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Showing papers on "Riccati equation published in 2016"


Journal ArticleDOI
TL;DR: A Max-Min theorem is provided in order to guarantee the saddle-point Nash equilibrium, and when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic, the existence of saddle- point Nash equilibrium is obtained via the solvability of a corresponding Riccati equation.
Abstract: Uncertain differential game investigates interactive decision making of players over time, and the system dynamics is described by an uncertain differential equation. This paper goes further to study the two-player zero-sum uncertain differential game. In order to guarantee the saddle-point Nash equilibrium, a Max–Min theorem is provided. Furthermore, when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic, the existence of saddle-point Nash equilibrium is obtained via the solvability of a corresponding Riccati equation. Finally, a resource extraction problem is analyzed by using the theory proposed in this paper.

139 citations


Journal ArticleDOI
TL;DR: In this paper, a stochastic linear quadratic (LQ) optimal control problem is considered and the notions of open-loop and closed-loop solvabilities are introduced.
Abstract: This paper is concerned with a stochastic linear quadratic (LQ) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue of how negative the control weighting matrix-valued function $R(\cdot)$ can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, some illustrative examples are presented.

129 citations


Journal ArticleDOI
TL;DR: In this paper, exact traveling wave solutions of the Bogoyavlenskii equation using a modified extended tanh-function method were considered. But their method has a broad applicability to many other nonlinear evolution equations in mathematical physics.

122 citations


Journal ArticleDOI
TL;DR: The stabilizing property of the solution to MARE is presented, and the uniqueness is proved for the almost stabilizing and positive semi-definite solution.
Abstract: This technical note deals with a modified algebraic Riccati equation (MARE) and its corresponding inequality and difference equation, which arise in modified optimal control and filtering problems and are introduced into the cooperative control problems recently. The stabilizing property of the solution to MARE is presented. Then, the uniqueness is proved for the almost stabilizing and positive semi-definite solution. Next, the parameter dependence of MARE is analyzed. An obtained parameter dependence result is finally applied to the study of semi-global synchronization of leader-following networks with discrete-time linear dynamics subject to actuator saturation.

93 citations


Posted Content
TL;DR: In this article, the authors compute the characteristic function of the log-price in rough Heston models, where the Riccati equation is replaced by a fractional Riccaci equation.
Abstract: It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing. Using an original link between nearly unstable Hawkes processes and fractional volatility models, we compute the characteristic function of the log-price in rough Heston models. In the classical Heston model, the characteristic function is expressed in terms of the solution of a Riccati equation. Here we show that rough Heston models exhibit quite a similar structure, the Riccati equation being replaced by a fractional Riccati equation.

86 citations


Journal ArticleDOI
TL;DR: A finite-time H∞ controller for uncertain robotic manipulators that achieves high-precision tracking performance without requiring the solution of a Hamilton-Jacobi equation or a Riccati equation is derived using the backstepping method.
Abstract: In this study, a finite-time ${H_\infty }$ controller for uncertain robotic manipulators that achieves high-precision tracking performance without requiring the solution of a Hamilton–Jacobi equation or a Riccati equation is derived using the backstepping method. First, a theory of robust finite-time stability for a class of uncertain nonlinear systems is studied. This theory is then used to develop a simple robust tracking controller for robotic manipulators that provides high precision, strong robustness, and fast response. Not only is the closed-loop system globally finite-time stable, but the L 2 gain is also less than or equal to γ. Simulations and experiments indicate that the proposed control approach is highly effective.

64 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the optimal control problem for a linear conditional McKean-Vlasov equation with quadratic cost functional, where the coefficients of the system and the weigh-ting matrices in the cost functional are adapted processes with respect to the common noise filtration.
Abstract: We consider the optimal control problem for a linear conditional McKean-Vlasov equation with quadratic cost functional. The coefficients of the system and the weigh-ting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration. Semi closed-loop strategies are introduced, and following the dynamic programming approach in [32], we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations. We present several financial applications with explicit solutions, and revisit in particular optimal tracking problems with price impact, and the conditional mean-variance portfolio selection in incomplete market model.

