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

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