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A linear system with time-delay which varies with time is considered as a plant. A feedback structure which uses only the instantaneous internal variables is assumed, and the feedback gain is determined with a solution of some matrix inequalities. It is shown that the resulting closed loop system is asymptotically stable, and moreover, it is the optimal regulator minimizing a quadratic cost functional, which contains time-varying weights. The required solution of the matrix inequalities can be calculated with a solution of some linear matrix inequalities, which can be solved numerically. A numerical example is shown to demonstrate the design procedure.