ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 49, 60, 93 Presman, Ernst L., CEMI, Moscow, Russia Optimality for the stochastic linear-quadratic regulator We consider a linear-quadratic regulator disturbed by a white noise with time dependent coefficients and the horizon T tending to infinity. It is well known (see [1]) that under natural conditions there exists the linear feedback control û which is optimal in deterministic case for T = ∞. Consider Δ_{T} (u) = J_{T} (û) − J_{T} (u), where J_{T} (u) is the random variable corresponding to the value of functional under admissible (adaptive) control u. Traditional result states that lim_{T →∞} E(Δ_{T} (u)/T) ≤ 0 for any admissible u. We show that lim sup_{T →∞} Δ_{T} (u)/ ln T < c_{1} a. s. for any admissible u, and construct u^{∗} such that lim sup_{T →∞} Δ_{T} (u^{∗})/ ln T > c_{2} > 0 a. s. Preliminary result was obtained in [2], u^{∗} was constructed in [3], general case is proved by T. Belkina, Yu. Kabanov, E. Presman (under preparation). References: [1] H. Kwakernaak, R. Sivan, Linear optimal control systems, Wiley-Interscience, 1972; [2] E. Presman, Theory of probability and its applications, 42 (1997), n.3; [3] S. Nagaev, E. Presman, Theory of probability and its applications, 43 (1998), n.2.