Optimal model matching problem for stochastic signals with an unknown fast decreasing spectral density (Q656263)

The paper studies the optimal control with a linear-quadratic ergodic functional (average cost per unit of time) which penalizes deviations of the process \((z(t))\) from a benchmark (a stationary process from a certain class) and tracks the costs of applying controls. The state \((x(t), z(t))\) is described by a linear stochastic differential equation depending on two sources of noise (one of them drives the benchmark process as well). The contribution of the paper is the design of \(\varepsilon\)-optimal controls when the spectral density of the noise is not known -- only the upper bound is given and it decreases quickly at infinity. This paper borrows from previous results of the second authors: [Dokl. Akad. Nauk 338, 19--24 (1994; Zbl 0859.93040)] and [\textit{A. V. Proskurnikov} and \textit{V. A. Yakubovich}, Dokl. Math. 74, No. 1, 614--618 (2006); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 409, No. 4, 461--466 (2006; Zbl 1138.93053), Dokl. Akad. Nauk, Ross. Akad. Nauk 421, No. 3, 321--325 (2008); translation in Dokl. Math., Control Theory 78, No. 1, 631--635 (2008; Zbl 1164.93008)].