Backlund Transformations for the Isospectral and Non-isospectral Matrix Kdv Hierarchies

By transforming the usual Lax pairs of isospectral and non-isospectral matrix Kdv hierarchies into Lax pairs Riccati form, a unified explicit from of Backlund transformations and superposition formulas for these two kinds of hierarchies of equations can be ohtaincd.     

Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation

In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Bäcklund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations.     

Algebraic and Differential Nonlinear Superposition Formulas

Knowledge of the Lax pair and the Darboux transformation for a completely integrable system provides an iterative approach for generating exact solutions. This approach involves solving for the eigenfunction of the Lax pair at each step. But this process can be considerably simplified using the Bäcklund transformation and Bianchi's permutability theorem. This allows constructing the so-called nonlinear superposition formula, which provides a new solution of the system in terms of three previous solutions. The advantage of this approach is that the differential order of the nonlinear superposition formulas is lower than that of the Lax pairs, and in some cases, these formulas reduce to algebraic equations. We consider the construction of new nonlinear superposition formulas in the form of both differential equations and algebraic equations.     

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.     

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.     

Sur le principe de superposition et l'équation de Riccati

In this Note we study the conditions under which a system of ordinary differential equations admits a nonlinear superposition of the solutions, and also the linearization of such systems. A particular study is established for the Riccati equation. To cite this article: S. Rezzag et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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.     

Coupled nonlinear stochastic differential equations

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.     

Global adapted solution of one-dimensional backward stochastic Riccati equations,with application to the mean–variance hedging

Backward stochastic Riccati equations are motivated by the solution of general linear quadratic optimal stochastic control problems with random coefficients, and the solution has been open in the general case. One distinguishing difficult feature is that the drift contains a quadratic term of the second unknown variable. In this paper, we obtain the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation. This solves the one-dimensional case of Bismut–Peng's problem which was initially proposed by Bismut (Lecture Notes in Math. 649 (1978) 180). We use an approximation technique by constructing a sequence of monotone drifts and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean–variance hedging problem with general random market conditions.     

Coupling Bäcklund trasnsformation of Riccati equation and division theorem method for traveling wave solutions of some class of NLPDEs

In this paper, a effective method for searching infinite sequence periodic and solitary wave solutions to nonlinear partial differential equations (NLPDEs) is proposed. A simple transformation technique and division theorem are used to reduce some class of NLPDEs to the Riccati equation, and then the infinite sequence periodic and solitary wave solutions of some class of NLPDEs are constructed by using Bäcklund transformation of Riccati equation and nonlinear superposition principle. As illustrative examples, we obtain the infinite sequence travelling-wave solutions of the three special equations, respectively.     

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.     

Numerical solution of the finite horizon stochastic linear quadratic control problem

The treatment of the stochastic linear quadratic optimal control problem with finite time horizon requires the solution of stochastic differential Riccati equations. We propose efficient numerical methods, which exploit the particular structure and can be applied for large‐scale systems. They are based on numerical methods for ordinary differential equations such as Rosenbrock methods, backward differentiation formulas, and splitting methods. The performance of our approach is tested in numerical experiments.     

Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping.

We show that the existence of a bounded on R ₊ and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem.

If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R ₊ solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

Existence of solutions of two generalized riccati operator equations and their representation

The solutions of two generalized Riccati operator equations are discussed in terms of two critical parameter values, which are related to the application of optimal control under unknown disturbances. Explicit formulas for calculating these two critical parameters as well as the closedform solutions of these two generalized Riccati operator equations are given. The connection between these two parameters and a zero-sum differential game is also investigated.     

On analytical solutions of matrix Riccati equations

Matrix Riccati equations appear in numerous applications, especially in control engineering. In this paper we derive analytical formulas for exact solutions of algebraic and differential matrix Riccati equations. These solutions are expressed in terms of matrix transfer functions of appropriate linear dynamical systems.     

Equivalent Conditions for the Solvability of Nonstandard and Singular LQ-Problems with Application to Parabolic Equations

We consider an optimal control problem with indefinite cost for an abstract model, which covers, in particular, parabolic systems in a general bounded domain. Necessary and sufficient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory extends also a finite-dimensional frequency theorem to the infinite-dimensional setting. Applications include the heat equation with Dirichlet and Neumann controls, as well as the strongly damped Euler–Bernoulli and Kirchhoff equations with the control in various boundary conditions.     

On the direct solution of Riccati equations arising in boundary control theory

Two classes of Riccati equations arising in the boundary control of parabolic systems are studied by direct methods. The new feature with respect to previous works on this subject is the low regularity of the final data. The classes considered here generalize those of [7]and [5]on one side, and of [14]on the other one. Completely new methods are used to obtain the solution of the Riccati equations, in both cases. The central theme is the dependence of the solutions on a «symmetric» norm of the final data, yielding these new results as well as a new proof of existence for the related algebraic Riccati equation under more general assumptions. The synthesis of the associated linear-quadratic-regulator problems is easily solved using these results.     

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.