Showing papers on "Riccati equation published in 2010"
17 Aug 2010
TL;DR: The Fokker-Planck equation as mentioned in this paper describes the evolution of conditional probability density for given initial states for a Markov process, which satisfies the Ito stochastic differential equation.
Abstract: In 1984, H. Risken authored a book (H. Risken, The Fokker-Planck Equation: Methods of Solution, Applications, Springer-Verlag, Berlin, New York) discussing the Fokker-Planck equation for one variable, several variables, methods of solution and its applications, especially dealing with laser statistics. There has been a considerable progress on the topic as well as the topic has received greater clarity. For these reasons, it seems worthwhile again to summarize previous as well as recent developments, spread in literature, on the topic. The Fokker-Planck equation describes the evolution of conditional probability density for given initial states for a Markov process, which satisfies the Ito stochastic differential equation. The structure of the Fokker-Planck equation for the vector case is
1,762 citations
TL;DR: The control of each agent using local information is designed and detailed analysis of the leader-following consensus is presented for both fixed and switching interaction topologies, which describe the information exchange between the multi-agent systems.
1,252 citations
TL;DR: In this article, a traveling-wave solution of the class of equations ∑ p = 1 n 1 α p ∂ p Q ∂ t p + ∑ q = 1 N 2 β q ∂ q Q ∆ x q + ∆ m = 1 M μ m Q m = 0 where α p, β q and μ m are parameters.
203 citations
TL;DR: A general p-shift linear optimal finite impulse response (FIR) estimator intended for solving universally the problems of filtering, smoothing, and prediction of discrete time-invariant models in state space is addressed.
Abstract: This paper addresses a general p-shift linear optimal finite impulse response (FIR) estimator intended for solving universally the problems of filtering (p = 0), smoothing (p 0) of discrete time-invariant models in state space. An optimal solution is found in the batch form with the initial mean square state function self-determined by solving the discrete algebraic Riccati equation. An unbiased solution represented both in the batch and recursive forms does not involve any knowledge about noise and initial state. The mean square errors in both the optimal and unbiased estimates are found via the noise power gain (NPG) and a recursive algorithm for fast computation of the NPG is supplied. Applications are given for FIR filtering with fixed, receding, and full averaging horizons.
169 citations
TL;DR: It is shown that a unified optimal solution to the FDF can be obtained by solving the discrete time Riccati equation and the optimal FDF is not unique.
135 citations
TL;DR: In this paper, a tensor potential is brought into a well-known form of Schrodinger-like problem possessing known solutions via the methodology of supersymmetry (SUSY).
105 citations
01 Jan 2010
103 citations
29 Jul 2010
TL;DR: In this paper, a new observer design technique for Lipschitz nonlinear systems is presented, where necessary and sufficient conditions for existence of a stable observer gain are developed using a S-Procedure Lemma.
Abstract: This paper presents a new observer design technique for Lipschitz nonlinear systems. Necessary and sufficient conditions for existence of a stable observer gain are developed using a S-Procedure Lemma. The developed condition is expressed in terms of the existence of a solution to an Algebraic Riccati Equation in one variable. Thus, the need to solve Linear Matrix Inequalities in multiple variables is eliminated. The advantage of the developed approach is that it is significantly less conservative than other previously published results for Lipschitz systems. It yields a stable observer for larger Lipschitz constants than other techniques previously published in literature.
96 citations
TL;DR: In this scheme, particle swarm optimization is used as a tool for the rapid global search method, and simulating annealing for efficient local search, and the scheme is equally capable of solving the integer order or fractional order Riccati differential equations.
Abstract: In this article, a stochastic technique has been developed for the solution of nonlinear Riccati differential equation of fractional order. Feed-forward artificial neural network is employed for accurate mathematical modeling and learning of its weights is made with heuristic computational algorithm based on swarm intelligence. In this scheme, particle swarm optimization is used as a tool for the rapid global search method, and simulating annealing for efficient local search. The scheme is equally capable of solving the integer order or fractional order Riccati differential equations. Comparison of results was made with standard approximate analytic, as well as, stochastic numerical solvers and exact solutions.
91 citations
TL;DR: In this technical note, a solution of the matrix differential Riccati equation that plays an important role in the linear quadratic optimal control problem is investigated and the approach employs the negative definite anti-stabilizing solution ofThe matrix algebraic RicCati equation and the solution ofthe matrix differential Lyapunov equation.
Abstract: In this technical note, we investigate a solution of the matrix differential Riccati equation that plays an important role in the linear quadratic optimal control problem. Unlike many methods in the literature, the approach that we propose employs the negative definite anti-stabilizing solution of the matrix algebraic Riccati equation and the solution of the matrix differential Lyapunov equation. An illustrative numerical example is provided to show the efficiency of our approach.
86 citations
TL;DR: In this paper, the authors studied the boundary stabilization of the Navier-Stokes equations about an unstable stationary solution by controlling finite dimension in feedback form, and they showed that the linear feedback control law is determined by solving an optimal control problem of finite dimension.
