Controllability of the semilinear beam equation

Abstract:

In this paper, we prove the approximate controllability of the following semilinear beam equation: {δ²y(t,x)/δt² = 2βΔ δy(t,x)\δt - 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) × δΩ, y(0,x) = y<inf>0</inf>(x),y <inf>t</inf>(x)=v<inf>0</inf>(x), x ε Ω, in the states space Z <inf>1</inf>=D(Δ)×L<inf>2</inf>(Ω) with the graph norm, where β > 1, Ω is a sufficiently regular bounded domain in IR ^N, the distributed control u belongs to L ²([0,τ];U) (U = L ²(Ω)), and the nonlinear function f:[0,τ]× × × → is smooth enough and there are a,c ε ℝ such that a< λ<inf>1</inf>² and sup(t,y,v,u)εQ <inf>τ</inf> f(t,y,v,u) - ay -cu\ <∞, where Q <inf>τ</inf> = [0,τ]×IR×IR×IR. We prove that for all τ > 0, this system is approximately controllable on [0,τ]. © 2013 Springer Science+Business Media New York.

Año de publicación:

2013

Keywords:

• Approximate controllability
• Beam equations
• strongly continuous semigroups

Fuente:

scopus