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Input shaping is an established technique to generate prefilters that move flexible mechanical systems with little or no residual vibration. While traditional input shaping design strategies are often analytical, the present paper introduces a design method based on numerical optimization. It is shown that, through a careful selection of the optimization variables, objective function and constraints, it is possible to obtain a linear optimization problem. As a result, it is guaranteed that the globally optimal input shaper be found in a few seconds of computational time. The presented optimization framework is able to handle higher-order, linear time-invariant dynamic systems, as opposed to traditional input shapers, which are mainly based on second- order systems. Moreover, constraints on input, output and state variables are easily accounted for, as well as robustness against parametric uncertainty. Numerical results illustrate the capability of the proposed design approach to reproduce existing input shaping design approaches, while experimental results illustrate its potential for higher-order systems.