Generalized cofactors and nonlinear superposition principles

It is known from Lie's works that the only ordinary differential equation of first order in which the knowledge of a certain number of particular solutions allows the construction of a fundamental set of solutions is, excepting changes of variables, the Riccati equation. For planar complex polynomial differential systems, the classical Darboux integrability theory exists based on the fact that a sufficient number of invariant algebraic curves permits the construction of a first integral or an inverse integrating factor. In this paper, we present a generalization of the Darboux integrability theory based on the definition of generalized cofactors.     

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.     

Analysis of Nonlinear Quadratic Control in Infinite Time

This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrices, function of the state variable. Dynamic constraints are represented by a bilinear differential systems of the form. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, respectively the adjoint system corresponding to those functions. Thus it will be obtained a matrix differential equation where the solution representing the positive defined symmetric matrix P ( x ), verifies the Riccati algebraic equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control

In this paper, we investigate the synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. Using a combination of Riccati differential equation approach, Lyapunov-Krasovskii functional, inequality techniques, some sufficient conditions for exponentially stability of the error system are formulated in form of a solution to the standard Riccati differential equation. The designed controller ensures that the synchronization of non-autonomous chaotic systems are proposed via delayed feedback control and intermittent linear state delayed feedback control. Numerical simulations are presented to illustrate the effectiveness of these synchronization criteria.     

Integrability theorems for a class of integral transforms

It is shown under weak hypotheses that systems of 2n linear differential equations in 2n variables generate sets of identities similar in structure to the classical trigonometric identities. For clarity of exposition only the case n = 1 is actually treated, but all final equations are written in such a manner as to be directly applicable to matrix systems (n > 1). These identities allow one to avoid, in a very simple way, certain difficulties which often occur in the integration of the Riccati equations arising from application of the invariant imbedding method to two point boundary value problems associated with such linear systems. The overall usefulness of the imbedding method is thereby considerably extended. One analytical and one numerical example are given to illustrate the actual use of these identities.     

Riccati equation arising in the boundary control of stochastic hyperbolic systems

A Riccati equation of stochastic control theory is studied directly. The equation arises in the synthesis of a linear quadratic regulator problem for systems governed by stochastic partial differential equations of hyperbolic type, with contorl acting on the boundary through Dirichlet of Neumann conditions     

A modification of the invariant imbedding method for a singular boundary value problem

We consider a dynamically-consistent analytical model of a 3D topographic vortex. The model is governed by equations derived from the classical problem of the axisymmetric Taylor–Couette flow. Using linear expansions, these equations can be reduced to a differential sixth-order equation with variable coefficients. For this differential equation, we formulate a boundary value problem, which has a number of issues for numerical solving. To avoid these issues and find the eigenvalues and eigenfunctions of the boundary value problem, we suggest a modification of the invariant imbedding method (the Riccati equation method). In this paper, we show that such a modification is necessary since the boundary conditions possess singular matrices, which sufficiently complicate the derivation of the Riccati equation. We suggest algebraic manipulations, which permit the initial problem to be reduced to a problem with regular boundary conditions. Also, we propose a method for obtaining a numerical solution of the matrix Riccati equation by means of recurrence relations, which allow us to obtain a matrizer converging to the required eigenfunction. The suggested method is tested by calculating the corresponding eigenvalues and eigenfunctions, and then, by constructing fluid particle trajectories on the basis of the eigenfunctions.     

Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping

In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered.The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen.This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type.Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions.Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too.Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves.     

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.     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.     

ϵ-Optimal and Optimal Controls for the Stochastic Linear-Quadratic Problem

We study the stochastic regulator problem in Hilbert spaces for systems governed by linear stochastic differential equations with retarded controls and with state and control dependent noise. We use integral Riccati equations and no reference to a Riccati differential equation or to the Ito formula is made.     

Classical Solutions of Nonautonomous Riccati Equations Arising in Parabolic Boundary Control Problems

An abstract linear-quadratic regulator problem over finite time horizon is considered; it covers a large class of linear nonautonomous parabolic systems in bounded domains, with boundary control of Dirichlet or Neumann type. The associated differential Riccati equation is studied from the point of view of semigroup theory; it is shown to have a classical, explicitly represented solution for very general final data; weighted H?lder regularity results for the optimal pair are deduced. Accepted 10 December 1997     

Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

In this paper, we consider a linear–quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward–backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear–quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.     

Existence and comparison theorems for partial differential equations of Riccati type

In this paper, we discuss the partial differential equation of Riccati type that describes the optimal filtering error covariance function for a linear distributed-parameter system with pointwise observations. Since this equation contains the Dirac delta function, it is impossible to apply directly the usual methods of functional analysis to prove existence and uniqueness of a bounded solution. By using properties of the fundamental solution and the classical technique of successive approximation, we prove the existence and uniqueness theorem. We then prove the comparison theorem for partial differential equations of Riccati type. Finally, we consider some applications of these theorems to the distributed-parameter optimal sensor location problem.     

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.     

Solutions of Riccati-Abel equation in terms of third order trigonometric functions

Solutions of the generalized Riccati equations with third order nonlinearity, named as Riccati-Abel equation, are expressed via third order trigonometric functions. It is shown, as the ordinary Riccati equation, also the Riccati-Abel equation has a relationship with a linear differential equations. A summation formula for solutions of Riccati-Abel equation is established. Possible applications of this formula in the generalized dynamics is outlined. The method admits an extension to the case of generalized Riccati equations with any order of nonlinearity     

伊藤-泊松型随机微分方程的线性二次控制

本文研究伊藤-泊松型随机微分方程的线性二次控制问题,利用动态规划方法、伊藤公式等技巧,通过解HJB方程,我们得到了随机Riccati方程及另外两个微分方程,求出控制变量,解决了线性二次最优控制最优问题.     

Ibragimov-type invariants for a system of two linear parabolic equations

We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented.