Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution

We consider the initial value problem for a nonsymmetric matrix Riccati differential equation, where the four coefficient matrices form an M-matrix. We show that for a wide range of initial values the Riccati differential equation has a global solution X(t) on [0,∞) and X(t) converges to the stable equilibrium solution as t goes to infinity.     

On multivariable encryption schemes based on simultaneous algebraic Riccati equations over finite fields

New multivariable asymmetric public-key encryption schemes based on the NP-complete problem of simultaneous algebraic Riccati equations over finite fields are suggested. We also provide a systematic way to describe any set of quadratic equations over any field, as a set of algebraic Riccati equations. This has the benefit of systematic algebraic crypt-analyzing any encryption scheme based on quadratic equations, to any possible vulnerable hidden structure, in view of the fact that the set of all solutions to any given single algebraic Riccati equation is fully described in terms of all the T-invariant subspaces of some restricted dimension, where T is the matrix of coefficients of the related algebraic Riccati equation.     

Discrete matrix Riccati equations with superposition formulas

An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati equations. Here we shall describe discretizations of Riccati equations that preserve the superposition formulas. The approach is general enough to include q-derivatives and standard discrete derivatives.     

On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

An invariant imbedding method for computation of eigenlengths of singular two-point boundary-value problems

In this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular TPBVP with the singularity at the origin. The invariant imbedding yields a first-order nonlinear equation called a Riccati equation and also gives the initial conditions at the origin for this equation. With the aid of Theorem 8 in Section 3 we numerically integrate the Riccati equation to “blowup” which gives our computed eigenlength.In closing, we would like to comment on the numerical merits of the integration-to-blowup technique. On the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results. The feature of a variable step size is essential as the value of z approaches the actual eigenlength. However, it is desirable to have a priori estimate or bounds of the eigenlength similar to those of Boland and Nelson [2] for the nonsingular case. The singular system, however, presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds. Hopefully an investigation of the matrix R(z) will yield these results.     

一类Riccati矩阵方程广义自反解的双迭代算法

本文研究了一类Riccati矩阵方程广义自反解的数值计算问题.利用牛顿算法将Riccati矩阵方程的广义自反解问题转化为线性矩阵方程的广义自反解或者广义自反最小二乘解问题,再利用修正共轭梯度法计算后一问题,获得了求Riccati矩阵方程的广义自反解的双迭代算法.拓宽了求解非线性矩阵方程的迭代算法.数值算例表明双迭代算法是有效的.     

Representation of solutions of n-order Riccati equation via generalized trigonometric functions

In this work we suggest a systematic method of construction of solutions of the n-order Riccati equation with constant coefficients in a field from the set of generalized trigonometric functions. The generalized trigonometric functions satisfy the system of evolution equations generated by the companion matrix of n  -order polynomial. The set of trigonometric functions depend of (n−1)$(n-1)$ variables formally expressed by series of exponential functions. In a particular case, the second order Riccati equation with constant coefficients is isomorphic to the evolution equation generated by the companion matrix of the associated quadratic polynomial. It is shown that the n>2$n>2$ order Riccati equation with coefficients in a field is derived from a linear system of evolution equations generated by companion matrix of the associated n  -order polynomial under (n−2)$(n-2)$ constraints.     

On the Minimal Nonnegative Solution of Nonsymmetric Algebraic Riccati Equation

We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired M-matrix solution of the quadratic matrix equation X^2 - EX - F = 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M-matrix.     

ON THE MINIMAL NONNEGATIVE SOLUTION OF ONSYMMETRIC ALGEBRAIC RICCATI EQUATION

We study perturbation bound and structured condition number about the minimalnonnegative solution of nonsymmetric algebraic Riccati equation,obtaining a sharp per-turbation bound and an accurate condition number.By using the matrix sign functionmethod we present a new method for finding the minimal nonnegative solution of this al-gebraic Riccati equation.Based on this new method,we show how to compute the desiredM-matrix solution of the quadratic matrix equation X~2-EX-F=0 by connecting itwith the nonsymmetric algebraic Riccati equation,where E is a diagonal matrix and F isan M-matrix.     

Fundamental solution for linear two-point boundary value problem

The quadratic functional minimization with differential restrictions represented by the command linear systems is considered. The optimal solution determination implies the solving of a linear problem with two points boundary values. The proposed method implies the construction of a fundamental solution S(t)—a n×n matrix—and of a vector h(t) defining an adjoint variable λ(t) depending of the state variable x(t). From the extremum necessary conditions it is obtained the Riccati matrix differential equation having the S(t) as unknown fundamental solution is obtained. The paper analyzes the existence of the Riccati equation solution S(t) and establishes as the optimal solution of the proposed optimum problem. Also a superior limit of the minimum for the considered quadratic functionals class are evaluated.     

Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations

For the algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix K, the minimal nonnegative solution can be found by Newton’s method and the doubling algorithm. When the two diagonal blocks of the matrix K have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton’s method. In those cases, Newton’s method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton’s method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton–Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton–Shamanskii method converges monotonically to the minimal nonnegative solution.We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.     

Stability of Riccati's Equation with Random Stationary Coefficients

The purpose of this paper is to study under weak conditions of stabilizability and detectability, the asymptotic behavior of the matrix Riccati equation which arises in stochastic control and filtering with random stationary coefficients. We prove the existence of a stationary solution of this Riccati equation. This solution is attracting, in the sense that if P _t is another solution, then onverges to 0 exponentially fast as t tends to +∞ , at a rate given by the smallest positive Lyapunov exponent of the associated Hamiltonian matrices. Accepted 13 January 1998     

Riccati equations and Lie series

Lie series and a special matrix notation for first-order differential operators are used to show that the Lie group properties of matrix Riccati equations arise in a natural way. The Lie series notation makes it evident that the solutions of a matrix Riccati equation are curves in a group of nonlinear transformations that is a generalization of the linear fractional transformations familiar from the classical complex analysis. It is easy to obtain a linear representation of the Lie algebra of the nonlinear group of transformations and then this linearization leads directly to the standard linearization of the matrix Riccati equations. We note that the matrix Riccati equations considered here are of the general rectangular type.     

Sensitivity of convergent matrices and polynomials

A matrix A is said to be convergent if and only if all its characteristic roots have modulus less than unity. When A is real an explicit expression is given for real matrices B such that A + B is also convergent, this expression depending upon the solution of a quadratic matrix equation of Riccati type. If A and A + B are taken to be in companion form, then the result becomes one of convergent polynomials (i.e., polynomials whose roots have modulus less then unity), and is much easier to apply. A generalization is given for the case when A and A + B are complex and have the same number of roots inside and outside a general circle.     

The Jacobi Matrix for Functions in Noncommutative Algebras

We develop a general tool for constructing the exact Jacobi matrix for functions defined in noncommutative algebraic systems without using any partial derivative. The construction is applied to solving nonlinear problems of the form f(x) = 0 with the aid of Newton’s method in algebras defined in \({\mathbb{R}^N}\) . We apply this to eight (commutative and noncommutative) algebras in \({\mathbb{R}^4}\) . The Jacobi matrix is explicitly constructed for polynomials in x?a and for polynomials in the reciprocals (x?a)¹ such that Jacobi matrices for functions defined by Taylor and Laurent expansions can be constructed in general algebras over \({\mathbb{R}^N}\) . The Jacobi matrix for the algebraic Riccati equation with matrix elements from an algebra in \({\mathbb{R}^N}\) is presented, and one particular algebraic Riccati equation is numerically solved in all eight algebras over \({\mathbb{R}^4}\) . Another case treated was the exponential function with algebraic variables including a numerical example. For cases where the computation of the exact Jacobi matrix for finding solutions of f(x) = 0 is time consuming, a hybrid method is recommended, namely to start with an approximation of the Jacobi matrix in low precision and only when \({\|f(x)\|}\) is sufficiently small, to switch to the exact Jacobi matrix.     

Convergence and existence results for continuous- and discrete-time Riccati equations

We give a new proof for the convergence of the solution of a terminal value problem for the periodic Riccati differential equation towards its strong solution as t → ?∞. The proof is mainly based on well-known comparison results and also on an explicit representation formula for the solution that reflects precisely the dependence on the terminal value. Moreover, we give sufficient conditions for the existence of a periodic solution of the differential equation. Similar results are derived for the discrete-time Riccati equation.     

Atkinson's super-linear oscillation theorem for matrix dynamic equations on a time scale

By applying the Riccati technique and operator theory, we establish on a time scale T both oscillation and non-oscillation criteria for Atkinson's super-linear matrix dynamic equation XΔ²+σ[Xm(t)Q(t)X*m(t)]Xσ(t)=0. These results extend and unify earlier results for the differential and difference equation case.     

Solution of the matrix Riccati equation for the linear quadratic control problems

This paper is concerned with the solution of the matrix Riccati differential equation with a terminal boundary condition. The solution of the matrix Riccati equation is given by using the solution of the algebraic form of the Riccati equation. An illustrative example for the proposed method is given.     

Spectrum transformation and conservation laws of lattice potential KdV equation

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation whose solutions have nonzero backgrounds. The derivation is based on the fact that the scattering data a(z) is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schrödinger spectral problem. The obtained conserved densities are asymptotic to zero when |n| (or |m|) tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.