World Academy of Science, Engineering and Technology
Published July 24, 2013 | Version 16412
Journal article Open

Implicit Lyapunov Control of Multi-Control Hamiltonians Systems Based On the State Error

Authors/Creators

Description

In the closed quantum system, if the control system is
strongly regular and all other eigenstates are directly coupled to the
target state, the control system can be asymptotically stabilized at the
target eigenstate by the Lyapunov control based on the state error.
However, if the control system is not strongly regular or as long as
there is one eigenstate not directly coupled to the target state, the
situations will become complicated. In this paper, we propose an
implicit Lyapunov control method based on the state error to solve the
convergence problems for these two degenerate cases. And at the same
time, we expand the target state from the eigenstate to the arbitrary
pure state. Especially, the proposed method is also applicable in the
control system with multi-control Hamiltonians. On this basis, the
convergence of the control systems is analyzed using the LaSalle
invariance principle. Furthermore, the relation between the implicit
Lyapunov functions of the state distance and the state error is
investigated. Finally, numerical simulations are carried out to verify
the effectiveness of the proposed implicit Lyapunov control method.
The comparisons of the control effect using the implicit Lyapunov
control method based on the state distance with that of the state error
are given.

Files

16412.pdf

Files (255.2 kB)

Name Size
md5:232c77a081b790719258e64c621fb523
255.2 kB Preview Download

Additional details

References

  • <p>
  • S. Cong and S. Kuang, "Quantum control strategy based on state distance," Acta Automatica Sinica, vol. 33, no. 1, pp. 28-31, 2007.
  • S. Kuang and S. Cong, "Lyapunov control methods of closed quantum systems," Automatica, vol. 44, no. 1, pp. 98-108, 2008.
  • M. Mirrahimi, P. Rouchon and G. Turinici, "Lyapunov control of bilinear Schrödinger equations," Automatica, vol. 41, pp. 1987-1994, 2005.
  • S. Grivopoulos and B. Bamieh, "Lyapunov-based control of quantum systems," In Proceedings of the 42nd IEEE Conference on Decision and Control, Maui. Hawaii, December 2003, pp. 434-438.
  • K. Beauchard, J. Coron, M. Mirrahimi and P. Rouchon, "Implicit Lyapunov control of finite dimensional Schrödinger equations," Systems & Control Letters, vol. 56, pp. 388-395, 2007.
  • S. Zhao, H. Lin, J. Sun and Z. Xue, "Implicit Lyapunov control of closed quantum systems," Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai. China, December 2009, pp. 3811-3815.
  • S. Zhao, H. Lin, J. Sun and Z. Xue, "An implicit Lyapunov control for finite-dimensional closed quantum systems," International Journal of Robust and Nonlinear control, vol. 22, Issue 11, pp. 1212-1228, 2012.
  • Fangfang Meng, Shuang Cong and Sen Kuang, "Implicit Lyapunov Control of Multi-Control Hamiltonian Systems Based on State Distance," The 9th World Congress on Intelligent Control and Automation, Beijing, pp. 5127-5232, 2012.
  • S. Krantz, H. Parks, The implicit function theorem: history, theory, and applications. Boston: Birkhauser, 2002. [10] J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method with Applications. New York: Academic Press, 1961 .</p>