On application of Rothe’s fixed point theorem to study the controllability of fractional semilinear systems with delays

The paper presents finite-dimensional dynamical control systems described by semilinear fractional-order state equations with multiple delays in the control and nonlinear function f. The relative controllability of the presented semilinear system is discussed. Rothe’s fixed point theorem is applied to study the controllability of the fractional-order semilinear system. A control that steers the semilinear system from an initial complete state to a final state at time t > 0 is presented. A numerical example is provided to illustrate the obtained theoretical results and a practical example is given to show a possible application of the study.