Fractional controllability of semilinear hyperbolic systems

This research considered the fractional controllability problem of internally controlled semilinear hyperbolic systems. In the first method, we show the controllability of the linear system using Hilbert uniqueness method HUM, and the nonlinear fractional controllability problem is solved applying Schauder’s fixed point theorem. Secondly, the analytics study is then attempted by employing generalized inverse methods and changed also into a fixed point problem. Hence, we give an approximate approach to find a control that brings the Riemann-Liouville fractional attained position \(^{RL}\mathcal {D}_x^{\beta }z(x, T)\) ${}^{\mathit{RL}}{\mathcal{D}}_x^{\beta }z(x,T)$ (resp. Caputo speed \(^C\mathcal {D}_t^{\alpha }z(x, T)\) ${}^C{\mathcal{D}}_t^{\alpha }z(x,T)$ ) to the required position \(y_1^{d}\) $y_1^d$ (resp. speed \(y_2^{d}\) $y_2^d$ ). Finally, computing simulations are used to confirm the obtained results.