an:03946025
Zbl 0588.93070
Mandl, Petr
Asymptotic ordering of probability distributions for linear controlled systems with quadratic cost
EN
Stochastic differential systems, Proc. 3rd Bad Honnef Conf. 1985, Lect. Notes Control Inf. Sci. 78, 277-283 (1986).
1986
a
93E20 62E20 93C05 60H10 93C99
Autonomous linear controlled systems; white noise disturbances; quadratic cost; asymptotic distributions; stochastic ordering
[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.
Zbl 0579.00014