A symmetric structure of variational and adjoint systems of stochastic Hamiltonian systems

Authors

Abstract

The authors have extended deterministic port-Hamiltonian systems into stochastic dynamical systems which are described by stochastic differential equations written in the sense of Itô, called stochastic port-Hamiltonian systems. This paper introduces variational systems and their adjoint ones for the stochastic port-Hamiltonian systems. We also reveal some of their properties, particularly an extension of a self-adjoint property of deterministic Hamiltonian systems, which plays an important role in learning optimal control for the deterministic Hamiltonian systems.

Citation

Journal: 49th IEEE Conference on Decision and Control (CDC)
Year: 2010
Volume:
Issue:
Pages: 1423–1428
Publisher: IEEE
DOI: 10.1109/cdc.2010.5718033

BibTeX

@inproceedings{Satoh_2010,
  title={{A symmetric structure of variational and adjoint systems of stochastic Hamiltonian systems}},
  DOI={10.1109/cdc.2010.5718033},
  booktitle={{49th IEEE Conference on Decision and Control (CDC)}},
  publisher={IEEE},
  author={Satoh, Satoshi and Fujimoto, Kenji},
  year={2010},
  pages={1423--1428}
}

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References

Yong, J. & Zhou, X. Y. Stochastic Controls. (Springer New York, 1999). doi:10.1007/978-1-4612-1466-3 – 10.1007/978-1-4612-1466-3
Fujimoto, K., Scherpen, J. M. A. & Gray, W. S. Hamiltonian realizations of nonlinear adjoint operators. Automatica vol. 38 1769–1775 (2002) – 10.1016/s0005-1098(02)00079-1
øksendal, Stochastic differential equations, An introduction with applications. (1998)
Kushner, H. J. On the stochastic maximum principle: Fixed time of control. Journal of Mathematical Analysis and Applications vol. 11 78–92 (1965) – 10.1016/0022-247x(65)90070-3
Peng, S. A General Stochastic Maximum Principle for Optimal Control Problems. SIAM Journal on Control and Optimization vol. 28 966–979 (1990) – 10.1137/0328054
Bismut, J.-M. An Introductory Approach to Duality in Optimal Stochastic Control. SIAM Review vol. 20 62–78 (1978) – 10.1137/1020004
karoui, Backward Stochastic Differential Equations. (1997)
satoh, Biped gait generation via iterative learning control including discrete state transitions. Proc 17th IFAC World Congress (2008)
Satoh, S., Fujimoto, K. & Hyon, S.-H. A framework for optimal gait generation via learning optimal control using virtual constraint. 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems 3426–3432 (2008) doi:10.1109/iros.2008.4650860 – 10.1109/iros.2008.4650860
Fujimoto, K. & Sugie, T. Iterative learning control of hamiltonian systems: I/O based optimal control approach. IEEE Transactions on Automatic Control vol. 48 1756–1761 (2003) – 10.1109/tac.2003.817908
van der schaft, Mathematical modeling of constrained Hamiltonian systems. Proc 3rd IFAC Symp Nonlinear Control Systems (1995)
ikeda, Stochastic differential equations and diffusion processes. (1989)
Arimoto, S., Kawamura, S. & Miyazaki, F. Bettering operation of Robots by learning. Journal of Robotic Systems vol. 1 123–140 (1984) – 10.1002/rob.4620010203
satoh, Stabilization of time-varying stochastic port-Hamiltonian systems based on stochastic passivity. Proc IFAC Symp Nonlinear Control Systems (2010)
Satoh, S. & Fujimoto, K. On passivity based control of stochastic port-Hamiltonian systems. 2008 47th IEEE Conference on Decision and Control 4951–4956 (2008) doi:10.1109/cdc.2008.4738733 – 10.1109/cdc.2008.4738733
maschke, Port-controlled Hamiltonian systems: modelling origins and system theoretic properties. Proc 2nd IFAC Symp Nonlinear Control Systems (1992)
Florchinger, P. A Passive System Approach to Feedback Stabilization of Nonlinear Control Stochastic Systems. SIAM Journal on Control and Optimization vol. 37 1848–1864 (1999) – 10.1137/s0363012997317478
Variational and Hamiltonian Control Systems. Lecture Notes in Control and Information Sciences (Springer Berlin Heidelberg, 1987). doi:10.1007/bfb0042858 – 10.1007/bfb0042858