### Abstract

We consider the one-dimensional Schroedinger equation in which the control is a time-dependent rectangular potential barrier/well. This is a bilinear control problem, as the potential multiplies the state. Differential geometric methods have been used to treat the bilinear control of systems of finitely many ODEs, and have been applied to the Schroedinger equation (quantum systems). In this paper we will calculate, using MATLAB, explicit controls which steer localized initial data to localized terminal data. These will be obtained using the Crank-Nicolson approximation, in which both space and time are discretized. If one semi-discretizes, in space, one obtains a bilinear control problem for a system of finitely many ODEs. One may pass from the semi-discretized system to CrankNicolson using the trapezoid rule. Thus the controls we calculate may be used to construct approximations to controls for the system of ODEs.

Original language | English (US) |
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Title of host publication | Proc. of the ASME Int. Des. Eng. Tech. Conf. and Comput. and Information in Engineering Conferences - DETC2005 |

Subtitle of host publication | 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control |

Pages | 623-626 |

Number of pages | 4 |

State | Published - Dec 1 2005 |

Event | DETC2005: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - Long Beach, CA, United States Duration: Sep 24 2005 → Sep 28 2005 |

### Publication series

Name | Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - DETC2005 |
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Volume | 6 A |

### Conference

Conference | DETC2005: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference |
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Country | United States |

City | Long Beach, CA |

Period | 9/24/05 → 9/28/05 |

### Fingerprint

### Keywords

- Control
- MATLAB
- Numerical
- Potential
- Schroedinger

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proc. of the ASME Int. Des. Eng. Tech. Conf. and Comput. and Information in Engineering Conferences - DETC2005: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control*(pp. 623-626). (Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - DETC2005; Vol. 6 A).

**Numerical approximation of bilinear control of the schroedinger equation.** / Kime, Katherine A.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proc. of the ASME Int. Des. Eng. Tech. Conf. and Comput. and Information in Engineering Conferences - DETC2005: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control.*Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - DETC2005, vol. 6 A, pp. 623-626, DETC2005: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, CA, United States, 9/24/05.

}

TY - GEN

T1 - Numerical approximation of bilinear control of the schroedinger equation

AU - Kime, Katherine A.

PY - 2005/12/1

Y1 - 2005/12/1

N2 - We consider the one-dimensional Schroedinger equation in which the control is a time-dependent rectangular potential barrier/well. This is a bilinear control problem, as the potential multiplies the state. Differential geometric methods have been used to treat the bilinear control of systems of finitely many ODEs, and have been applied to the Schroedinger equation (quantum systems). In this paper we will calculate, using MATLAB, explicit controls which steer localized initial data to localized terminal data. These will be obtained using the Crank-Nicolson approximation, in which both space and time are discretized. If one semi-discretizes, in space, one obtains a bilinear control problem for a system of finitely many ODEs. One may pass from the semi-discretized system to CrankNicolson using the trapezoid rule. Thus the controls we calculate may be used to construct approximations to controls for the system of ODEs.

AB - We consider the one-dimensional Schroedinger equation in which the control is a time-dependent rectangular potential barrier/well. This is a bilinear control problem, as the potential multiplies the state. Differential geometric methods have been used to treat the bilinear control of systems of finitely many ODEs, and have been applied to the Schroedinger equation (quantum systems). In this paper we will calculate, using MATLAB, explicit controls which steer localized initial data to localized terminal data. These will be obtained using the Crank-Nicolson approximation, in which both space and time are discretized. If one semi-discretizes, in space, one obtains a bilinear control problem for a system of finitely many ODEs. One may pass from the semi-discretized system to CrankNicolson using the trapezoid rule. Thus the controls we calculate may be used to construct approximations to controls for the system of ODEs.

KW - Control

KW - MATLAB

KW - Numerical

KW - Potential

KW - Schroedinger

UR - http://www.scopus.com/inward/record.url?scp=33244477897&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33244477897&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33244477897

SN - 0791847438

SN - 9780791847435

T3 - Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - DETC2005

SP - 623

EP - 626

BT - Proc. of the ASME Int. Des. Eng. Tech. Conf. and Comput. and Information in Engineering Conferences - DETC2005

ER -