target state.

Three extremal trajectories solutions of the smooth continuation method for , 0, and 0.1. The black dot indicates the position of the target state.

Evolution of the continuation parameters as a function of the number of steps for the discrete method (solid line) and the smooth one (dashed line).

Evolution of the continuation parameters as a function of the number of steps for the discrete method (solid line) and the smooth one (dashed line).

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