The paper presents a method for computing the response of a 1D elastic continuum supported by a set of semi-active viscous dampers and induced by a load travelling over it. The magnitude of the moving force has been assumed to be constant by neglect of the inertia forces. Full analytical solution is based on the power series method and is given in an arbitrary time interval. The time-marching scheme allows us to continue a solution in successive layers with initial conditions taken from the end of previous stages. The semi-active open loop control strategy is proposed. Shapes of damping functions are defined as a form of piecewise constant function. The control strategy is suboptimal and it outperforms the passive case. Numerical results are presented for the cases of a string and a Bernoulli–Euler beam.