Global exact controllability on H1Γ0 (Ω) x L2 (Ω) of semilinear wave equations with Neumann L2(0, T; L2 (Γ))-boundary control

We provide four proofs that the semilinear wave equation with globally Lipschitz nonlinear term is exactly controllable on the finite energy space H-T0(1)(Omega) x L-2(Omega) within the class of L-2[0, T; L-2 (T-1) ]-boundary controls exercised in the Neumann boundary conditions, whenever the corresponding linear equation satisfies the same property; and on the same time interval. One proof is based on the global inversion approach applied to the original, controlled problem, which was proposed by Lasiecka and Triggiani [29]. In contrast, the other three proofs are based, on a uniform continuous observability inequality of the dual uncontrolled problem. Precise links are exhibited and exploited among these four approaches. All proofs are essentially operator-theoretic with a partial differential equation (PDE) interpretation. A common thread of the four proofs is an analysis of suitable families of collectively compact operators, which then admit uniform inversions in the style of Reference 29. This way, compactness and uniqueness arguments are entirely dispensed with.