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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 beused. The distribution of $Y$ depends on some unknown parameter $\theta$, andwe consider the problem of testing multiple hypotheses $H_1: \theta=\theta_1$,$H_2: \theta=\theta_2, ...$, $H_k: \theta=\theta_k$ allowing the data to becontrolled by $X$, in the following sequential context.  The experiment starts with assigning a value $X_1$ to the control variableand observing $Y_1$ as a response. After some analysis, another value $X_2$ forthe control variable is chosen, and $Y_2$ as a response is observed, etc. It issupposed that the experiment eventually stops, and at that moment a finaldecision in favor of one of the hypotheses $H_1,...$, $H_k$ is to be taken. Inthis article, our aim is to characterize the structure of optimal sequentialtesting procedures based on data obtained from an experiment of this type inthe case when the observations $Y_1, Y_2,..., Y_n$ are independent, givencontrols $X_1,X_2,..., X_n$, $n=1,2,...$.

Sequential multiple hypothesis testing in presence of control variables