Null Controllability with Constraints on the State for the Semilinear Heat Equation

We consider a null controllability problem for the semilinear heat equation with finite number of constraints on the state. Interpreting each constraint by means of adjoint state notion, we transform the linearized problem into an equivalent linear problem of null controllability with constraint on the control. Using inequalities of observability adapted to the constraint, we solve the equivalent problem. Then, by a fixed-point method, we prove the main result.