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 Hamiltonian Regularization of the Burgers Equation

Abstract: We consider a quasilinear equation that consists of the inviscid Burgers equation plus O(?2) nonlinear terms. As we show, these extra terms regularize the Burgers equation in the following sense: for smooth initial data, the ? > 0 equation has classical solutions globally in time. Furthermore, in the zero-? limit, solutions of the regularized equation converge strongly to weak solutions of the Burgers equation. We present numerical evidence that the zero-? limit satisfies the Oleinik entropy inequality. For all ? ? 0, the regularized equation possesses a nonlocal Poisson structure. We prove the Jacobi identity for this generalized Hamiltonian structure.

Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points

Abstract: The boundary value and expansion problems for the equation of the nth order with boundary conditions at two points have been studied by Birkhoff.t Bochert has suggested the generalization of these results to the equation with auxiliary conditions at more than two points. Such generalization of the essential properties of the differential system has been carried out by the author, and in this paper is given the proof of the convergence of the expansion, which may be studied quite independently of the other results. The formal development of the boundary problem and a more detailed discussion of the form of the series will be presented in other papers. The differential equation is taken in the form

Riccati Observers for the Nonstationary PnP Problem

Abstract: This paper revisits the problem of estimating the pose (position and orientation) of a body in 3-D space with respect to (w.r.t.) an inertial frame by using 1) the knowledge of source points positions in the inertial frame, 2) the measurements of the body angular velocity expressed in the body's frame, 3) the measurements of the body translational velocity, either in the body frame or in the inertial frame, and 4) source points bearing measurements performed in the body frame. An important difference with the much studied static Perspective-n-Point problem addressed with iterative algorithms is that body motion is not only allowed but also used as a source of information that improves the estimation possibilities. With respect to the probabilistic framework commonly used in other studies that develop extended Kalman filter solutions, the deterministic approach here adopted is better suited to point out the observability conditions, that involve the number and disposition of the source points in combination with body motion characteristics, under which the proposed observers ensure robust estimation of the body pose. These observers are here named Riccati observers because of the instrumental role played by the continuous Riccati equation in the design of the observers and in the Lyapunov stability and convergence analysis that we develop independently of the well-known complementary (either deterministic or probabilistic) optimality properties associated with Kalman filtering. The set of these observers also encompasses extended Kalman filter solutions. Another contribution of this study is to show the importance of using body motion to improve the observers performance and, when this is possible, of measuring the body translational velocity in the inertial frame rather than in the body frame to allow for the body pose estimation from a single source point taken as the origin of the inertial frame. This latter possibility finds a natural extension in the Simultaneous Localisation And Mapping (SLAM) problem in Robotics.