The linear-quadratic optimal control problem is considered for time-varying descriptor systems of the formE(t+1)x(t+1)=A(t)x(t)+B(t)u(t)+C(t)f(t),whereEis a square matrix with constant rankr(which may be singular) andf(t) is a known vector. By using variational techniques, necessary conditions are derived for the existence of optimal control. By transforming to a coordinate system which can be computed by performing a non-singular linear transformation, necessary con- ditions for the existence of optimal control can be changed to a two-point boundary value problem which looks like the usual two-point boundary problem of linear- quadratic optimal control. By applying the sweep method, such a two-point boundary value problem results in a matrix Riccati equation and a linear difference equation. At the same time, the optimal control is uniquely determined in terms of the ‘state’ x1(t), which is the dynamic part of the descriptor vector x(t). The optimal cost is also given in terms of x1(t).