Showing papers on "Riccati equation published in 2009"

A general approach for the exact solution of the schrodinger equation

Abstract: The Schrodinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schrodinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.

The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics

Abstract: I the present paper, we construct the traveling wave solutions involving parameters of the combined Korteweg-de Vries–modified Korteweg-de Vries equation, the reaction-diffusion equation, the compound KdV–Burgers equation, and the generalized shallow water wave equation by using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters take special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions.

On the stability of functional equations

Abstract: We give some theorems on the stability of the equation of homomorphism, of Lobacevski’s equation, of almost Jensen’s equation, of Jensen’s equation, of Pexider’s equation, of linear equations, of Schroder’s equation, of Sincov’s equation, of modified equations of homomorphism from a group (not necessarily commutative) into a $${\mathbb{Q}}$$ -topological sequentially complete vector space or into a Banach space, of the quadratic equation, of the equation of a generalized involution, of the equation of idempotency and of the translation equation. We prove that the different definitions of stability are equivalent for the majority of these equations. The boundedness stability and the stability of differential equations and the anomalies of stability are considered and open problems are formulated too.

State-Dependent Riccati-Equation-Based Guidance Law for Impact-Angle-Constrained Trajectories

Abstract: IN MANY advanced guidance applications, it is required to intercept the target from a particular direction: that is, to achieve a certain impact angle [1–3]. Closed-form solutions for energyoptimal impact-angle-constrained guidance laws have been proposed for a stationary target by Ryoo et al. [4], who used the linear quadratic regulator technique after linearizing the engagement kinematics. The guidance law proposed by them captures all impact angles from any initial launch angle in a planar engagement scenario. Lu et al. [5] solved the problem of guiding a hypersonic gliding vehicle in its terminal phase to a stationary target using adaptive proportional navigation guidance. Ohlmeyer and Phillips [6] extended the idea of explicit guidance (proposed by Cherry [7]) to include a terminal impact angle constraint. However, the simulations by Ohlmeyer and Phillips [6] are carried out only for a vertical impact against a stationary target, and the impact angle errors encountered are sensitive to the launch altitude.

Series solutions of non-linear riccati differential equations with fractional order

Abstract: In this paper, based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve non-linear Riccati differential equation with fractional order. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ℏ . Besides, it is proved that well-known Adomian’s decomposition method is a special case of the homotopy analysis method when ℏ = −1. This work illustrates the validity and great potential of the homotopy analysis method for the non-linear differential equations with fractional order. The basic ideas of this approach can be widely employed to solve other strongly non-linear problems in fractional calculus.

Optimal control of unknown affine nonlinear discrete-time systems using offline-trained neural networks with proof of convergence

Abstract: The optimal control of linear systems accompanied by quadratic cost functions can be achieved by solving the well-known Riccati equation. However, the optimal control of nonlinear discrete-time systems is a much more challenging task that often requires solving the nonlinear Hamilton―Jacobi―Bellman (HJB) equation. In the recent literature, discrete-time approximate dynamic programming (ADP) techniques have been widely used to determine the optimal or near optimal control policies for affine nonlinear discrete-time systems. However, an inherent assumption of ADP requires the value of the controlled system one step ahead and at least partial knowledge of the system dynamics to be known. In this work, the need of the partial knowledge of the nonlinear system dynamics is relaxed in the development of a novel approach to ADP using a two part process: online system identification and offline optimal control training. First, in the system identification process, a neural network (NN) is tuned online using novel tuning laws to learn the complete plant dynamics so that a local asymptotic stability of the identification error can be shown. Then, using only the learned NN system model, offline ADP is attempted resulting in a novel optimal control law. The proposed scheme does not require explicit knowledge of the system dynamics as only the learned NN model is needed. The proof of convergence is demonstrated. Simulation results verify theoretical conjecture.

Applied Pseudoanalytic Function Theory

Abstract: Introduction.- I. Pseudoanalytic function theory and second-order elliptic equations.- 1. Definitions and results from Bers' theory.- 2. Second order equations.- 3. Formal powers.- 4. Cauchy's integral formula.- 5. Complex Riccati equation.- II. Applications to Sturm-Liouville theory.- 6. Sturm-Liouville equation.- 7. Spectral problems and Darboux transformation.- III. Applications to real first-order systems.- 8. Beltrami fields.- 9. Static Maxwell system.- IV. Hyperbolic pseudoanalytic functions.- 10. Hyperbolic numbers and analytic functions.- 11. Hyperbolic pseudoanalytic functions.- 12. Klein-Gordon equation.- V. Bicomplex and biquaternionic pseudoanalytic functions and applications.- 13. The Dirac equation.- 14. Complex second order elliptic equations and bicomplex pseudoanalytic functions.- 15. Multidimensional second order equations.- Open problems.- Bibliography.- Index.

