Control of quasi non-integrable Hamiltonian systems for targeting a specified stationary probability density

ABSTRACT

An innovative design procedure for the feedback control of quasi non-integrable Hamiltonian systems to target a specified stationary probability density function (SPDF) is proposed based on the averaged Fokker–Planck–Kolmogorov equation. The control force is split into a conservative part and a dissipative part. The conservative part is designed to make the controlled system and the target SPDF having the same Hamiltonian structure. The dissipative part is then determined to make the target SPDF to be the stationary solution of the controlled system. Then, a Lyapunov function method is adopted for proving that the transient solution of the controlled system approaches to the target SPDF as time t → ∞. The simulation result for an example shows the effectiveness of the proposed control strategy.

KEYWORDS:

• Stochastic nonlinear systems
• stationary probability density function
• feedback control
• stochastic average
• FPK equation

Acknowledgments

The work reported in this paper is supported by the National Natural Science Foundation of China under Grant No.11302123, 11321202, 11432012, Innovation Program of Shanghai Municipal Education Commission under Grant No.14YZ163 and “Chen Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation under Grant No.13CG63.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Innovation Program of Shanghai Municipal Education Commission [grant number 14YZ163]; the National Natural Science Foundation of China [grant number 11302123], [grant number 11321202], [grant number 11432012]; Shanghai Municipal Education Commission and Shanghai Education Development Foundation [grant number 13CG63].