Interior Controllability of a Reaction-Diffusion System with Cross-Diffusion Matrix

We prove the interior approximate controllability for the following reaction-diffusion system with cross-diffusion matrix in , in , , on , , , , where is a bounded domain in , , the diffusion matrix has semisimple and positive eigenvalues , is an arbitrary constant, is an open nonempty subset of , denotes the characteristic function of the set , and the distributed controls . Specifically, we prove the following statement: if (where is the first eigenvalue of ), then for all and all open nonempty subset of the system is approximately controllable on .