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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.


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Journal ArticleDOI
TL;DR: In this paper, Chen et al. proposed a series of non-travelling wave and coefficient function solutions for the (2+1)-dimensional Broer-Kaup-Kupershmidt equation.
Abstract: By using the Chen et al. ansatz [Chen Y, Wang Q, Lang Y. Naturforsch 2005;60a:127] and by modifying our extended Fan sub-equation method [Yomba E. Phys Lett A 2005;336:463]. We have obtained new and more general solutions including a series of non-travelling wave and coefficient function solutions namely: soliton-like solutions, triangular-like solutions, single and combined non-degenerate Jacobi elliptic wave function-like solutions for the (2+1)-dimensional Broer–Kaup–Kupershmidt equation. The most important achievement of this method lies on the fact that we have succeeded in one move to give all the solutions which can previously be obtained by application of at least four methods (the method using the Riccati equation, or the first kind elliptic equation, or the auxiliary ordinary equation, or the generalized Riccati equation as mapping equation).

140 citations

Journal ArticleDOI
A. van Harten1
TL;DR: In this paper, a mathematically rigorous proof of the validity of G-L's equation for a general situation of one space variable and a quadratic nonlinearity is given.
Abstract: The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(e2) away from the critical valueRc for which the system loses stability. Heree>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/e2)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(e2) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.

139 citations

Journal ArticleDOI
TL;DR: In this paper, a continuous-time linear system with finite jumps at discrete instants of time is considered and an iterative method to compute the √ L 2 -induced norm of the system with jumps is presented.
Abstract: This paper considers a continuous-time linear system with finite jumps at discrete instants of time. An iterative method to compute the ${\cal L}_2$-induced norm of a linear system with jumps is presented. Each iteration requires solving an algebraic Riccati equation. It is also shown that a linear feedback interconnection of a continuous-time finite-dimensional linear time-invariant (FDLTI) plant and a discrete-time finite-dimensional linear shift-invariant (FDLSI) controller can be represented as a linear system with jumps. This leads to an iterative method to compute the ${\cal L}_2$-induced norm of a sampled-data system.

139 citations

Journal ArticleDOI
TL;DR: A Max-Min theorem is provided in order to guarantee the saddle-point Nash equilibrium, and when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic, the existence of saddle- point Nash equilibrium is obtained via the solvability of a corresponding Riccati equation.
Abstract: Uncertain differential game investigates interactive decision making of players over time, and the system dynamics is described by an uncertain differential equation. This paper goes further to study the two-player zero-sum uncertain differential game. In order to guarantee the saddle-point Nash equilibrium, a Max–Min theorem is provided. Furthermore, when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic, the existence of saddle-point Nash equilibrium is obtained via the solvability of a corresponding Riccati equation. Finally, a resource extraction problem is analyzed by using the theory proposed in this paper.

139 citations

Journal ArticleDOI
TL;DR: This paper considers the simultaneous fault detection and control (SFDC) problem and its solution is presented in terms of two coupled Riccati equations, formulated as a mixed H"2/H"~ optimization problem.

138 citations


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