Decomposition of linear-quadratic optimal control problems for two-steps systems

The paper deals with problems with state equations of two sequentially acting systems. Matching conditions for trajectories at the switch point are absent, however the minimized functional depends on values of a state trajectory in the left and right sides from the switch point. State trajectories have fixed left and right points, and, in general, they are discontinuous functions. The algorithm for solving this problem is given, which is based on sequential solving eight independent initial value problems. The formula for the minimal value of the performance index is also presented.