Abstract: We study the boundary stabilization of the two-dimensional
Navier-Stokes equations about an unstable stationary solution
by controls of finite dimension in feedback form. The main novelty
is that the linear feedback control law is determined by solving
an optimal control problem of finite dimension. More precisely,
we show that, to stabilize locally the Navier-Stokes equations,
it is sufficient to look for a boundary feedback control of
finite dimension, able to stabilize the projection of
the linearized equation onto the unstable subspace of
the linearized Navier-Stokes operator. The feedback operator
is obtained by solving an algebraic Riccati equation in a space
of finite dimension, that is to say a matrix Riccati equation.
TL;DR: Modification of truncated expansion method is used for obtaining exact solution of the Kudryashov–Sinelshchikov equation for describing the pressure waves in liquid with gas bubbles.
TL;DR: In this paper, sufficient conditions for the existence of a solution to the problem u ∈ [0, ω ], u ( 0 ) = u ( ω ), u ǫ ( 0) = uǫ( ω ).
TL;DR: In this paper, Sinc-Collocation method for solving Lane-Emden Equation (LESE) is proposed and it is found that Sinc procedure converges with the solution at an exponential rate.
TL;DR: In this paper, a new form of homotopy perturbation method (NHPM) has been adopted for solving the quadratic Riccati differential equation, where the solution is considered as a Taylor series expansion converges rapidly to the exact solution of the nonlinear equation.
TL;DR: A hybrid method which combines the Adomian decomposition method, the Laplace transform algorithm and the Pade approximant is introduced to solve the approximate analytic solutions of the nonlinear Riccati differential equations.
Abstract: In this paper, a hybrid method which combines the Adomian decomposition method (ADM), the Laplace transform algorithm and the Pade approximant is introduced to solve the approximate analytic solutions of the nonlinear Riccati differential equations. This hybrid method demonstrates accurate and reliable results, and has a great improvement in the ADM truncated series solution which diverges rapidly as the applicable domain increases. Three examples herein are given to demonstrate good accuracy and fast convergence in comparison with the exact solution.
01 Jan 2010
TL;DR: In this paper, the results derived by differential transform method were compared with the results of homotopy analysis method and Adomian decomposition method and an efficient recurrence relation for solving these equations was obtained.
Abstract: In this article differential transform method (DTM) is considered to solve quadratic Riccati dif- ferential equation. The results derived by differential transform method will be compared with the results of homotopy analysis method and Adomian decomposition method. It would be shown that this method used for quadratic Riccati differential equation is more effective and promising than homotopy analysis method and Adomain decomposition method. An efficient recurrence relation for solving these equations will be obtained.
TL;DR: The homogeneous balance method and symbolic computation are used to construct new exact traveling wave solutions for the Benjamin–Bona–Mahoney (BBM) equation, which contain rational and periodic-like solutions.
TL;DR: In this article, a variational iteration method for solving fractional Riccati differential equation is proposed, which is based on the use of Lagrange multipliers for determining optimal value of a parameter in a functional.
Abstract: We will consider He's variational iteration method for solving fractional
Riccati differential equation. This method is based on the use of Lagrange multipliers for
identification of optimal value of a parameter in a functional. This technique provides a
sequence of functions which converges to the exact solution of the problem. The present
method performs extremely well in terms of efficiency and simplicity.
TL;DR: Two numerical techniques are presented for solving the solution of Riccati differential equation by expanding the required approximate solution as the elements of cubic B-spline scaling function or Chebyshev cardinal functions.
TL;DR: It is proved that this system coupling the incompressible Navier-Stokes equations in a 2D rectangular-type domain with a damped Euler-Bernoulli beam equation is exponentially stabilizable, locally about the null solution, with any prescribed decay rate, by a feedback control corresponding to a force term in the beam equation.
Abstract: We study a system coupling the incompressible Navier-Stokes equations in a 2D rectangular-type domain with a damped Euler-Bernoulli beam equation, where the beam is a part of the upper boundary of the domain occupied by the fluid. Due to the deformation of the beam, the fluid domain depends on time. We prove that this system is exponentially stabilizable, locally about the null solution, with any prescribed decay rate, by a feedback control corresponding to a force term in the beam equation. The feedback is applied on the whole structure, and it is determined, via a Riccati equation, by solving an infinite time horizon control problem for the linearized model. A crucial step in this analysis consists of showing that this linearized system can be rewritten thanks to an analytic semigroup of which the infinitesimal generator has a compact resolvent.
TL;DR: An improved Lyapunov–Krasovskii functional combined with Newton–Leibniz formula, a memoryless switched controller design for exponential stabilization of switched systems with time-varying delays is proposed.
Abstract: This paper deals with the problem of stabilization for a class of hybrid systems with time-varying delays. The system to be considered is with nonlinear perturbation and the delay is time varying in both the state and control. Using an improved Lyapunov–Krasovskii functional combined with Newton–Leibniz formula, a memoryless switched controller design for exponential stabilization of switched systems is proposed. The conditions for the exponential stabilization are presented in terms of the solution of matrix Riccati equations, which allow for an arbitrary prescribed stability degree.
