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Abstract.We prove the approximate controllability of the semilinear heat equation in R N , when the nonlinear term is globally Lipschitz and depends both on the state u and its spatial gradient u. The approximate controllability is viewed as the limit of a sequence of optimal control problems. In order to avoid the difficulties related to the lack of compactness of the Sobolev embeddings, we work with the similarity variables and use weighted Sobolev spaces.Mathematical subject classification: 93B05, 73C05, 35B37.