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This paper presents a practical approach to the problem of optimal-boundary-control synthesis in 1-dimensional, linear, stationary, distributed-parameter systems. The optimal-control function is obtained directly from a generalised quadratic performance index, by gradient methods extended to the functional space. Algorithms for two gradient methods, namely, the steepest-descent and the conjugate-gradient, are developed. As an example, these algorithms are applied to a specific 1-dimensional heat-conduction system. The computational results clearly demonstrate the feasibility of both methods and, in particular, the accelerated convergence of the conjugate-gradient method.