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We study the linear-quadratic optimal control problem with the state equation consisting of two sequentially acting descriptor systems. Matching conditions for trajectories at the switch point are absent. However, the minimized functional depends on values of a state trajectory at the left and right sides of the switch point. State trajectories have partially fixed left and right points and, in general, they are discontinuous functions. We present the algorithm for solving this problem, which avoids the use of boundary value problems. It is based on the sequential solving of eight initial value problems for differential-algebraic equations. The formula for the minimal value of the performance index is also given.