Showing papers on "Riccati equation published in 1986"
TL;DR: The fundamental idea behind the algorithm presented involves constructing an upper bound for the Lyapunov derivative corresponding to the closed loop system, a quadratic form, which can be found by solving a certain matrix Riccati equation.
825 citations
TL;DR: In this paper, the authors characterized the transition to chaos of the solutions to the Kuramoto-Sivashinsky equation through extensive numerical simulation, and showed that the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points, invariant tori, traveling wave trains, and homoclinic orbits.
310 citations
TL;DR: In this paper, the authors generalize the notion of stability radius introduced in [1] to allow for structured perturbations and relate the stability radius to the existence of Hermitian solutions of an algebraic Riccati equation.
295 citations
TL;DR: Lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix Riccati and Lyapunov equation are derived and results in a tighter bound as compared to the Upper bound for the maximal eigenvalue.
Abstract: Lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix Riccati and Lyapunov equation are derived. The upper trace bound obtained in this note in many cases results in a tighter bound as compared to the Upper bound for the maximal eigenvalue proposed in [1] and [2].
187 citations
TL;DR: In this article, two difference schemes for the sine-Gordon equation were proposed and proved to be stable and convergent in the presence of a discrete energy which is conserved.
185 citations
TL;DR: In this article, the interaction of radio-frequency waves with a plasma is described by a Fokker-Planck equation with an added quasilinear term, and methods for solving this equation on a computer are discussed.
178 citations
TL;DR: In this paper, it was shown that the stabilizability condition can be removed and convergence of the Riccati difference equation can be established under restrictive assumptions including the requirement that the state transition matrix be nonsingular.
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.
177 citations
TL;DR: On etudie les etats fondamentaux singuliers de l'equation de Poisson semilineaire Δu+f(u)=0 and de son analogue quasi-lineaire div(A(|Du|)Du)+f(U)=0 ou Δ est le laplacien a n dimensions and A(p) et f(u) sont des fonctions donnees as discussed by the authors.
Abstract: On etudie les etats fondamentaux singuliers de l'equation de Poisson semilineaire Δu+f(u)=0 et de son analogue quasi-lineaire div(A(|Du|)Du)+f(u)=0 ou Δ est le laplacien a n dimensions et A(p) et f(u) sont des fonctions donnees
151 citations
TL;DR: A method is presented to solve the real algebraic Riccati equation - XNX + XA + A^{T}X + K = 0, where K = K T and N = N T .
Abstract: A method is presented to solve the real algebraic Riccati equation - XNX + XA + A^{T}X + K = 0 , where K = K^{T} and N = N^{T} . The solution for the corresponding eigenvalue problem Mx = \lambda x , where M is a Hamiltonian matrix, is computed by an algorithm similar to the QR algorithm. Special symplectic matrices are used for the transformation of M such that the Hamiltonian form is preserved during the computations.
131 citations
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TL;DR: In this article, the authors apply approximation techniques to the study of one of the fundamental equations of mathematical analysis, the first order nonlinear ordinary differential equation (ODE), and show that their approach can be applied to the analysis of ODEs.
Abstract: In this chapter we wish to apply approximation techniques to the study of one of the fundamental equations of mathematical analysis, the first order nonlinear ordinary differential equation.
TL;DR: The Riccati equation as mentioned in this paper is a quadratic differential equation on the space of real symmetric matrices, which is closely related, via compactification of the phase space, to the differential equations on the Grassmann manifold and the Lagrange-Grassmann manifold.
Abstract: The matrix Riccati equation which arises from optimal control and filtering problems is a quadratic differential equation on the space of real symmetric $n \times n$ matrices. It is closely related, via compactification of the phase space, to the differential equations on the Grassmann manifold and the Lagrange–Grassmann manifold whose flows are generated by the action of one-parameter subgroups of the general linear group and of the symplectic group respectively. We. determine the complete phase portraits of the Riccati equations on all three spaces. The asymptotic behavior of every solution is described. The phase portraits are characterized topologically as well as set-theoretically. Although the Riccati equation is not generally a Morse–Smale vector field, we are able to show that it possesses suitable generalizations of many of the important properties of Morse–Smale vector fields. In particular, the Riccati equation satisfies a generalized version of the Morse inequalities for a Morse–Smale dynamica...
