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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 theframework of a linear-quadratic optimal sampled-data control problem. As afirst result, we prove that, as the sampling periods tend to zero, the optimalsampled-data controls converge pointwise to the optimal permanent control.Then, we extend the classical Riccati theory to the sampled-data controlframework, by developing two different approaches: the first one uses arecently established version of the Pontryagin maximum principle for optimalsampled-data control problems, and the second one uses an adequate version ofthe dynamic programming principle. In turn, we obtain a closed-loop expressionfor optimal sampled-data controls of linear-quadratic problems.

Linear-quadratic optimal sampled-data control problems: convergence result and Riccati theory