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=DΔz+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 RNRN, D   is an n×nn×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))u∈L2([0,r];U)=L2([0,r];L2(Ω,Rm)) and B∈L(U,Z)B∈L(U,Z) with U=L2(Ω,Rm)U=L2(Ω,Rm), Z=L2(Ω;Rn)Z=L2(Ω;Rn). The standard notation zt(x)zt(x) defines a function from [−τ,0][−τ,0] to RnRn (with x   fixed) by zt(x)(s)=z(t+s,x)zt(x)(s)=z(t+s,x), −τ⩽s⩽0−τ⩽s⩽0. Here τ⩾0τ⩾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∈Zϕ0∈Z, ϕ∈L2([−τ,0];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.