Asymptotic distribution of some quadratic functionals of linear stochastic evolution systems

A function space asymptotic distribution of quadratic functionals induced from an unknown system is obtained in terms of a multi-dimensional Wiener process where the control is a linear transformation of the state that depends smoothly on the unknown parameters. The result is easily specialized to the asymptotic distribution of the family of random variables formed as the upper limit of the integrals of the quadratic terms is varied.The result provides a measure of the dependence of such a quadratic functional on a family of strongly consistent estimates of the unknown parameters, and in some cases it provides an interesting contrast with the case of all known parameters. In this paper, it is shown that, for some linear stochastic evolution systems, there are special feedback control laws where the variance of the asymptotic normal distribution of the average costs is less for the control law based on the estimates of the parameters than for the control law based on the true parameter values. This phenomenon does not occur if the feedback control laws are optimal stationary controls.This research was supported by NSF Grants Nos. ECS-87-18026 and ECS-9113029.The author thanks Professor Alain Benssousan for his great hospitality in INRIA, where this paper was written, and Professors Tyrone Duncan, Pravin Varaiya, and the anonymous reviewer for their very useful comments.