Trajectory and null controllability of a control system governed by fractional-order composite relaxation differential equations

Abstract

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 (a, k)-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. © Indian Academy of Sciences 2025.