Optimal Sequential Multiple Hypothesis Testing in Presence of Control Variables 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 ₁: θ=\thet₁, ₂: θ=\thet₂, łdots , _k: θ=\thet_k allowing the data to be controlled by X, in the following sequential context. The experiment starts with assigning a value ₁ to the control variable and observing ₁ as a response. After some analysis, another value ₂ for the control variable is chosen, and 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 ₁,łdots , _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 ₁, ₂,łdots , _n are independent, given controls ₁,₂,łdots , _n, n=1,2,łdots .