Approximate controllability of a system of parabolic equations with delay

In this paper we give necessary and sufficient conditions for the approximate controllability of the following system of parabolic equations with delay: [GRAPHICS] where Omega is bounded domain in R-N, D is an n x n nondiagonal matiex whose eigenvalues are semi-simple with nonnegative real part, the control u is an element of L-2 (Omega; R-m); U) = L-2([0, r]; L-2 (Omega, R-m)) and B is an element of L(U, Z) with U = L-2 (Omega, R-m), Z = L-2 (Omega; R-n). The standard notation zt (x) defines a function from [-tau, 0] to R-n (with x fixed) by z(t) (x) (s) = z(t + s, x), -tau <= s <= 0. Here tau >= 0 is the maximum delay, which is supposed to be finite. We assume that the operator L : L-2 ([-tau, 0]; Z) -> Z is linear and bounded, and phi(0) is an element of Z, phi is an element of L-2 ([-tau, 0]; Z). To this end: First, we reformulate this system into a standard first-order delay equation. Secondly, the semigroup associated with the first-order delay equation on an appropriate product space is expressed as a series of strongly continuous sernigroups and orthogonal projections related with the eigenvalues of the Laplacian operator (A = 2); this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations. Finally, we use the well-known result on the rank condition for the approximate controllability of delay system to derive our main result. (c) 2008 Elsevier Inc. All rights reserved.