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The problems considered here have cost , a quadratic in , wherexois interpreted as an initial condition and as an unconstrained control variable. The aim is to choose rationally when the only measurement available is the partial linear measurement and when should be desirable for every xoconsistent with the measurement yo. A dominance concept is introduced which facilitates suitable choice of . Properties of such are studied. The results are applied to the standard linear regulator, giving an open-loop control dependent on yo, and to the problem of choosing the initial conditions for dynamic compensators.