62 citations


Journal ArticleDOI
TL;DR: In this article, the space-time fractional nonlinear Schrodinger equation is solved by mean of on the fractional Riccati expansion method, which can be useful for further understanding of mechanisms of the complicated nonlinear physical phenomena and fractional differential equations.

61 citations


Journal ArticleDOI
TL;DR: In this article, Lie symmetry analysis of the seventh-order time fractional Sawada-Kotera-Ito (FSKI) equation with Riemann-Liouville derivative is performed.
Abstract: In this paper Lie symmetry analysis of the seventh-order time fractional Sawada–Kotera–Ito (FSKI) equation with Riemann–Liouville derivative is performed. Using the Lie point symmetries of FSKI equation, it is shown that it can be transformed into a nonlinear ordinary differential equation of fractional order with a new dependent variable. In the reduced equation the derivative is in Erdelyi–Kober sense. Furthermore, adapting the Ibragimov’s nonlocal conservation method to time fractional partial differential equations, we obtain conservation laws of the underlying equation. In addition, we construct some exact travelling wave solutions for the FSKI equation using the sub-equation method.

61 citations


Journal ArticleDOI
TL;DR: In this paper, the dynamical interactions of a double pendulum arm and an electromechanical shaker are investigated, and the robustness of these two controllers is tested by a sensitivity analysis to parametric uncertainties.
Abstract: In this paper the dynamical interactions of a double pendulum arm and an electromechanical shaker is investigated. The double pendulum is a three degree of freedom system coupled to an RLC circuit based nonlinear shaker through a magnetic field, and the capacitor voltage is a nonlinear function of the instantaneous electric charge. Numerical simulations show the existence of chaotic behavior for some regions in the parameter space and this behaviour is characterized by power spectral density and Lyapunov exponents. The bifurcation diagram is constructed to explore the qualitative behaviour of the system. This kind of electromechanical system is frequently found in robotic systems, and in order to suppress the chaotic motion, the State-Dependent Riccati Equation (SDRE) control and the Nonlinear Saturation control (NSC) techniques are analyzed. The robustness of these two controllers is tested by a sensitivity analysis to parametric uncertainties.

60 citations


Journal ArticleDOI
01 Feb 2016-Optik
TL;DR: In this paper, the Riccati equation was used to construct the soliton solutions for pulse propagation equation with z-dependent coefficients, and the periodic wave and the optical soliton solution were obtained for the pulse propagation equations with dependent coefficients.

Journal ArticleDOI
TL;DR: Using the Lie group analysis method of fractional differential equations, Lie symmetries are derived for the FDLSS equation and conservation laws are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

Book
26 Apr 2016
TL;DR: This paper is a treaty of the infinity-Laplace equation as discussed by the authors, which has many features from the ordinary Laplace Equation and is based on lectures by the author, and has been applied to image processing and to mass transfer problems.
Abstract: This BCAM SpringerBriefs is a treaty of the Infinity-Laplace Equation, which has inherited many features from the ordinary Laplace Equation, and is based on lectures by the author. The Infinity-Laplace Equation has delightful counterparts to the Dirichlet integral, the mean value property, the Brownian motion, Harnack's inequality, and so on. This "fully non-linear" equation has applications to image processing and to mass transfer problems, and it provides optimal Lipschitz extensions of boundary values.

Journal ArticleDOI
TL;DR: It is shown that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view.
Abstract: In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse, and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton--Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual, and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure. In the context of linear-quadratic regulator problems, we show that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view. Finally, the new results provide theoretical ground for recently proposed modifications of projection methods onto rational Krylov subspaces.

Journal ArticleDOI
TL;DR: In this paper, an extended Fan sub-equation method is used to seek some new and more general traveling wave solutions of nonlinear Schrodinger equation (NLSE), which is preferable to use this method to solve NLSE because this method gives us all the solutions obtained previously by the application of at least four methods (the method of using Riccati equation, or auxiliary ordinary differential equation method, or first kind elliptic equation or the generalized Ricciati equation as mapping equation) in a unified manner.
Abstract: An extended Fan sub-equation method is used to seek some new and more general traveling wave solutions of nonlinear Schrodinger equation (NLSE). The important fact of this method is to take the full advantage of clear relationship among general elliptic equation involving five parameters and other existing sub-equations involving three parameters. It is preferable to use this method to solve NLSE because this method gives us all the solutions obtained previously by the application of at least four methods (the method of using Riccati equation, or auxiliary ordinary differential equation method, or first kind elliptic equation or the generalized Riccati equation as mapping equation) in a unified manner. So it is shown that this method is concise and its applications are promising.

