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.

A study of wave propagation in varying cross-section waveguides by modal decomposition. Part I. Theory and validation

Abstract: The propagation of acoustic waves in waveguides with variable cross section is considered using multimodal decomposition. The approach adopted is to construct two infinite first‐order differential equations for the components of the pressure and the velocity projected over the normal modes. From these an infinite matricial Riccati equation is derived for the impedance matrix. These equations are ordinary differential equations that can be integrated after truncation at a sufficient number of modes and take into account the coupling between modes. The stiffness of the pressure‐velocity equations induced by the presence of evanescent modes is avoided by first calculating the impedance matrix along the guide. The method is checked using different examples where the solutions of the plane‐wave approximation or the finite element method are known. Results show the method provides simple and accurate means to obtain the acoustic field with correct boundary conditions in a nonuniform guide with no restriction on the flare.

Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices

Abstract: Until recently, it was believed that a necessary and sufficient condition for convergence of the Riccati difference equation of optimal filtering was that the system be both delectable and stabilizable. Recently, it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular. The present paper generalizes these results in several directions. First, properties of the algebraic Riccati equation are established for the case of singular state transition matrix. Second, several assumptions previously imposed in establishing convergence of the Riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability, weakening the conditions on the initial covariance, and allowing the state transition matrix to be singular. Third, results on the convergence and properties of the Riccati equations are expressed as both necessary and sufficient conditions, whereas previous results were only sufficient. These extensions mean that the results have wider applicability, including fixed-lag smoothing problems and filtering for systems with time delays. The implications of the results in the dual problem of optimal control are also studied.

Indefinite Stochastic Linear Quadratic Control and Generalized Differential Riccati Equation

Abstract: A stochastic linear quadratic (LQ) control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite. Indefinite stochastic LQ theory has been extensively developed and has found interesting applications in finance. However, there remains an outstanding open problem, which is to identify an appropriate Riccati-type equation whose solvability is {\it equivalent} to the solvability of the indefinite stochastic LQ problem. This paper solves this open problem for LQ control in a finite time horizon. A new type of differential Riccati equation, called the generalized (differential) Riccati equation, is introduced, which involves algebraic equality/inequality constraints and a matrix pseudoinverse. It is then shown that the solvability of the generalized Riccati equation is not only sufficient, but also necessary, for the well-posedness of the indefinite LQ problem and the existence of optimal feedback/open-loop controls. Moreover, all of the optimal controls can be identified via the solution to the Riccati equation. An example is presented to illustrate the theory developed.