Null controllability with constraints on the state for nonlinear heat equations

We consider the null controllability problem with a finite number of constraints on the state for a nonlinear heat equation involving gradient terms in a bounded domain of R-N. The control is distributed along a bounded subset of the domain and the nonlinearity is assumed to be of class C-1 and globally Lipschitz. Interpreting each constraint in terms of the notion of adjoint state, we transform the linearized problem into an equivalent null controllability problem with constraint on the control. Using a Carleman inequality adapted to the constraint, we prove first the null controllability of the linearized problem. Then, by a fixed-point method, we show that the same result holds when the nonlinearity is of class C-1 and globally Lipschitz.