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Controllability of semilinear systems of parabolic equations with delay on the state

Abstract:

In this paper we prove the approximate controllability of the following semilinear system parabolic equations with delay on the state variable, where Ω is a bounded domain in RN,D is a n × n non diagonal matrix whose eigenvalues are semi-simple with non negative real part, the control u belongs to L2([0,r];U)(U=L2(Ω,Rm)) 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:L2([-τ,0];Z)→Z is linear and bounded with Z=L2(Ω;Rn) and the nonlinear function f:[0,r] × IRn×IRm→IRn is smooth and bounded.