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 θ, and we consider the problem of testing multiple hypotheses H1:θ=θ1, H2:θ=θ2,…, Hk:θ=θk allowing the data to be controlled by X, in the following sequential context. The experiment starts with assigning a value X1 to the control variable and observing Y1 as a response. After some analysis, another value X2 for the control variable is chosen, and Y2 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 H1,…, Hk 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 Y1,Y2,…,Yn are independent, given controls X1,X2,…,Xn, n=1,2,….