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An alternative method is presented for designing optimal linear systems with quadratic performance criteria. The method is based on the Pontryagin maximum principle, and, after deriving the trajectory equation satisfying the optimality condition, an asymptotic stability constraint allows the characteristic equation governing the optimal stable system to be derived. This equation, combined with the system equation, leads directly to the optimum-feedback configuration. Application of the method to a position-control system, in which the mean-square error is to be minimised without using too much power, gives results which agree directly with more complex methods of Noton and also Athans and Falb. Further, by phrasing the problem using state-space terminology, the method appears to be more readily applicable than the classical approach adopted by S. L. Chang.