Desensitizing controls for semilinear wave equations

The desensitizing control problems for semilinear wave equations are studied. When the nonlinearity f satisfies lim|s|→∞f(s)|s|ln2|s|=lim|s|→∞f′(s)ln2|s|=0, we obtain the existence of a control which desensitizes the boundary observation. The proof is based on the combination of fixed-point arguments and explicit observability estimates for the linearized wave equation with a potential that depends on both x and t. The work solves the problem proposed by Tebou (2011) in the one-dimensional setting. For the interior observation, we assume that the nonlinearity is globally Lipschitz, under which the existence of ε-desensitizing controls is proved. Our proof is based on the fixed-point arguments and the unique continuation property.