Let us consider the system
where is a bounded domain of with -boundary , is a (probably small) non-empty open subset of , denotes the characteristic function of , is fixed and is a function in . Hence, the control is acting only on . It is well known that this system is approximately controllable at any fixed time by taking controls in the sense that, for each and , there exists such that the corresponding solution of the system satisfies . Of course, there exist infinite controls satisfying this property. The objective of the paper is to obtain explicit bounds on the cost of approximate controllability for the system, i.e. the infimum of over all satisfying the previous property. It is proved that this cost is of order . The simultaneous finite-approximate controllability is also investigated. The proofs combine global Carleman estimates, energy estimates for parabolic equations and the variational approach to approximate controllability. When the coefficient is constant, a different approach is used to show that the cost is of order and that this estimate is sharp.