SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 489-508, January 2021.
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: \beginequation* \xymatrix@R=1.5cm@C=3.5cm (SD-DRE) & E^T,\Delta \ar[r]^T \to +\infty \ar[d]_\Vert \Delta \Vert \to 0 & E^\infty,\Delta \ar[d]^\Vert \Delta \Vert \to 0 & (SD-ARE) (P-DRE) & E^T \ar[r]_T \to +\infty & E^\infty & (P-ARE) \endequation* i.e., that (i) 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 ${(P{-}ARE)}$; (ii) when the maximal step $\Vert \Delta \Vert$ 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.

中文翻译：

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有限和无限时间范围内最优永久和采样数据控制问题的统一Riccati理论

SIAM控制与优化杂志，第59卷，第1期，第489-508页，2021年1月。
我们重新审视并扩展了Riccati理论，统一了有限和无限时间范围内的连续时间线性二次最优永久和采样数据控制问题。简而言之，我们证明下图可互换：\ beginequation * \ xymatrix @ R = 1.5cm @ C = 3.5cm（SD-DRE）＆E ^ T，\ Delta \ ar [r] ^ T \ to + \ infty \ ar [d] _ \ Vert \ Delta \ Vert \ to 0＆E ^ \ infty，\ Delta \ ar [d] ^ \ Vert \ Delta \ Vert \ to 0＆（SD-ARE）（P-DRE） ＆E ^ T \ ar [r] _T \ to + \ infty＆E ^ \ infty＆（P-ARE）\ endequation *，即（i）当时间范围$ T $趋于$ + \ infty $时，一个从采样数据差Riccati方程$ {（SD {-} DRE）} $传递到采样数据代数Riccati方程$ {（SD {-} ARE）} $，然后从永久微分Riccati方程$ { （P {-} DRE）} $到永久代数Riccati方程$ {（P {-} ARE）} $; （ii）当时间分区$ \ Delta $的最大步长$ \ Vert \ Delta \ Vert $趋于0时，一个从$ {（SD {-} DRE）} $传递到$ {（P {-} DRE ）} $，以及从$ {（SD {-} ARE）} $到$ {（P {-} ARE）} $。上图中的$ E $符号（带有各种上标）指的是上面列出的每个Riccati方程的解。我们的符号和分析提供了一个统一的框架，以解决所有相应的结果。