Null Controllability Of A Semilinear Parabolic Equation Governed By The Bilinear Control

This paper consists of two parts. In part 1, we are concerned with the bilinear controllability and the existence of the optimal control for a class of semilinear parabolic equations with gradient quadratic growth and Dirichlet boundary condition. In part 2, we discuss the bilinear controlla-bility of the steady-state solution and the existence of the optimal control for a class of semilinear parabolic equations with gradient quadratic growth and Neumann boundary condition.It is worth pointing out that there is no restriction on the growth of the nolinearity function with respect to the variable in the problem under consideration, which is a remarkable difference comparing with the situation that the control functions acts on the right-hand side of the equations.The technique used in this paper is the combinations of the Hopf-Cole transformations, the prior estimates for solutions of elliptic and parabolic equations, upper and lower solutions method, the strategy of stepwise control and the estimates for the cost of controllability.