Existence and Finite Approximate Controllability of Nonlinear Systems Involving Two Fractional Derivatives

ABSTRACT In real life, there are many processes which involve several memory channels such as biological tissues, viscoelastic media, neuron dynamics, and so forth. Systems having exactly one derivative term are not capable to handle such issues. However, these problems can be modeled by systems involving several derivative terms. This paper investigates the issue of existence and finite‐approximate controllability properties for semi‐linear systems with two fractional derivatives in the Riemann‐Liouville sense without assuming the Lipschitz continuity of the nonlinear operator. First, we derive the existence result by using Schaefer's fixed‐point theorem, compactness of the fractional resolvent, and properties of the inner product. For this, we verify the compactness of the solution map by using the Ascoli‐ArzelÃČÆŠ theorem. Then, with the help of the obtained existence result, we show that the nonlinear system is finite‐approximately controllable if the associated linear system is approximately controllable. The fractional resolvent as well as Riemann‐Liouville derivatives have a singularity at , and due to this fact, we consider the Banach space instead of the usual function space . At the end, an illustrative example is presented for the obtained results.