In this paper, we consider the cost of fast controls for a large class of linear equations of parabolic or dispersive type in one space dimension in small time. By extending the work of Tenenbaum and Tucsnak [J. Differential Equations, 243 (2007), pp. 70-100], we are able to give precise upper bounds on the time-dependance of the cost of fast controls when the time of control T tends to 0. We also give a lower bound of the cost of fast controls for the same class of equations, which proves the optimality of the power of T involved in the cost of the control. These general results are then applied to treat notably the case of linear KdV equations and fractional heat or Schrodinger equations.