TL;DR: In this article, bifurcations of attractors in a driven damped nonlinear Schrodinger equation were studied, and a chaotic attractor in a low-dimensional subspace spanned by a soliton and long-wavelength radiation was found.
TL;DR: In this paper, a new class of solvable nonlinear dynamical systems has been identified by the requirement that the ordinary differential equations describing each member of this class possess nonlinear superposition principles.
Abstract: A new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE’s) describing each member of this class possess nonlinear superposition principles. These systems of ODE’s are generally not derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE’s are integrated in a unified way by finding explicit integrals for them and relating them all to a ‘‘pivotal’’ member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincare surface of section—in the form of sensitive dependence on initial conditions—near a boundary separating bounded from unbounded motion.
TL;DR: In this paper, the structure of the set of hermitian solutions of the matrix quadratic equation (XDX - XA - A^ * X - C = 0) is studied under the conditions that $C = C^ * $, D is positive semidefinite and $(A,D)$ is stabilizable.
Abstract: The structure of the set of hermitian solutions of the matrix quadratic equation $XDX - XA - A^ * X - C = 0$ is studied under the conditions that $C = C^ * $, D is positive semidefinite and $(A,D)$ is stabilizable. New features (e.g., nonexistence of the minimal solution) appear in contrast with the known case when $(A,D)$ is controllable.
TL;DR: In this paper, the authors considered a linear quadratic differential game in which the weighting on the minimizing control is allowed to approach zero and showed that if a certain minimum phase condition is satisfied then the value of the game will approach zero.
TL;DR: In this paper, the eigenvalues and sums and products of the Eigenvalues of the solution of the discrete Riccati and Lyapunov matrix equations and the continuous Lyapinov matrix equation were studied.
Abstract: We present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete Riccati and Lyapunov matrix equations and the continuous Lyapunov matrix equation. Nearly all of our bounds for the discrete Riccati equation are new. The bounds for the discrete and continuous Lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation.
TL;DR: In this article, the authors derived a sine-Gordon-like equation from the Riccati equation for soliton potentials by iterating once the auto-Backlund transformation and showed that the discrete eigenfunctions of the sixth-order recursion operator can be written in terms of the solutions of the isospectral equation.
Abstract: The solution of the third-order isospectral equation of the Caudrey-Dodd-Gibbon-Sawada-Kotera equation (CDGSKE) for soliton potential is obtained recursively from the Riccati equation derived by iterating once the auto-Backlund transformation. It is then shown that the discrete eigenfunctions of the sixth-order recursion operator for this equation can be written in terms of the solutions of the isospectral equation. The behaviour of the 1-soliton solution which has certain novel features is studied. A sine-Gordon-like equation resembling the double-sine-Gordon equation is derived from the CDGSKE.
TL;DR: In this paper, it was shown that a periodic Riccati equation admits a unique positive semidefinite periodic solution and the solution is stabilizing if and only if the underlying system is stabilizable and detectable.
Abstract: A new proof is given of the following result: a periodic Riccati equation admits a unique positive semidefinite periodic solution and the solution is stabilizing if and only if the underlying system is stabilizable and detectable. The proof hinges on the decomposition of the Riccati equation induced by the system canonical decomposition. The solution structure is therefore naturally pointed out.
TL;DR: In this article, the authors describe the set of hermitian solutions in various ways: in terms of factorizations of rational matrix functions which take hermitians values on the unit circle, the invariant subspaces of a matrix which is unitary in an indefinite scalar product, and all invariants of a certain matrix.
