Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains

We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives in , in , on , , in where is a smooth bounded domain, , the diffusion matrix has semisimple and positive eigenvalues , , is an open nonempty set, and is the characteristic function of . Specifically, we prove that under some conditions over the coefficients , the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all the system is approximately controllable on .