Null-controllability for the beam equation with structural damping. Part 1. Distributed control
arxiv(2024)
Abstract
Let Δ be the Dirichlet Laplacian on the interval (0,π). The null
controllability properties of the equation
u_tt+Δ^2 u+ρ
(Δ)^α u_t=F(x,t)
are studied. Let T>0, and assume initial
conditions (u^0,u^1)∈ Dom(Δ)× L^2(0,π). We first prove finite
dimensional null control results: suppose F(x,t)=f^1(t)h^1(x)+f^2(t)h^2(x)
with h^1,h^2 given functions. For α∈ [0,3/2), we prove that there
exist h^1,h^2∈ L^2(0,π) such that for any (u^0,u^1), there exist L^2
null controls (f^1,f^2). For α< 1 and ρ <2, we prove null
controllability with f^2=0 and h^1 belonging to a large class of functions.
For α∈ [3/2,2), we prove spectral and null controllability both
generally fail, but two dimensional weak controllability holds. Our second set
of results pertains to F(x,t)=χ_Ω(x)f(x,t), with Ω any open
subset of (0,π). For any α∈ [0,3/2), we prove there exists a null
control f∈ L^2(Ω×(0,T)) To prove our main results, we use the
Fourier method to rewrite the control problems as moment problems. These are
then solved by constructing biorthogonal sets to the associated exponential
families. These constructions seem to be non-standard and may be of independent
interest.
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