2024/01/26 by Sergei Avdonin, Avdonin, Sergei, Julian Edward +3
Computer Science · Engineering · Mathematics · #93B05 #93C20 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2401.14987
openalex publication_date 2024/01/26 · openalex created_date 2024/01/30 · openalex updated_date 2026/07/28
Let Δ be the Dirichlet Laplacian on the interval (0,π). The null controllability properties of the equation utt+Δ2 u+ρ(Δ)αut=F(x,t) are studied. Let T>0, and assume initial conditions (u0,u1)∈ Dom(Δ)× L2(0,π). We first prove finite dimensional null control results: suppose F(x,t)=f1(t)h1(x)+f2(t)h2(x) with h1,h2 given functions. For α∈ [0,3/2), we prove that there exist h1,h2∈ L2(0,π) such that for any (u0,u1), there exist L2 null controls (f1,f2). For α< 1 and ρ<2, we prove null controllability with f2=0 and h1 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)=χΩ(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∈ L2(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.