The forced linear and nonlinear systems considered are assumed to be asymptotically stable at large when the input is absent. Given prescribed bounds on the state variables and the output, the dynamic range of the system is defined as the maximum allowable range of the magnitude (norm) of the input that will keep the system variables within those prescribed bounds. In this work, the Liapunov functions for the unforced system are used to obtain an estimate of its dynamic range. The method developed is based on a theorem of Malkin. An estimate of the dynamic range of linear time invariant lumped-parameter systems is obtained by means of a quadratic form Liapunov function. It is shown that the best estimate results from a quadratic form generated by a diagonal matrix having all its elements equal (the common value of those elements being an arbitrary positive number). For the class of nonlinear control systems, using a Lurie-Postnikov Liapunov function, the best estimate is obtained by choosing the same Liapunov function as the one which determines the weakest requirements for absolute stability of the unforced system. A computer algorithm based on the above method is described and used in the solution of an example.