Journal Article•
TL;DR: In this article, the traveling wave solutions involving parameters of nonlinear evolution equations in the mathematical physics via the (3+1)- dimensional potential- YTSF equation, the ( 3+ 1)- dimensional generalized shallow water equation, a Kadomtsev- Petviashvili equation, and a Jimbo-Miwa equation are derived from the travelling waves.
Abstract: In the present paper, we construct the traveling wave solutions involving parameters of nonlinear evolution equations in the mathematical physics via the (3+1)- dimensional potential- YTSF equation, the (3+1)- dimensional generalized shallow water equation, the (3+1)- dimensional Kadomtsev- Petviashvili equation, the (3+1)- dimensional modified KdV-Zakharov- Kuznetsev equation and the (3+1)- dimensional Jimbo-Miwa equation by using a simple method which is called the ()- expansion method, where satisfies a second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the travelling waves. The travelling wave solutions are expressed by hyperbolic, trigonometric and rational functions.
TL;DR: The Hyers–Ulam stability of the linear differential equation of higher order with constant coefficients in Aoki–Rassias sense is obtained and a connection with dynamical sytems perturbation is established.
TL;DR: A modified variational iteration method for solving Riccati differential equations that can enlarge the convergence region of iterative approximate solutions and give good approximations for a larger interval, rather than a local vicinity of the initial position.
Abstract: In this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. The solutions of Riccati differential equations obtained using the traditional variational iteration method (VIM) give good approximations only in the neighborhood of the initial position. The main advantage of the present method is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval, rather than a local vicinity of the initial position. Numerical results show that the method is simple and effective.
01 Dec 2010
TL;DR: An Approximate/Adaptive Dynamic Programming (ADP) algorithm that finds online the Nash equilibrium for two-player nonzero-sum differential games with linear dynamics and infinite horizon quadratic cost is presented.
Abstract: This paper presents an Approximate/Adaptive Dynamic Programming (ADP) algorithm that finds online the Nash equilibrium for two-player nonzero-sum differential games with linear dynamics and infinite horizon quadratic cost. Each of the game players is using the procedure of Integral Reinforcement Learning (IRL) to calculate online the infinite horizon value function that it associates with every given set of feedback control policies. It will be shown that the online algorithm is mathematically equivalent to an offline iterative method, previously introduced in the literature, that solves the set of coupled algebraic Riccati equations (ARE) underlying the game problem using complete knowledge on the system dynamics. Here we show how the ADP techniques will enhance the capabilities of the offline method allowing an online solution without the requirement of complete knowledge of the system dynamics. The two participants in the continuous-time differential game are competing in real-time and the feedback Nash control strategies will be determined based on online measured data from the system. The algorithm is built on interplay between a learning phase, where each of the players is learning online the value that they associate with a given set of play policies, and a policy update step, performed by each of the payers towards decreasing the value of their cost. The players are learning concurrently. The feasibility of the ADP scheme is demonstrated in simulation.
TL;DR: In this paper, it was shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of non-negative solution for a steady state advection type linear partial differential equation.
TL;DR: In this paper, an analytic solution beyond adiabatic approximation was obtained by transferring the 1D Schrodinger equation into the Ricatti equation, which is more accurate than JWKB approximation.
Abstract: We obtain an analytic solution beyond adiabatic approximation by transferring the 1D Schrodinger equation into the Ricatti equation. Then we show that our solution is more accurate than JWKB approximation. The generalizations of the approach to 3D are suggested, and possible applications of obtained solutions are discussed.
TL;DR: It is shown that the structure preserving doubling algorithm of Anderson is in fact the cyclic reduction algorithm of Hockney and Buzbee and applied to a suitable UQME, and a new algorithm obtained by complementing the transformations with the shrink-and-shift technique of Ramaswami.
Abstract: The problem of reducing an algebraic Riccati equation XCX − AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind PX 2 + QX + R = 0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Anderson (Int J Control 28(2):295–306, 1978) is in fact the cyclic reduction algorithm of Hockney (J Assoc Comput Mach 12:95–113, 1965) and Buzbee et al. (SIAM J Numer Anal 7:627–656, 1970), applied to a suitable UQME. A new algorithm obtained by complementing our transformations with the shrink-and-shift technique of Ramaswami is presented. The new algorithm is accurate and much faster than SDA when applied to some examples concerning fluid queue models.
TL;DR: In this paper, a two-component generalization of the generalized Hunter-Saxton equation was proposed, which can be viewed as a bi-variational Euler equation, and is shown to be bi-hamiltonian.
Abstract: In this paper, we propose a two-component generalization of the generalized Hunter-Saxton equation obtained in \cite{BLG2008}. We will show that this equation is a bihamiltonian Euler equation, and also can be viewed as a bi-variational equation.