Exterior Controllability Properties for a Fractional Moore-Gibson-Thompson Equation

The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore-Gibson-Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. Assuming that b > 0 and alpha - tau c(2)/b, we show that if 0 < s < 1 and Omega subset of R-N (N >= 1) is a bounded domain with a Lipschitz continuous boundary partial derivative Omega, then there is no control function g such that the following system {tau u(ttt )+ alpha u(tt) + c(2)(-Delta)(s) u + b(-Delta)(s) u(t) = 0 in Omega x (0, T), u = g chi O in (R-N\Omega) x (0, T), u(., 0) = u(0), u(t) (., 0) = u(1), u(tt)(., 0) = u(2) in Omega, is exactly or null controllable in time T > 0. However, we prove that for 0 < s < 1, the system is approximately controllable for every g is an element of H-1 ((0, T); L-2(O)), where O subset of R-N\(Omega) over bar is an arbitrary non-empty open set.