Controllability of Semilinear Systems of Parabolic Equations with Delay on the State

In this paper we prove the approximate controllability of the following semilinear system parabolic equations with delay on the state variable∂z ( t , x ) ∂t= D Δz + L z t + Bu ( t , x ) + f ( t , z ( t , x ) , u ( t , x ) ) ,t ∈0 , r,∂z ∂η= 0 ,t ∈0 , r,x ∈ ∂ Ω ,z ( 0 , x ) = φ 0 ( x ) ,x ∈ Ω ,z ( s , x ) = ψ 0 ( s , x ) ,s ∈− τ , 0,x ∈ Ω ,where Ω is a bounded domain inR N , D is a n × n non diagonal matrix whose eigenvalues are semi‐simple with non negative real part, the control u belongs toL 2 ( [ 0 , r ] ; U )( U = L 2 ( Ω , R m ) ) and B is a n × m matrix. Here τ ≥0 is the maximum delay, which is supposed to be finite. We assume that the operator L : L 2 ([− τ ,0]; Z )→ Z is linear and bounded with Z = L 2 ( Ω ; R n ) and the nonlinear function f :[0, r ] × I R n × I R m → I R n is smooth and bounded.