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This paper discusses the controllability problem of biosystems based on the piecewise-affine system model representation. First, we consider what kind of controllability problems are useful for analyzing and controlling biosystems, and then the controllable set problem, that is, a problem of finding a state set in which each state can be driven to a given target state, is formulated. It is shown that this problem will be very useful for many kinds of problems on control of biosystems such as the input allocation problem and the stabilization problem. Next, based on our previous probabilistic controllability analysis technique for hybrid dynamical systems, a more sophisticated method for solving the above complex problem in an approximated and suitable way for biosystem analysis is proposed. Finally, the proposed framework is applied to the quorum sensing system of the pathogen Pseudomonas aeruginosa for explaining how it is formulated and what solutions are obtained.