THE SOLUTION FOR THE GENERALIZED RICCATIALGEBRAIC EQUATIONS OF LINEAR EQUALITY CONSTRAINT SYSTEM

Based on the dynamic equation, the performance functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper.

Conservation Laws of the Differential-Difference KP Equation

An infinite number of semi-discrete and continuous conservation laws for the differential-difference KP equation were obtained by using a solvable generalized Riccati equation.

Soliton solutions for nonlinear variable-order fractional Korteweg-de Vries(KdV)equation arising in shallow water waves

Nonlinear fractional differential equations provide suitable models to describe real-world phenomena and many fractional derivatives are varying with time and space.The present study considers the advanced and broad spectrum of the nonlinear(NL)variable-order fractional differential equation(VO-FDE)in sense of VO Caputo fractional derivative(CFD)to describe the physical models.The VO-FDE transforms into an ordinary differential equation(ODE)and then solving by the modified(G/G)-expansion method.For ac-curacy,the space-time VO fractional Korteweg-de Vries(KdV)equation is solved by the proposed method and obtained some new types of periodic wave,singular,and Kink exact solutions.The newly obtained solutions confirmed that the proposed method is well-ordered and capable implement to find a class of NL-VO equations.The VO non-integer performance of the solutions is studied broadly by using 2D and 3D graphical representation.The results revealed that the NL VO-FDEs are highly active,functional and convenient in explaining the problems in scientific physics.

(2+1)维Burgers方程新的复合解

In this paper, a new generalized compound Riccati equations rational expansion method （GCRERE） is proposed. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method is not only recover some known solutions, but also find some new and general complexiton solutions. Being concise and straightforward, it is applied to the （2＋1）-dimensional Burgers equation. As a result, eight families of new exact analytical solutions for this equation are found. The method can also be applied to other nonlinear partial differential equations.

Time-inconsistent stochastic linear-quadratic control problem with indefinite control weight costs

A time-inconsistent linear-quadratic optimal control problem for stochastic differential equations is studied.We introduce conditions where the control cost weighting matrix is possibly singular.Under such conditions,we obtain a family of closed-loop equilibrium strategies via multi-person differential games.This result extends Yong’s work(2017) in the case of stochastic differential equations,where a unique closed-loop equilibrium strategy can be derived under standard conditions(namely,the control cost weighting matrix is uniformly positive definite,and the other weighting coefficients are positive semidefinite).

Analytical and approximate solutions of nonlinear Schrödinger equation with higher dimension in the anomalous dispersion regime

The generalized Riccati equation mapping method(GREMM)is used in this paper to obtain different types of soliton solutions for nonlinear Schrödinger equation with higher dimension that existed in the regimes of anomalous dispersion.Later,we use the q-homotopy analysis method combined with the Laplace transform(q-HATM)to obtain approximate solutions of the bright and dark optical solitons.The q-HATM illustrates the solutions as a rapid convergent series.In addition,to show the physical behavior of the solutions obtained by the proposed techniques,the graphical representation has been provided with some parameter values.The findings demonstrate that the proposed techniques are useful,efficient and reliable mathematical method for the extraction of soliton solutions.

New sub-equation method to construct solitons and other solutions for perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials

In a previous work,Zayed and Al-Nowehy have applied the Riccati equation mapping method combined with the generalized extended(G/G)-expansion method and found new exact solutions of the nonlinear KPP equation.In the present article,we propose a different method,namely,a new sub-equation method consists of the Riccati equation mapping method and the(G/G,1/G)-expansion method to find new exact solutions of the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials.This proposed method is not found elsewhere.Hyperbolic,trigonometric and rational function solutions are given.New solutions of the generalized Riccati equation are presented for the first time which are not reported previously.The solutions of the given nonlinear equation can be applied in ocean engineering for calculating the height of tides in the ocean.

Data-Driven Direct Adaptive Risk-Sensitive Control of Stochastic Systems

The authors propose a data-driven direct adaptive control law based on the adaptive dynamic programming(ADP) algorithm for continuous-time stochastic linear systems with partially unknown system dynamics and infinite horizon quadratic risk-sensitive indices.The authors use online data of the system to iteratively solve the generalized algebraic Riccati equation(GARE) and to learn the optimal control law directly.For the case with measurable system noises,the authors show that the adaptive control law approximates the optimal control law as time goes on.For the case with unmeasurable system noises,the authors use the least-square solution calculated only from the measurable data instead of the real solution of the regression equation to iteratively solve the GARE.The authors also study the influences of the intensity of the system noises,the intensity of the exploration noises,the initial iterative matrix,and the sampling period on the convergence of the ADP algorithm.Finally,the authors present two numerical simulation examples to demonstrate the effectiveness of the proposed algorithms.

Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

By taking advantage of three different computational analytical methods:the Lie symmetry analysis,the generalized Riccati equation mapping approach,and the modified Kudryashov approach,we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation(NCDR)with power-law nonlinearity and density-dependent diffusion.Lie symmetry analysis is one of the pow-erful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables.By the Lie group technique,we obtain one-parameter in-variant transformations,determining equations and corresponding vectors for the considered convection-diffusion-reaction equation.By treating the parameters of the governing equation as constants,the ap-plied methods yield a variety of new closed-form solutions,including inverse function solutions,periodic solutions,exponential function solutions,dark solitons,singular solitons,combo bright-singular solitons,and the combine of bright-dark solitons and dark-bright solitons.Moreover,using the Bäcklund transfor-mation of the generalized Riccati equation and modified Kudryashov method,we can construct multiple solitons and other solutions of the considered equation.The obtained new solutions of this work demon-strate that the used approaches are powerful and effective in dealing with nonlinear equations,and that these solutions are required to explain many biological and physical phenomena.Comparing our obtained solutions of this paper with the ones obtained in the literature,we see that our solutions are new and not reported elsewhere.These newly formed soliton solutions will be more beneficial in the various dis-ciplines of ocean engineering,plasma physics,and nonlinear sciences.

Feedback Stackelberg Solution for Mean-Field Type Stochastic Systems with Multiple Followers

This paper discusses feedback Stackelberg strategies for the continuous-time mean-field type stochastic systems with multiple followers in infinite horizon.First,optimal control problems of the followers are studied in the sense of Nash equilibrium.With the help of a set of generalized algebraic Riccati equations(GAREs),sufficient conditions for the solvability are put forward.Then,the leader faces a constrained optimal control problem by transforming the cost functional into a trace criterion.Employing the Karush-Kuhn-Tucker(KKT)conditions,necessary conditions are presented in term of the solvability of the cross-coupled stochastic algebraic equations(CSAEs).Moreover,feedback Stackelberg strategies are obtained based on the solutions of the CSAEs.In addition,an iterative scheme is introduced to obtain efficiently the solutions of the CSAEs.Finally,an example is given to shed light on the effectiveness of the proposed results.

Infinite horizon indefinite stochastic linear quadratic control for discrete-time systems

This paper discusses discrete-time stochastic linear quadratic （LQ） problem in the infinite horizon with state and control dependent noise, where the weighting matrices in the cost function are assumed to be indefinite. The problem gives rise to a generalized algebraic Riccati equation （GARE） that involves equality and inequality constraints. The well-posedness of the indefinite LQ problem is shown to be equivalent to the feasibility of a linear matrix inequality （LMI）. Moreover, the existence of a stabilizing solution to the GARE is equivalent to the attainability of the LQ problem. All the optimal controls are obtained in terms of the solution to the GARE. Finally, we give an LMI -based approach to solve the GARE via a semidefinite programming.