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Asymptotic ordering of probability distributions for linear controlled systems with quadratic cost. (English) Zbl 0588.93070
Stochastic differential systems, Proc. 3rd Bad Honnef Conf. 1985, Lect. Notes Control Inf. Sci. 78, 277-283 (1986).
[For the entire collection see Zbl 0579.00014.]
Autonomous linear controlled systems subject to white noise disturbances are considered. By $$C_ T$$ the associated quadratic cost up to time T is denoted. Under the optimal stationary control the average cost attains its minimal value $$\theta$$ as $$T\to \infty$$. Moreover, $$(C_ T-\theta T)/\sqrt{T}$$ has asymptotically the normal distribution N(0,$$\Delta)$$ where $$\Delta$$ is a variance parameter.
It is shown that under rather general conditions this is the best result achievable. Namely, N(0,$$\Delta)$$ is the lower bound for the asymptotic distributions of $$(C_ T-\theta T)/\sqrt{T}$$ in the sense of stochastic ordering. Conditions for the attainability of this lower bound and extensions to the nonautonomous case are presented.

##### MSC:
 93E20 Optimal stochastic control 62E20 Asymptotic distribution theory in statistics 93C05 Linear systems in control theory 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 93C99 Model systems in control theory