is it possible to use an experimental and a control group in a mixed method design?
Right away: it is very hard to give a substantial answer without knowing more about your concrete research interest.
The whole idea about mixed methods is that you reflect about the objectives of your research, translate complex research into components and think about the most coherent and productive strategy to answer these component questions. Therefore a first very general answer is: yes, of course, depending on your research question, experiments can be part of a mixed methods design.
Classical ("orthodox") experimental research also knows a form of mixed-methods-design: do exploratory research (qual) first, develop your idea and then implement your experiment (quan). In recent debates, the sequence is often turned around: quantitative research comes first, then we use qualitative research do understand what is going on.
So the first question to answer is: mixed methods what for? Then go into the details. These details include: if you have control and treatment group, you usually don't want participants to know what group they are in, and often you do not want them to know too much about your research interest. In this case, it could be detrimental to do qualitative phases of research first. But that's just one example of issues you'd have to consider.
I agree with Kenneth Horvath that we need to know more your overall goals for using mixed methods. In particular, you did not mention a qualitative component for project, so the key question is what purpose that would serve.
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