Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ at this stage can be used. The distribution of $Y$ depends on some unknown parameter $\theta$, and we consider the problem of testing multiple hypotheses $H_1: \theta=\theta_1$, $H_2: \theta=\theta_2, \ldots $, $H_k: \theta=\theta_k$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, another value $X_2$ for the control variable is chosen, and $Y_2$ as a response is observed, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favor of one of the hypotheses $H_1,\ldots $, $H_k$ is to be taken. In this article, our aim is to characterize the structure of optimal sequential testing procedures based on data obtained from an experiment of this type in the case when the observations $Y_1, Y_2,\ldots , Y_n$ are independent, given controls $X_1,X_2,\ldots , X_n$, $n=1,2,\ldots $.

sequential analysis, multiple hypotheses, optimal stopping, sequential probability ratio test, sequential hypothesis testing, control variable, independent observations, optimal control, optimal decision, optimal sequential testing procedure, Bayes

62L10, 62L15, 60G40, 62C99, 93E20