The Taylor series is used for the solution of the optimal-control problem for time-varying linear systems. Instead of solving the state transition matrix from the state equation with a terminal condition, the present approach first transforms the terminal condition into an initial condition, and then solves the initial-value problem to find the transition matrix. This approach leads lo a recursive algebraic formulation for the transition matrix, and only an inverse matrix of small dimension 2n× 2nappears in this formulation. Thus a closed-loop control law is obtained without solving the non-linear Riccati equation, and the matrix to be inverted has only small dimension 2n× 2n. The present approach is of great interest because of its simplicity and numerical stability.