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Riccati equation

About: Riccati equation is a research topic. Over the lifetime, 10428 publications have been published within this topic receiving 210015 citations. The topic is also known as: Riccati's differential equation.


Papers
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Journal ArticleDOI
TL;DR: In this paper, the Conley index theory is used to describe the dynamics on the global attractors of bistable gradient-like evolution equations via a semiconjugacy onto the dynamics of a simple system of ordinary differential equations.
Abstract: The dynamics on the global attractors of bistable gradient-like evolution equations are described via a semiconjugacy onto the dynamics of a simple system of ordinary differential equations. The fact that the semiconjugacy is “onto” implies that, given any solution to the ordinary differential equation, there exists a corresponding orbit on the attractor of the evolution equation. It is also shown that these results apply to the damped wave equation, a viscoelastic beam equation, the Fitz–Hugh–Nagumo equations, the Cahn–Hilliard equation, and the phase-field equations. The proofs are based on the Conley index theory.

53 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied the infinite horizon quadratic cost minimization problem for a well-posed linear system in the sense of Salamon and Weiss, and showed that the feedback operator can be expressed in terms of the Riccati operator.
Abstract: We study the infinite horizon quadratic cost minimization problem for a well-posed linear system in the sense of Salamon and Weiss. The quadratic cost function that we seek to minimize need not be positive, but it is convex and bounded from below. We assume the system to be jointly stabilizable and detectable and give a feedback solution to the cost minimization problem. Moreover, we connect this solution to the computation of either a (J,S)-inner or an S-normalized coprime factorization of the transfer function, depending on how the problem is formulated. We apply the general theory to get factorization versions of the bounded and positive real lemmas. In the case where the system is regular it is possible to show that the feedback operator can be expressed in terms of the Riccati operator and that the Riccati operator is a stabilizing self-adjoint solution of an algebraic Riccati equation. This Riccati equation is nonstandard in the sense that the weighting operator in the quadratic term differs from the expected one, and the computation of the correct weighting operator is a nontrivial task.

52 citations

Journal ArticleDOI
TL;DR: The Thomas-Fermi equation is solved by the Sinc-Collocation method that converges to the solution at an exponential rate and is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations.

52 citations

Journal ArticleDOI
TL;DR: In this paper, a linear continuous-time system is given whose input and output disturbances and initial conditions are unknown but bounded by known convex sets, together with the system dynamics and any available observation, determine at any time a set of all possible states, containing the true state of the system.
Abstract: A linear continuous-time system is given whose input and output disturbances and initial conditions are unknown but bounded by known convex sets. These sets, together with the system dynamics and any available observation, determine at any time a set of all possible states, containing the true state of the system. An ellipsoidal bound on this set is obtained. The positive-definite matrix and the center which describe the bounding ellipsoid are found to obey two coupled differential equations: a Riccati matrix differential equation and a vector differential equation. They are similar in structure to the Kalman filter equations except that the matrix part of the solution is not precomputable. A precomputable bound can be obtained, however. The cases with no output and no input disturbances are discussed. An "almost-precomputable" bound is described. Computational results show the applicability and the limitation of the approach.

52 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023153
2022335
2021203
2020240
2019223
2018231