Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games

In this paper 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.

[1]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[2]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[3]  Frank L. Lewis,et al.  Optimal Control , 1986 .

[4]  B. Anderson,et al.  A game theoretic approach to H ∞ control for time-varying systems , 1992 .

[5]  M. R. James,et al.  Asymptotic Series and Exit Time Probabilities , 1992 .

[6]  P. Whittle Risk-sensitive linear/quadratic/gaussian control , 1981, Advances in Applied Probability.

[7]  G. Barles,et al.  Discontinuous solutions of deterministic optimal stopping time problems , 1987 .

[8]  A. Schaft On a state space approach to nonlinear H ∞ control , 1991 .

[9]  Hitoshi Ishii,et al.  A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities , 1985 .

[10]  W. Fleming,et al.  Deterministic and Stochastic Optimal Control , 1975 .

[11]  W. Fleming Stochastic Control for Small Noise Intensities , 1971 .

[12]  W. Fleming,et al.  Asymptotic expansions for Markov processes with lévy generators , 1989 .

[13]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[14]  K. Glover,et al.  State-space formulae for all stabilizing controllers that satisfy and H ∞ norm bound and relations to risk sensitivity , 1988 .

[15]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[16]  J. Helton,et al.  H" CONTROL FOR NONLINEAR PLANTS: CONNECTIONS WITH DIFFERENTIAL GAMES , 1989 .

[17]  P. Souganidis,et al.  Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations. , 1983 .

[18]  B. Heimann,et al.  Fleming, W. H./Rishel, R. W., Deterministic and Stochastic Optimal Control. New York‐Heidelberg‐Berlin. Springer‐Verlag. 1975. XIII, 222 S, DM 60,60 , 1979 .

[19]  Rhodes,et al.  Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games , 1973 .

[20]  S. Sheu Stochastic Control and Exit Probabilities of Jump Processes , 1985 .

[21]  P. Whittle Risk-Sensitive Optimal Control , 1990 .

[22]  P. Whittle A risk-sensitive maximum principle , 1990 .