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.

On solving periodic Riccati equations

Abstract: Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to single shooting periodic generator methods, the multiple shooting-type methods have the main advantage of being able to address problems with larger periods. Three methods are discussed to solve DPREs. Two of the methods represetn extensions of a quotient-product swapping and collapsing "fast" algorithm. All proposed approches are completely general, being applicable to periodic Riccati equations with time-varying dimensions as well as with singular weighting matrices.

The Riccati method for the Helmholtz equation

Abstract: The operator Riccati equation for the Dirichlet‐to‐Neumann map is derived from the exact operator factorization of the two‐dimensional variable coefficient Helmholtz equation. Numerical schemes are developed for the operator Riccati equation and its variant using a local eigenfunction expansion. This leads to a practical computational method for acoustic wave propagation over large range distances, since the boundary value problem of the Helmholtz equation is reduced to ‘‘initial’’ value problems that are solved by marching in the range. The efficiency and accuracy of the method is demonstrated by numerical experiments including the plane‐parallel range‐dependent waveguide benchmark problem proposed by the Acoustical Society of America.

When the telegrapher's equation furnishes a better approximation to the transport equation than the diffusion approximation

Abstract: It has been suggested that a solution to the transport equation which includes anisotropic scattering can be approximated by the solution to a telegrapher’s equation @A.J. Ishimaru, Appl. Opt. 28, 2210 ~1989!#. We show that in one dimension the telegrapher’s equation furnishes an exact solution to the transport equation. In two dimensions, we show that, since the solution can become negative, the telegrapher’s equation will not furnish a usable approximation. A comparison between simulated data in three dimensions indicates that the solution to the telegrapher’s equation is a good approximation to that of the full transport equation at the times at which the diffusion equation furnishes an equally good approximation. @S1063-651X~97!04205-0#

A multishift algorithm for the numerical solution of algebraic riccati equations

Abstract: We study an algorithm for the numerical solution of algebraic matrix Riccati equa- tions that arise in linear optimal control problems. The algorithm can be considered to be a multishift technique, which uses only orthogonal symplectic similarity transformations to compute a Lagrangian invariant subspace of the associated Hamiltonian matrix. We describe the details of this method and compare it with other numerical methods for the solution of the algebraic Riccati equation.