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In this paper, we consider the cost of null controllability for a large classof linear equations of parabolic or dispersive type in one space dimension insmall time. By extending the work of Tenenbaum and Tucsnak in "New blow-uprates for fast controls of Schr\"odinger and heat equations`", we are able togive precise upper bounds on the time-dependance of the cost of fast controlswhen the time of control T tends to 0. We also give a lower bound of the costof fast controls for the same class of equations, which proves the optimalityof the power of T involved in the cost of the control. These general resultsare then applied to treat notably the case of linear KdV equations andfractional heat or Schr\"odinger equations.

On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension