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,y<inf>t</inf>,u), in (0,τ) × Ω, y(t,x) = Δ y(t,x)= 0, on(0,τ) × ∂Ω, y(0,x) = y<inf>0</inf>(x), y<inf>t</inf>}(x)=v<inf>0</inf>(x), x ∈ Ω, in the states space Z<inf>1</inf> = D(-Δ) × L²(Ω) endowed with the graph norm, where β >1, Ω is a sufficiently regular bounded domain in ℝ^N, 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].

Año de publicación:

2016

Keywords:

• strongly continuous semigroups
• semilinear beam equation
• Approximate controllability
• Rothe's fixed point theorem

Fuente:

scopus

google