Abstract: Hermitian solutions of the discrete algebraic Riccati equation play an important role in the least-squares optimal control problem for discrete linear systems. In this paper we describe the set of hermitian solutions in various ways: in terms of factorizations of rational matrix functions which take hermitian values on the unit circle; in terms of certain invariant subspaces of a matrix which is unitary in an indefinite scalar product; and in terms of all invariant subspaces of a certain matrix. These results are inspired by known results for the algebraic Riccati equation arising in the least-squares optimal control problem for continuous linear systems.
TL;DR: In this paper, a collection of results on the Riccati equation is presented and the questions addressed are the existence of strong solutions of the algebraic version and the convergence of solutions of Riccaci difference equation to those of algebraic versions.
Abstract: A collection of results on the Riccati equation is presented. The questions addressed are the existence of strong solutions of the algebraic Riccati equation and the convergence of solutions of the Riccati difference equation to those of the algebraic equation. The results derived utilize detestability conditions only.
TL;DR: In this article, the existence, uniqueness and parametric dependence of solutions of the algebraic Riccati equation are considered. And different criteria for the solubility of this equation are obtained with the help of symplectic algebra.
Abstract: Questions of existence, uniqueness and the parametric dependence of solutions of the algebraic Riccati equation are considered. The different criteria for the solubility of this equation are obtained with the help of symplectic algebra.
TL;DR: In this article, a method for solving the asymmetric coupled Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic games is presented, where the state weighting matrices in player's cost functionals are proportional to each other.
Abstract: A method for solving the asymmetric coupled Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic games is presented. The class of games studied here is one in which the state weighting matrices in player's cost functionals are proportional to each other. By writing in a special order the necessary conditions for open-loop Nash strategy, a matrix with specific properties is derived. These properties are then exploited to solve the two-point boundary-value problem. Some special cases are discussed and a simple example is given to illustrate the solution procedure.
TL;DR: In this article, a wide range of parameter cases for the Ginzburg-Landau equation were considered, including primary and secondary bifurcation, limit cycle solutions, and nonlinear dispersion relations relating spatial and temporal frequencies.
TL;DR: In this article, the optimal control of a class of stochastic parabolic systems is studied, which includes systems with noise depending on spatial derivatives of the state, Neumann boundary control, and Dirichlet boundary observation.
Abstract: The optimal control of a class of stochastic parabolic systems is studied. This class includes systems with noise depending on spatial derivatives of the state, Neumann boundary control, and Dirichlet boundary observation, and extends a class of stochastic systems with distributed control studied by Da Prato [3] and Da Prato and Ichikawa [4]. The work is based on the direct study of the Riccati equation arising in the optimal control problem over finite time horizon. The problem over infinite time horizon and the corresponding algebraic Riccati equation are also considered.
TL;DR: In this article, the Lyapunov and Riccati differential equations with periodically time-varying coefficients are considered under the assumption of detectability of the underlying periodic system and two inertia theorems are provided linking the inertia of the solution to the so-called monodromy matrix.
Abstract: The Lyapunov and Riccati differential equations with periodically time-varying coefficients are considered. Under the assumption of detectability of the underlying periodic system, two inertia theorems are provided linking the inertia of the solution to the one of the so-called monodromy matrix.
TL;DR: In this article, an interpretation of the bilinear transformation of the Riccati equation is presented, and a physical interpretation and derivation of the Bilinear Transformation of a continuous type algebraic R-RICCati equation (ARE) is given.
Abstract: An interpretation of the bilinear transformation of the Riccati equation is presented. The bilinear transformation of a continuous type algebraic Riccati equation (ARE) to a discrete type ARE, and its inverse transformation, have been used for solving ARE numerically. However, its physical meaning has not been clear, and the problem formulation was complicated. This note gives a physical interpretation and derivation of the bilinear transformation of the ARE. Finally, cheap control for continuous-time linear systems is considered from the view of the bilinearly transformed system.
TL;DR: In this paper, the effects of initial stresses, in the form of inplane stress resultants in an equivalent plate model, on the controllability of transverse dynamic response are studied.