<p>The purpose of this paper is to study the trajectory controllability (which is said to be the strongest notion rather than other controllabilities) and null controllability of a control system commanded by fractional-order composite relaxation semi-linear differential equation. The linear control system under consideration addresses a classical problem in fluid dynamics, specifically focussing on the dynamics of a particle experiencing unsteady movement within a viscous fluid while subjected to gravitational forces. Initially, we derive the mild solution of the considered control system employing the results on (<i>a</i>,&#xa0;<i>k</i>)-regularised families of bounded linear operator (matrices). Under some appropriate assumptions regarding non-linear terms and by utilising the Gronwall’s inequality, we establish the trajectory controllability and null controllability with constrained controls via fixed point technique. Finally, a few examples are shown to illustrate and explore the applications of the obtained findings.</p>