Abstract
The problem of controlling a near-Hamiltonian noisy system so as to keep it within a domain of bounded oscillations has been studied intensively in the last decade. This paper considers a new class of problems associated with control against large deviation in a weakly perturbed system. An exponential risk-sensitive residence time criterion is introduced as a performance measure, and a related HJB equation is constructed. An averaging procedure is developed for deriving an approximate solution of the risk-sensitive control problem in the small noise hmit. It is shown that the averaged HJB equation is reduced to a first order PDE with the coefficients dependent on the noise intensity in the leading order term, though this intensity tends to zero in the original system. Near optimal control is constructed as a nonlinear time-independent feedback with parameters dependent on the noise intensity in the small noise limit. An example illustrates an application of this method to a system with resonance dynamics and with non-white noise perturbations.
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References
O. Alvarez and M. Bardi, “Viscosity solutions methods for singular perturbations in deterministic and stochastic control.” SI AM J. Control Optim., vol. 40, pp. 1159–1188, 2001.
A. Bensoussan, Perturbation Methods in Optimal Control, John Wiley, New York, 1988.
M. G. Crandall, H. Ishii, and P. L. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bill Amer Math. Soc, vol. 27, pp. 1–67, 1992.
P. Dupuis and H. J. Kushner, “Minimizing escape probabilities: A large deviation approach,” SIAMJ. Contr Optim., vol. 27, pp. 432–445, 1989.
P. Dupuis and W. M. McEneaney, “Risk-sensitive and robust escape criteria.” SIAM J. Control Optim., vol. 35, pp. 2021–2049, 1997.
W. H. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975.
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1992.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Veriag, New York, 1986.
A. S. Kovaleva, “Asymptotic solution of the optimal control problem for nonlinear oscillations in the neighborhood of a resonance,” J. Appl. Math. Mech., vol. 62, pp. 843–852, 1998.
A. S. Kovaleva, “Near-resonance motion in systems with random perturbations,” J. Appl. Math. Mech, vol. 62, pp. 43–49, 1998.
A. S. Kovaleva, Optimal Control of Mechanical Oscillations, Springer-Verlag, Berlin, 1999.
H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhausen Boston, 1990.
M. I. Freidhn and A. D. Wentzell, Random Perturbations of Dynamical Systems. Second Ed., Springer-Verlag. New York, 1998.
P. Whittle, Risk-Sensitive Optimal Control, J. Wiley, New York, 1990.
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Kovaleva, A. (2003). Control Against Large Deviation for Oscillatory Systems. In: Namachchivaya, N.S., Lin, Y.K. (eds) IUTAM Symposium on Nonlinear Stochastic Dynamics. Solid Mechanics and Its Applications, vol 110. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0179-3_21
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DOI: https://doi.org/10.1007/978-94-010-0179-3_21
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-3985-7
Online ISBN: 978-94-010-0179-3
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