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We revisit and extend the Riccati theory, unifying continuous-timelinear-quadratic optimal permanent and sampled-data control problems, in finiteand infinite time horizons. In a nutshell, we prove that:-- when the timehorizon T tends to $+\infty$, one passes from the Sampled-Data DifferenceRiccati Equation (SD-DRE) to the Sampled-Data Algebraic Riccati Equation(SD-ARE), and from the Permanent Differential Riccati Equation (P-DRE) to thePermanent Algebraic Riccati Equation (P-ARE);-- when the maximal step of thetime partition $\Delta$ tends to $0$, one passes from (SD-DRE) to (P-DRE), andfrom (SD-ARE) to (P-ARE).Our notations and analysis provide a unified frameworkin order to settle all corresponding results.

Unified Riccati theory for optimal permanent and sampled-data control problems in finite and infinite time horizons