Journal ArticleDOI
TL;DR: Some results on existence of solutions for a quadratic Volterra integral equation of fractional order in two independent variables are presented and it is shown that solutions of this integral equation are locally attractive.

Journal ArticleDOI
TL;DR: The resulting fault-tolerant compensation control scheme is designed based on the closed- loop systems, and therefore has more practical significance than the existing FTC methodologies developed in terms of the open-loop systems.
Abstract: For digital proportional–integral–derivative control systems with unknown dynamics, the data-driven output-feedback fault-tolerant control (FTC) problem is studied in this paper. In a framework of active FTC, the issue of online recursive identification of the residual generator, the state observer, and the observability canonical form of the plant under consideration is addressed; the problem of reconfiguration of the data-driven fault-tolerant compensation controller with $L_2$ -gain properties is also dealt with by means of the above-obtained results, the prefilter and the Riccati equation related to $H_{\infty }$ control so as to accommodate faults and ensure tracking performance. The resulting fault-tolerant compensation control scheme is designed based on the closed-loop systems, and therefore has more practical significance than the existing FTC methodologies developed in terms of the open-loop systems. Finally, the effectiveness of the proposed FTC approach is validated by the speed control experiment on a dc motor.

Journal ArticleDOI
TL;DR: In this article, the generalized Riccati equation mapping method was used to solve a continuous nonlinear model associated with the previous nonlinear transmission line and obtain miscellaneous travelling wave solutions including trigonometric, hyperbolic, and rational functions.
Abstract: In this paper, we investigate exact soliton solutions to a nonlinear transmission line. Using a concise and simple method known as the generalized Riccati equation mapping method, we solve a continuous nonlinear model associated with the previous nonlinear transmission line. As a result, we obtain miscellaneous travelling wave solutions including trigonometric, hyperbolic, and rational functions.

Journal ArticleDOI
TL;DR: This paper is devoted to both theoretical and numerical study of Riccati equation with fractional order and a formulation to the fractional-order Legendre operational matrix of fractional integration is constructed.

Journal ArticleDOI
TL;DR: In this paper, the exact solution of some nonlinear partial differential equations (NLPDEs) such as, Kodomtsev-Petviashvili (KP) equation, the (2 + 1)-dimensional breaking soliton equation and the modified generalized Vakhnenko equation by using the simple equation method is presented.

Journal ArticleDOI
TL;DR: In this article, the decay of the entries of the Riccati matrix is exploited to develop efficient methods for approximating the solution of the Lyapunov equation by a sparse matrix, where the coefficient matrices A and P are large symmetric banded matrices.

Journal ArticleDOI
TL;DR: An easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation is developed.
Abstract: This paper investigates the output regulation problem for a class of regular first-order hyperbolic partial differential equation (PDE) systems. A state feedback and an error feedback regulator are considered to force the output of the hyperbolic PDE plant to track a periodic reference trajectory generated by a neutrally stable exosystem. A new explanation is given to extend the results in the literature to solve the regulation problem associated with the first-order hyperbolic PDE systems. Moreover, in order to provide the closed-loop stability condition for the solvability of the regulator problems, the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper. This paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation.

Journal ArticleDOI
TL;DR: In this article, a physically important object, called the horizon function, is introduced in the study of geodesic collapse, which is closely related to the stellar characteristics and satisfies a simple Riccati equation.
Abstract: We introduce a physically important object, called the horizon function, in the study of geodesic collapse. It is closely related to the stellar characteristics and satisfies a simple Riccati equation. This equation is integrated and all of its solutions are found in terms of some generating function. Previous solutions are regained and further investigated.

