Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games. (English) Zbl 0761.93076
Summary: We consider a finite horizon, nonlinear, stochastic, risk-sensitive optimal control problem with complete state information, and show that it is equivalent to a stochastic differential game. Risk-sensitivity and small noise parameters are introduced, and the limits are analyzed as these parameters tend to zero. First-order expansions are obtained which show that the risk-sensitive controller consists of a standard deterministic controller, plus terms due to stochastic and game-theoretic methods of controller design. The results of this paper relate to the design of robust controllers for nonlinear systems.
MSC:
| 93E20 | Optimal stochastic control |
| 93C40 | Adaptive control/observation systems |
| 91A23 | Differential games (aspects of game theory) |
| 49N70 | Differential games and control |
| 49N75 | Pursuit and evasion games |
References:
| [1] | J. A. Ball and J. W. Helton, pp416-01 control for nonlinear plants: Connections with differential games,Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, FL, 1989, pp. 956–962. |
| [2] | G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems,Math. Model. Numer. Anal.,21 (1987), 557–579. · Zbl 0629.49017 |
| [3] | M. G. Crandall, L. C. Evans, and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations,Trans. Amer. Math. Soc.,282 (1984), 487–502. · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X |
| [4] | M. G. Crandall, H. Ishii, and P. L. Lions,User’s Guide to Viscosity Solutions of Second-Order Partial Differential Equations, CEREMADE Report No. 9039, 1990. |
| [5] | J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State-space solutions to standardH 2 andH control problems,IEEE Trans. Automat. Control,34 (1989), 831–847. · Zbl 0698.93031 · doi:10.1109/9.29425 |
| [6] | L. C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities,Ann. Inst. H. Poincaré,2 (1985), 1–20. · Zbl 0601.60076 |
| [7] | L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations,Indiana Univ. Math. J.,33 (1984), 773–797. · Zbl 1169.91317 · doi:10.1512/iumj.1984.33.33040 |
| [8] | W. H. Fleming, Stochastic control for small noise intensities,SIAM J. Control,9 (1971), 473–517. · Zbl 0218.93024 · doi:10.1137/0309035 |
| [9] | W. H. Fleming and M. R. James, Asymptotic series and exit time probabilities,Ann. Probab. (to appear). · Zbl 0771.60055 |
| [10] | W. H. Fleming and W. M. McEneaney, Risk sensitive optimal control and differential games,Proceedings of the Conference on Adaptive and Stochastic Control, University of Kansas, Sept. 1991, Lecture Notes on Control and Information Science, Springer-Verlag (to appear). · Zbl 0788.90097 |
| [11] | W. H. Fleming and R. W. Rishel,Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975. · Zbl 0323.49001 |
| [12] | W. H. Fleming and H. M. Soner, Asymptotic expansions for Markov processes with Levy generators,Appl. Math. Optim.,19 (1989), 203–223. · Zbl 0713.60085 · doi:10.1007/BF01448199 |
| [13] | W. H. Fleming and P. E. Souganidis, Asymptotic series and the method of vanishing viscosity,Indiana Univ. Math. J.,35 (1986), 425–447. · Zbl 0573.35034 · doi:10.1512/iumj.1986.35.35026 |
| [14] | W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games,Indiana Univ. Math. J.,38 (1989), 293–314. · Zbl 0686.90049 · doi:10.1512/iumj.1989.38.38015 |
| [15] | M. I. Friedlin and A. D. Wentzell,Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984. |
| [16] | K. Glover and J. C. Doyle, State space formulae for all stabilizing controllers that satisfy anH norm bound and relations to risk sensitivity,Systems Control Lett.,11 (1988), 167–172. · Zbl 0671.93029 · doi:10.1016/0167-6911(88)90055-2 |
| [17] | A. Isidori, Robust regulation of nonlinear systems,MTNS Abstracts, Kobe, Japan (1991) 216–219. |
| [18] | D. H. Jacobson, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games,IEEE Trans. Automat. Control,18 (1973), 124–131. · Zbl 0274.93067 · doi:10.1109/TAC.1973.1100265 |
| [19] | D. J. N. Limebeer, B. D. O. Anderson, P. P. Khargonekar, and M. Green, A game-theoretic approach toH control for time-varying systems,SIAMJ. Control Optim.,30 (1992), 262–283. · Zbl 0791.93026 · doi:10.1137/0330017 |
| [20] | A. J. van der Schaft, On a state space approach to nonlinearH control,Systems Control Lett.,16 (1991), 1–8. · Zbl 0737.93018 · doi:10.1016/0167-6911(91)90022-7 |
| [21] | S-J. Sheu, Stochastic control and exit probabilities of jump processes,SIAM J. Control Optim.,23 (1985), 306–328. · Zbl 0567.60032 · doi:10.1137/0323022 |
| [22] | P. Whittle, Risk-sensitive linear/quadratic/Gaussian control,Adv. in Appl. Probab.,13 (1981), 764–777. · Zbl 0489.93067 · doi:10.2307/1426972 |
| [23] | P. Whittle, A risk-sensitive maximum principle,Systems Control Lett.,15 (1990), 183–192. · Zbl 0724.93084 · doi:10.1016/0167-6911(90)90110-G |
| [24] | P. Whittle,Risk-Sensitive Optimal Control, Wiley, New York, 1990. · Zbl 0718.93068 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
