We study the local controllability of a mathematical model of an abstract object which "swims" in the two-dimensional ( 2D) nonstationary Stokes fluid. We assume that this object consists of finitely many subsequently connected small sets ("thick points"), each of which can act upon any of the adjacent sets in a rotation fashion with the purpose of generating its fish- or snake-like motion. We regard the magnitudes of the respective rotation forces, entering the system's equations as coefficients, as multiplicative ( or bilinear) controls. The structural integrity of the object is maintained by the elastic forces acting between the aforementioned adjacent sets according to Hooke's law. Models like this are of a interest in biology and engineering applications dealing with propulsion systems in fluids.