Exact Internal Controllability for the Semilinear Heat Equation

Using the Browder]Minty surjective theorem from the theory of monotone operators, we consider the exact internal controllability for the semilinear heat equation. We show that the system is exactly controllable in L2 V. if the nonlinearities are globally Lipschitz continuous. Furthermore, we prove that the controls depend Lipschitz continuously on the terminal states, and discuss the behaviour of the controls as the nonlinear terms tend to zero in some sense. A variant of the Hilbert Uniqueness Method is presented to cope with the nonlinear nature of the problem.