A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory

A large scale nonsymmetric algebraic Riccati equation XCX ? XE ? AX + B = 0 arising in transport theory is considered, where the n × n coefficient matrices B,C are symmetric and low-ranked and A, E are rank one updates of nonsingular diagonal matrices. By introducing a balancing strategy and setting appropriate initial matrices carefully, we can simplify the large-scale structure-preserving doubling algorithm (SDA_ls) for this special equation. We give modified large-scale structure-preserving doubling algorithm, which can reduce the flop count of original SDA_ls by half. Numerical experiments illustrate the effectiveness of our method.     

Delta算子Riccati方程研究的新结果

基于Delta算子描述，统一研究连续时间代数Riccati方程(CARE)和离散时间代数Riccati方程(DARE)的定界估计问题，提出了统一代数Riccati方程(UARE)解矩阵的上下界，给出UARE中P与R和Q的几个基本关系．     

Riccati方程与Bernoulli方程的一种解关系(英文)

给出Ｒｉｃｃａｔｉ方程和Ｂｅｒｎｏｕｌｌｉ方程的统一求积方法,揭示两类方程的一种解关系．     

New upper solution bounds of the discrete algebraic Riccati matrix equation

In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.     

The Algebraic Riccati Equation for Quaternions

We will present the general solution of the algebraic Riccati equation for the quaternionic case, where also one additional variation is treated. For computational purpose a very simple form of the exact Jacobi matrix for Riccati polynomials is presented. There are several examples.     

Riccati equation for infinite-dimensional discrete bilinear systems

The problem of quadratic optimal feedback control for infinite-dimensionaldiscrete bilinear systems is associated with an algebraic Riccati-likeoperator equation. The existence and uniqueness of a nonnegativesolution to such a Riccati-like operator equation is establishedas the strong limit of a sequence of solutions to Lyapunov-likeoperator equations. This research was supported in part by CNPq (Brazilian NationalResearch Council) and FAPESP (Research Council of the Stateof sào Paulo).     

Riccati方程周期解的分支

证明带参数λ的Riccati方程x′=x~2+(λ+Q(t))存在周期解的分支点λ_0,当λλ_0时有且仅有两个周期解,当λ=λ_0时有且仅有一个周期解,当λλ_0时所有解无界.     

Estimating Mahler measures using periodic points for the doubling map

The Mahler measure m(P)$m\left(P\right)$ of a polynomial P$P$ is a numerical value which is useful in number theory, dynamical systems and geometry. In this article we show how this can be written in terms of periodic points for the doubling map on the unit interval. This leads to an interesting algorithm for approximating m(P)$m\left(P\right)$ which we illustrate with several examples.     

Solution Bounds for the Discrete Riccati Equation and Its Applications

In this paper, we address the estimation problem for the solution of the discrete algebraic matrix Riccati equation. Both upper and lower bounds are measured. Compared to the majority of the approaches proposed in the literature, the present results are sharper. We also apply the results obtained to solve the robust stabilization problem of discrete time-delay systems. A robust stabilizability criterion and the corresponding state feedback control law are proposed. Furthermore, the tolerable bound of the delay term is also estimated. Finally, numerical examples are given to demonstrate the applications of the results.     

Riccati inequality and other results for discrete symplectic systems

In this paper we establish several new results regarding the positivity and nonnegativity of discrete quadratic functionals F associated with discrete symplectic systems. In particular, we derive (i) the Riccati inequality for the positivity of F with separated endpoints, (ii) a characterization of the nonnegativity of F for the case of general (jointly varying) endpoints, and (iii) several perturbation-type inequalities regarding the nonnegativity of F with zero endpoints. Some of these results are new even for the special case of discrete Hamiltonian systems.     

A numerical evaluation of solvers for the periodic Riccati differential equation

Efficient and accurate structure exploiting numerical methods for solving the periodic Riccati differential equation (PRDE) are addressed. Such methods are essential, for example, to design periodic feedback controllers for periodic control systems. Three recently proposed methods for solving the PRDE are presented and evaluated on challenging periodic linear artificial systems with known solutions and applied to the stabilization of periodic motions of mechanical systems. The first two methods are of the type multiple shooting and rely on computing the stable invariant subspace of an associated Hamiltonian system. The stable subspace is determined using either algorithms for computing an ordered periodic real Schur form of a cyclic matrix sequence, or a recently proposed method which implicitly constructs a stable deflating subspace from an associated lifted pencil. The third method reformulates the PRDE as a convex optimization problem where the stabilizing solution is approximated by its truncated Fourier series. As known, this reformulation leads to a semidefinite programming problem with linear matrix inequality constraints admitting an effective numerical realization. The numerical evaluation of the PRDE methods, with focus on the number of states (n) and the length of the period (T) of the periodic systems considered, includes both quantitative and qualitative results.     

利用Riccati方程求解Burgers方程

应用李群理论中的伸缩变换群,把非线性二阶偏微分方程-Burgers方程转化为非线性非齐次一阶常微分方程-Riccati方程,将Riccati方程转化为Bernoulli方程和齐次线性二阶常微分方程,从而找到了Riccati方程的许多解,最后进一步求出了Burgers方程许多新的解析解.     

一类Riccati型方程的通积分

给出 Riccati型方程 :f′(y) dydx=p(x) f 2 (y) +Q(x) f (y) +R(x) e∫Q( x) dx在条件 p(x) e∫Q( x) dx=21 ∫R(x) dx′下的通积分 ,由此 ,得到若干类 Riccati方程的通积分     

Joint discrete universality for periodic zeta-functions

Abstract

In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied.     

Attractors for discrete periodic dynamical systems

A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the union of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an autonomous map, properties that the nonautonomous attractor inherits from the autonomous attractor are discussed. Examples from population biology are presented.     

Asymptotic Expansions of Solutions to the Riccati Equation

Doklady Mathematics - Scalar real Riccati equations with coefficients expandable in convergent power series in a neighborhood of infinity are considered. Extendable solutions to equations of this...