Controllability of positive solutions of nonlinear fractional dynamic systems on time scales: An integral boundary value approach

This paper investigates the controllability of positive solutions for a fractional dynamic system endowed with integral boundary conditions on an arbitrary time scale. By combining the method of upper and lower solutions with Schauder’s fixed-point theorem, we derive a set of sufficient conditions that guarantee positivity of solutions to the proposed problem. Central to this investigation is the construction of an appropriate Green’s function, which serves as the principal analytical tool throughout the study of the fractional dynamic system. The theoretical findings are subsequently supported by illustrative examples in different arbitrary time scales, accompanied by MATLAB simulations that depict the conditions governing the positivity of the solution and elucidate the influence of the control function on its trajectories.