On error of estimation and minimum of cost for wide band noise driven systems

The suggested open problem concerns the error of estimation and the minimum of the cost in the filtering and optimal control problems for a partially observable linear system corrupted by wide band noise processes. Recent results allow to construct a wide band noise process in a certain integral form on the basis of its autocovariance function and design the optimal filter and the optimal control for a partially observable linear system corrupted by such wide band noise processes. Moreover, explicit formulae for the error of estimation and for the minimum of the cost have been obtained. But, the information about wide band noise contained in its autocovariance function is incomplete. Hence, every autocovariance function generates infinitely many wide band noise processes represented in the integral form. Consequently, the error of estimation and the minimum of the cost mentioned above are for a sample wide band noise process corresponding to the given autocovariance function. The following problem arises: given an autocovariance function, what are the least upper and greatest lower bounds of the respective error of estimation and the respective minimum of the cost? What are the distributions of the error of estimation and the minimum of the cost? What are the parameters of the wide band noise process producing the average error and the average minimum of the cost? Modern stochastic optimal control and filtering theories use white noise driven systems. Results such as the separation principle and the Kalman-Bucy filtering are based on the white noise model. In fact, white noise, being a mathematical idealization, gives only an approximate description of real noise. In some fields the parameters of real noise are near to the parameters of white noise and, so, the mathematical methods of control and filtering for white noise driven systems can be satisfactorily applied to them. But in many fields white noise is a crude approximation to real noise. Consequently, the theoretical optimal controls and the theoretical optimal filters for white noise driven systems become not optimal and, indeed, might be quite far from being optimal. It becomes important to develop the control and estimation theories for the systems driven by noise models that describe real noise more adequately. Such a noise model is the wide band noise model. The importance of wide band noise processes was mentioned by Fleming and Rishel [1]. An approach to wide band noise based on approximations by white noise was …