Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient

In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of R N . We first consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F( y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L 2-norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.