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Controllability of the perturbed Beam equation

Abstract:

In this paper, applying the Rothe's Fixed Point Theorem, we prove the approximate controllability of the following semilinear Beam equation: ∂2y(t,x)/∂t2 = 2β Δ ∂ y(t,x)/∂t - Δ2y(t,x)+u(t,x) +f(t,y,yt,u), in (0,τ) × Ω, y(t,x) = Δ y(t,x)= 0, on(0,τ) × ∂Ω, y(0,x) = y0(x), yt}(x)=v0(x), x ∈ Ω, in the states space Z1 = D(-Δ) × L2(Ω) endowed with the graph norm, where β >1, Ω is a sufficiently regular bounded domain in ℝN, the distributed control u belongs to L2([0,τ]; U)(U=L2(Ω)) and the non-linear function f:[0,τ] × ℝ× ℝ× ℝ→ℝ is smooth enough and there are η, b, c ∈ ℝ with 1/2 ≤ η ≤ 1 such that |f(t,y,v,u)|<b|u|η +c ∀ y,v,u∈ ℝ, We prove that for all τ > 0 this system is approximately controllable on [0,τ].