Topic
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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13 May 2009
TL;DR: In this article, the authors introduce pseudoanalytic function theory and second-order elliptic equations and apply them to real first-order systems, including the Sturm-Liouville equation and Dirac equation.
Abstract: Introduction.- I. Pseudoanalytic function theory and second-order elliptic equations.- 1. Definitions and results from Bers' theory.- 2. Second order equations.- 3. Formal powers.- 4. Cauchy's integral formula.- 5. Complex Riccati equation.- II. Applications to Sturm-Liouville theory.- 6. Sturm-Liouville equation.- 7. Spectral problems and Darboux transformation.- III. Applications to real first-order systems.- 8. Beltrami fields.- 9. Static Maxwell system.- IV. Hyperbolic pseudoanalytic functions.- 10. Hyperbolic numbers and analytic functions.- 11. Hyperbolic pseudoanalytic functions.- 12. Klein-Gordon equation.- V. Bicomplex and biquaternionic pseudoanalytic functions and applications.- 13. The Dirac equation.- 14. Complex second order elliptic equations and bicomplex pseudoanalytic functions.- 15. Multidimensional second order equations.- Open problems.- Bibliography.- Index.
122 citations
TL;DR: In this paper, exact traveling wave solutions of the Bogoyavlenskii equation using a modified extended tanh-function method were considered. But their method has a broad applicability to many other nonlinear evolution equations in mathematical physics.
122 citations
01 Jan 2001
TL;DR: In this paper, the authors present a statement of problem relations with the Calculus of Variations Controllability Solution of the Linear Regulator Problem Stability of the Riccati Equation General Solution of Riccaci Equation Bibliography Appendix: The Generalized Inverse of a Matrix
Abstract: This chapter contains sections titled: Introduction Notation and Terminology Statement of Problem Relations With the Calculus of Variations Controllability Solution of the Linear Regulator Problem Stability of the Riccati Equation General Solution of the Riccati Equation Bibliography Appendix: The Generalized Inverse of a Matrix
121 citations
15 Sep 1996
TL;DR: In this article, a new nonlinear filter referred to as the state-dependent Riccati equation filter (SDthis article) is presented, which is derived by constructing the dual of a little known nonlinear regulator control design technique which involves the solution of a state-dependent RICE (SDRE) and which has been appropriately called the SDRE control method.
Abstract: A new nonlinear filter referred to as the state-dependent Riccati equation filter (SDREF) is presented. The SDREF is derived by constructing the dual of a little known nonlinear regulator control design technique which involves the solution of a state-dependent Riccati equation (SDRE) and which has been appropriately called the SDRE control method. The resulting SDREF has the same structure as the continuous steady-state linear Kalman filter. In contrast to the linearized Kalman filter (LKF) and the extended Kalman filter (EKF) which are based on linearization, the SDREF is based on a parameterization that brings the nonlinear system to a linear structure having state-dependent coefficients (SDC). In a deterministic setting, before stochastic uncertainties are introduced, the SDC parameterization fully captures the nonlinearities of the system, It was shown in Cloutier et al. (1996) that, in the multivariable case, the SDC parameterization is not unique and that the SDC parameterization itself can be parameterized. This latter parameterization creates extra degrees of freedom that are not available in traditional filtering methods. These additional degrees of freedom can be used to either enhance filter performance, avoid singularities, or avoid loss of observability. The main intent of this paper is to introduce the new nonlinear filter and to illustrate the behaviorial differences and similarities between the new filter, the LKF, and the EKF using a simple pendulum problem.
121 citations
TL;DR: In this article, the (2 + 1)-dimensional Gardner equation is decomposed into the first two non-trivial soliton systems in the hierarchy, and two compatible Hamiltonian systems of ordinary differential equations.
Abstract: To decompose the (2 + 1)-dimensional Gardner equation, an isospectral problem and a corresponding hierarchy of (1 + 1)-dimensional soliton equations are proposed. The (2 + 1)-dimensional Gardner equation is separated into the first two non-trivial (1 + 1)-dimensional soliton systems in the hierarchy, and in turn into two new compatible Hamiltonian systems of ordinary differential equations. Using the generating function flow method, the involutivity and the functional independence of the integrals are proved. The Abel-Jacobi coordinates are introduced to straighten out the associated flows. The Riemann-Jacobi inversion problem is discussed, from which quasi-periodic solutions of the (2 + 1)-dimensional Gardner equation are obtained by resorting to the Riemann theta functions.
121 citations