<p>We deal with non-homogeneous unsteady Stokes equations in low regularity spaces under the action of a finite number of particles located inside the flow domain. The main targets of this task are the solvability in a very weak sense and the identification of point-source locations through an optimization process of a well-defined performance functional. The lack of regularity induced by the source term complicates the derivation of optimality conditions and numerical algorithms. Simulation results based on efficient algorithms are provided to illustrate the effectiveness of the established theory. This work unifies mathematical rigor, computational efficiency, and real-world applicability. It provides both a theoretical benchmark and a practical toolkit for inverse flow control.</p>