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 u(t) = a(1)Delta u + a(2)Delta v - c(1)(-Delta)(alpha 1)u - c(2)(-Delta)(alpha 2)v + 1(omega)f(1)(x, t) in Omega x ] 0, t* [, v(t) = b(1)Delta u + b(2)Delta v - d(1)(-Delta)(beta 1)u - d(2)(-Delta)(beta 2)v + 1(omega)f(2)(x, t) in Omega x ]0, t* [, u = v = 0 on partial derivative Omega x ]0, t*[, u (x, 0) = u(0) (x), v (x, 0) = v(0) (x) in x is an element of Omega, where Omega subset of R-N (N >= 1) is a smooth bounded domain, u(0), v(0) is an element of L-2 (Omega), the diffusion matrix M = [GRAPHICS] has semisimple and positive eigenvalues 0 < rho(1) <= rho(2), 0 < alpha(1), alpha(2), beta(1), beta(2) < 1, omega subset of Omega is an open nonempty set, and 1(omega) is the characteristic function of omega. Specifically, we prove that under some conditions over the coefficients a(i), b(i), c(i), d(i) (i = 1, 2), the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all t* > 0 the system is approximately controllable on [0, t*].