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: ⎧⎪ ⎪⎪⎪⎪⎨⎪ ⎪⎪⎪⎪⎩ ∂z(t, x) ∂t = D z + Lzt + Bu(t, x), t ∈ (0, r], ∂z ∂η = 0, x ∈ ∂Ω, t ∈ (0, r], z(0, x) = φ0(x), x ∈ Ω, z(s, x) = φ(s, x), s ∈ [−τ , 0), x ∈ Ω, where Ω is a bounded domain in RN, D is an n × n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the control u ∈ L2([0, r]; U) = L2([0, r]; L2(Ω,Rm)) and B ∈ L(U, Z) with U = L2(Ω,Rm), Z = L2(Ω;Rn). The standard notation zt (x) defines a function from [−τ , 0] to Rn (with x fixed) by zt (x)(s) = z(t + s, x), −τ s 0. Here τ 0 is the maximum delay, which is supposed to be finite. We assume that the operator L : L2([−τ , 0]; Z) → Z is linear and bounded, and φ0 ∈ Z, φ ∈ L2([−τ , 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 semigroups 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