On the relation between the sampled-data and the continuous optimal linear regulator problems

In this paper, the geometric relationship between the optimal solutions to the sampled-data and continuous linear quadratic regulator problems is investigated in a Hilbert space framework. It is shown that the optimal sampled-data solution, excluding the response due to the initial condition, is the projection of the optimal continuous solution onto the set of all solutions that satisfy the sampled-data constraint. That is, the optimal sampled-data solution is an optimal approximation to the continuous solution. In fact, it is shown that the sampled-data solution can be obtained by solving a sampled-data tracking problem with the continuous solution as the desired trajectory.