Abstract
Risk-sensitive stochastic control problems for nonlinear systems described by stochastic differential equations are considered. A logarithmic transformation is applied to the optimal cost function. The value function for a zero-sum, two-controller differential game is obtained in the limit, as a small parameter which represents noise intensity tends to zero. Convergence to the value function is proved by viscosity solution methods for nonlinear partial differential equations.
Partially supported by NSF under grant DMS-900038, by ARO under grant DAAL03-86-K-0171 and by AFOSR under grant AFOSR-89-0015
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References
J. S. Baras, A. Bensoussan and M. R. James, “Dynamic observers as asymptotic limits of recursive filters: special cases”, SIAM J. Appl. Math 48 (1988) 1147–1158.
E. N. Barron and R. Jensen, “Total risk aversion, stochastic optimal control and differential games”, Appl. Math. and Optimiz 19 (1989) 313–327.
T. Basar and P. Bernhard, “H ∞ — optimal Control and Related Minimax Design Problems” Birkhauser, Boston 1991.
M. D. Donsker and S. R. S. Varadhan, “Asymptotic evaluation of certain Markov process expectations for large time”, I, II, III, Comm. Pure Appl. Math. 28 (1975) 1–45, 279–301; 29 (1976) 389–461.
L. C. Evans and P. E. Souganidis, “Differential games and representation formulas for solutions of Hamilton-Jacobi equations”, Indiana Univ. Math. J. 33 (1984) 773–797.
W. H. Fleming and R. W. Rishel, “Deterministic and Stochastic Optimal Control”, Springer Verlag, 1975.
W. H. Fleming and H. M. Soner, “Controlled Markov Processes and Viscosity Solutions”, Springer Verlag, 1992.
W. H. Fleming and P. E. Souganidis, “PDE-viscosity solution approach to some problems of large deviations”, Annali Scuola Normale Sup. Pisa, Ser. IV 23 (1986) 171–192.
W. H. Fleming and C-P Tsai, “Optimal exit probabilities and differential games”, Applied Math. Optimiz. 7 (1981) 253–282.
M. I. Freidlin and A. D. Wentzell, “Random Perturbations of Dynamical Systems”, Springer Verlag, 1984.
K. Glover, “Minimum entropy and risk-sensitive control: the continuous time case”, Proc. 28th IEEE Conf. on Decision and Control, Dec. 1989, 388–391.
K. Glover and J. C. Doyle, “State-space formulae for all stabilizing controllers that satisfy an H ∞-norm bound and relations to risk sensitivity”, Systems and Control Letters 11, (1988) 167–172.
O. Hijab, “Minimum energy estimation,” Ph.D. dissertation, Univ of Calif. Berkeley, 1980.
D. H. Jacobson, “Optimal stochastic linear systems with exponential criteria and their relation to deterministic differential games”, IEEE Trans. Automat. Control AC — 18 (1973) 124–131.
P. Whittle, “Risk-sensitive Optimal Control”, Wiley, 1990.
P. Whittle, “A risk-sensitive maximum principle”, Systems and Control Lett. 15 (1990) 183–192.
M. R. James, “Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games”, Preprint.
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© 1992 Springer-Verlag
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Fleming, W.H., McEneaney, W.M. (1992). Risk sensitive optimal control and differential games. In: Duncan, T.E., Pasik-Duncan, B. (eds) Stochastic Theory and Adaptive Control. Lecture Notes in Control and Information Sciences, vol 184. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0113240
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DOI: https://doi.org/10.1007/BFb0113240
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