Hexagonal circle patterns with constant intersection angles mimicking holomorphic maps z and log(z) are studied. It is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete Painlevé and Riccati equations. The general solution of the Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solu...

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. We also study the Poincar...

Discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control and -control of Markovian jump linear systems are considered. First, the L_ equations that arise from the quadratic optimal control problem are studied. The matrix cost is only assumed to be hermitian. Conditions for existence of the maximal hermitian solution are derived in terms of the concept of mean sq...

Nonsymmetric algebraic Riccati equations for which the four coefficient matrices form an irreducible M -matrix M are considered. The emphasis is on the case where M is an irreducible singular M -matrix, which arises in the study of Markov models. The doubling algorithm is considered for finding the minimal nonnegative solution, the one of practical interest. The algorithm has been recently stud...

In the case of barotropic FRW cosmologies, the Hubble parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. We consider these cosmologies in the framework of nonrelativistic supersymmetry that has been effective in the area of supersymmetric quantum mechanics. Recalling that Faraoni [Amer. J. Phys. 67 (1999), 732–734] showed how to reduce the barotro...

In the paper a linear-quadratic optimization problem (LCTOR) with unseparated two-point boundary conditions is considered. To solve this problem is proposed a new sweep algorithm which increases doubles the dimension of the original system. In contrast to the well-known methods, here it refuses to solve linear matrix and nonlinear Riccati equations, since the solution of such multi-point optimi...

This paper investigates the use of state-dependent Riccati equation control for closed-loop guidance of the hypersonic phase of atmospheric entry. Included are a discussion of the development of the state-dependent Riccati equations, their outgrowth from Hamilton-Jacobi-Bellman theory, a discussion of the closed-loop nonlinear system’s closed-loop stability and robustness from both a theoretica...

We will investigate the stability behavior of quadratic maps in higher dimensions. To check stability, we will use infinitesimal V -stability of critical points of the map; since the infinitesimal V -stability of a map at all of its critical points is equivalent to the stability of the map. We will establish the connection between infinitesimal V-stability of solutions to the Algebraic Riccati ...

As in the finite-dimensional case, the appropriate state feedback for the infinitedimensional H disturbance-attenuation problem may be calculated by solving a Riccati equation. This operator Riccati equation can rarely be solved exactly. We approximate the original infinitedimensional system by a sequence of finite-dimensional systems and consider the corresponding finite-dimensional disturbanc...

In this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and extend the original solution method to four equations, translating Riccati approach terms Lie symmetries. To get complete solution, hypergeometric functions come into play, which, course, were unknown Riccati’s time. Re-discovering introduced c...