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The objective of this work is to study the approximate controllability with constraints on the control of the heat equation. To demonstrate the approximate controllability, it suffices to use Hahn Banach theorem and notice that the orthogonal of the set of reachable states at the moment t=0 is reduced to zero. Finally we adapted the method of duality (primal-dual) of Fenchel-Rockafellar to characterize the optimal control

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