Approximate Controllability of a Semilinear Heat Equation

Abstract

We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: zt(t,x)=Δz(t,x)+1ωu(t,x)+f(t,z(t,x),u(t,x)) in (0,τ]×Ω,z=0, on (0,τ)×∂Ω,z(0,x)=z0(x), x∈Ω, where Ω is a bounded domain in ℝN  (N≥1), z0∈L2(Ω), ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0,τ;L2(Ω)), and the nonlinear function f:[0,τ]×ℝ×ℝ→ℝ is smooth enough, and there are a,b,c∈ℝ, R>0 and 1/2≤β<1 such that |f(t,z,u)-az|≤c|u|β+b, for all u,z∈ ℝ,|u|,|z|≥R. Under this condition, we prove the following statement: for all open nonempty subset ω of Ω, the system is approximately controllable on [0,τ]. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state z0 to an ϵ neighborhood of the final state z1 at time τ>0.