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 the following diagram commutes: [GRAPHICS] i.e., that (i) when the time horizon T tends to +infinity, 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); (ii) when the maximal step parallel to Delta parallel to of the time partition Delta tends to 0, one passes from (SD-DRE) to (P-DRE), and from (SD-ARE) to (P-ARE). The notation E in the above diagram (with various superscripts) refers to the solution of each of the Riccati equations listed above. Our notation and analysis provide a unified framework in order to settle all corresponding results.