Sima, VasileVasileSima2025-03-262025-03-262019-09-2610.1007/978-1-4471-5102-9_100053-1https://dspace.apps.devcloud.ici.ro/handle/123456789/233Linear and quadratic (Riccati) matrix equations are fundamental for systems and control theory and its numerous applications. Generalized or standard Sylvester and Lyapunov equations and generalized Riccati equations for continuous- and discrete-time systems are considered. Essential applications in control are mentioned. The main solvability conditions and properties of these equations, as well as state-of-the-art solution techniques, are summarized. The continuous progress in this area paves the way for further developments and extensions to more complex control problems.enControllabilityGramianHankel singular valueLyapunov equationNumerical algorithmObservabilityPerformance indexRiccati equationSchur decompositionStabilityStein equationSylvester equationBook chapterhttps://link.springer.com/referenceworkentry/10.1007%2F978-1-4471-5102-9_100053-1