Approximate controllability for the semilinear heat equation involving gradient terms

This paper deals with the approximate controllability of the semilinear hea t equation, when the nonlinear term depends on both the state y and its spa tial gradient del y and the control acts on any nonempty open subset of the domain. Our proof relies on the fact that the nonlinearity is globally Lip schitz with respect to (y, del y). The approximate controllability is viewe d as the limit of a sequence of optimal control problems. Another key ingre dient is a unique continuation property proved by Fabre (Ref. 1) in the con text of linear heat equations. Finally, we prove that approximate controlla bility can be obtained simultaneously with exact controllability over finit e-dimensional subspaces.