An algorithm, based on the fixed-point contraction-mapping principle of functional analysis, for calculating the optimal controls for time-varying linear multivariable systems is considered. By an appropriate transformation of the operators involved, it is shown that the convergence rate may be maximised and that the resulting optimal algorithm will always converge. Comparison of the optimal algorithm with the steepest-descent and conjugate-gradient schemes shows the optimal algorithms to have significantly superior convergence rates, as well as requiring appreciably smaller computer storage.