  We consider a general linear control system and a general quadratic cost,
where the state evolves continuously in time and the control is sampled, i.e.,
is piecewise constant over a subdivision of the time interval. This is the
framework of a linear-quadratic optimal sampled-data control problem. As a
first result, we prove that, as the sampling periods tend to zero, the optimal
sampled-data controls converge pointwise to the optimal permanent control.
Then, we extend the classical Riccati theory to the sampled-data control
framework, by developing two different approaches: the first one uses a
recently established version of the Pontryagin maximum principle for optimal
sampled-data control problems, and the second one uses an adequate version of
the dynamic programming principle. In turn, we obtain a closed-loop expression
for optimal sampled-data controls of linear-quadratic problems.