Journal ArticleDOI
TL;DR: A contraction analysis of risk-sensitive Riccati equations is proposed, which can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.
Abstract: A contraction analysis of risk-sensitive Riccati equations is proposed. When the state-space model is reachable and observable, a block-update implementation of the risk-sensitive filter is used to show that the $N$-fold composition of the Riccati map is strictly contractive with respect to the Thompson's part metric of positive definite matrices, when $N$ is larger than the number of states. The range of values of the risk-sensitivity parameter for which the map remains contractive can be estimated a priori. It is also found that a second condition must be imposed on the risk-sensitivity parameter and on the initial error variance to ensure that the solution of the risk-sensitive Riccati equation remains positive definite at all times. The two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.

Journal ArticleDOI
TL;DR: In this article, the general theory of open quantum systems in the Gaussian regime is studied and a number of diverse ramifications and consequences of the theory are explored. But the main focus is on the parametrization of the most general Gaussian completely positive map, which is missing in the existing literature.
Abstract: This article focuses on the general theory of open quantum systems in the Gaussian regime and explores a number of diverse ramifications and consequences of the theory. We shall first introduce the Gaussian framework in its full generality, including a classification of Gaussian (also known as ‘general-dyne’) quantum measurements. In doing so, we will give a compact proof for the parametrisation of the most general Gaussian completely positive map, which we believe to be missing in the existing literature. We will then move on to consider the linear coupling with a white noise bath, and derive the diffusion equations that describe the evolution of Gaussian states under such circumstances. Starting from these equations, we outline a constructive method to derive general master equations that apply outside the Gaussian regime. Next, we include the general-dyne monitoring of the environmental degrees of freedom and recover the Riccati equation for the conditional evolution of Gaussian states. Our derivation ...

Journal ArticleDOI
TL;DR: In this article, the sensitivity to the choice of initial conditions in the evolution of Gaussian wave packets via the nonlinear Riccati equation has been investigated and the effects of the environment on the quantum uncertainties, correlation function, quantum energy contribution and tunnelling currents have been shown.

Journal ArticleDOI
TL;DR: Novel event-triggered low-gain and high–low-gain control algorithms based on Riccati equations are proposed to achieve semi-global stabilisation of null controllable systems subject to actuator saturation.
Abstract: This paper investigates the problem of event-triggered control for semi-global stabilisation of null controllable systems subject to actuator saturation. First, for a continuous-time system, novel event-triggered low-gain control algorithms based on Riccati equations are proposed to achieve semi-global stabilisation. The algebraic Riccati equation with a low-gain parameter is utilised to design both the event-triggering condition and the linear controller; a minimum inter-event time based on the Riccati ordinary differential equation is set a priori to exclude the Zeno behaviour. In addition, the high–low gain techniques are utilised to extend the semi-global results to event-based global stabilisation. Furthermore, for a discrete-time system, novel low-gain and high–low-gain control algorithms are proposed to achieve event-triggered stabilisation. Numerical examples are provided to illustrate the theoretical results.

Journal ArticleDOI
Yun-kai Liu1, Biao Li1
TL;DR: For the (2+1)-dimensional Gardner equation, the truncated Painleve method was developed to obtain the nonlocal residual symmetry and Backlund transformation in this paper, and the symmetry group transformation can be computed from the extended system.

Journal ArticleDOI
13 Feb 2016-Pramana
TL;DR: In this article, two integration schemes are employed to obtain solitons, singular periodic waves and other types of solutions of the Drinfel-d-Sokolov-Wilson equation.
Abstract: In this paper, two integration schemes are employed to obtain solitons, singular periodic waves and other types of solutions of the Drinfel’d–Sokolov–Wilson equation. The two schemes studied in this paper are the Backlund transformation of Riccati equation and the trial function approach. The corresponding constraint conditions of the solutions are also given.

Journal ArticleDOI
TL;DR: In this article, the Riccati equation is used to obtain bounds for modified Bessel functions of consecutive orders, and a procedure is considered in which the bounds obtained from the analysis of the RCE are used to define a new function satisfying a new RCE, which yields sharper bounds.