Numerical State-Dependent Riccati Equation Approach for Missile Integrated Guidance Control

Abstract: A x , B x = state-dependent system matrices of size n n and n m Q x , R x = state-dependent weighting matrices of sizes n n and m m r, _ r = range and range rate of the target with respect to the missile S = solution to the Riccati equation T = rocket motor thrust per unit mass acting along the longitudinal axis of the missile u = control vector of size m 1 upert = control perturbation vector of size m 1 x = state vector of size n 1 xpert = state perturbation vector of size n 1 X, Y, Z = relative position components of the target with respect to the missile y, z = position of the moving masses along the pitch and yaw axes with respect to the body yc, zc = moving-mass position commands , = pitch and yaw Euler angles of the missile y, z = line-of-sight angles

Estimation of Spatially Distributed Processes Using Mobile Spatially Distributed Sensor Network

Abstract: The problem of estimating a spatially distributed process described by a partial differential equation (PDE), whose observations are contaminated by a zero mean Gaussian noise, is considered in this work. The basic premise of this work is that a set of mobile sensors achieve better estimation performance than a set of immobile sensors. To enhance the performance of the state estimator, a network of sensors that are capable of moving within the spatial domain is utilized. Specifically, such an estimation process is achieved by using a set of spatially distributed mobile sensors. The objective is to provide mobile sensor control policies that aim to improve the state estimate. The metric for such an estimate improvement is taken to be the expected state estimation error. Using different spatial norms, two guidance policies are proposed. The current approach capitalizes on the efficient filter gain design in order to avoid intense computational requirements resulting from the solution to filter Riccati equations. Simulation studies implementing and comparing the two proposed control policies are provided.

On the Value Functions of the Discrete-Time Switched LQR Problem

Abstract: In this paper, we derive some important properties for the finite-horizon and the infinite-horizon value functions associated with the discrete-time switched LQR (DSLQR) problem. It is proved that any finite-horizon value function of the DSLQR problem is the pointwise minimum of a finite number of quadratic functions that can be obtained recursively using the so-called switched Riccati mapping. It is also shown that under some mild conditions, the family of the finite-horizon value functions is homogeneous (of degree 2), is uniformly bounded over the unit ball, and converges exponentially fast to the infinite-horizon value function. The exponential convergence rate of the value iterations is characterized analytically in terms of the subsystem matrices.

Time distributed-order diffusion-wave equation. I. Volterra-type equation

Abstract: A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.

Dynamic Output Integral Sliding-Mode Control With Disturbance Attenuation

Abstract: This technical note proposes a dynamic output feedback sliding-mode control algorithm for linear MIMO systems with mismatched norm-bounded uncertainties along with disturbances and matched nonlinear perturbations. A control law is first designed to ensure that the system behavior can satisfy the reaching and sliding condition. A scheme designed to combine the output-dependent integral sliding surface with a full-order compensator is then proposed. Through utilizing H infin control analytical technique, once the system is in the sliding mode, the proposed algorithm can guarantee robust stabilization and sustain the nature of performing disturbance attenuation when the solutions to two algebraic Riccati inequalities can be found. Finally, the feasibility of our proposed algorithm is illustrated using a numerical example.

Self-Tuning Decoupled Fusion Kalman Predictor and Its Convergence Analysis

Abstract: For the multisensor systems with unknown noise variances, by the correlation method, the information fusion noise variance estimators are presented by taking the average of the local noise variance estimators under the least squares fusion rule. They have the average accuracy and have consistency. A self-tuning Riccati equation with the fused noise variance estimators is presented, and then a self-tuning decoupled fusion Kalman predictor is presented based on the optimal fusion rule weighted by scalars for state component predictors. In order to prove their convergence, the dynamic variance error system analysis (DVESA) method is presented, which transforms the convergence problem of the self-tuning Riccati equation into a stability problem of a dynamic variance error system described by the Lyapunov equation. A stability decision criterion of the Lyapunov equation is presented. By the DVESA method, the convergence of the self-tuning Riccati equation is proved, and then it is proved that the self-tuning decoupled fusion Kalman predictor converges to the optimal decoupled fusion Kalman predictor in a realization, so it has asymptotic optimality. A simulation example for a tracking system with 3-sensor shows the effectiveness, and verifies the convergence.

Inexact Kleinman-Newton Method for Riccati Equations

Abstract: In this paper we consider the numerical solution of the algebraic Riccati equation using Newton's method. We propose an inexact variant which allows one control the number of the inner iterates used in an iterative solver for each Newton step. Conditions are given under which the monotonicity and global convergence result of Kleinman also hold for the inexact Newton iterates. Numerical results illustrate the efficiency of this method.

Integrable discretizations of the short pulse equation

Abstract: In the present paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key of the construction is the bilinear forms and determinant structure of solutions of the SP equation. We also give the determinant formulas of N-soliton solutions of the semi-discrete and full-discrete analogues of the SP equations, from which the multi-loop and multi-breather solutions can be generated. In the continuous limit, the full-discrete SP equation converges to the semi-discrete SP equation, then to the continuous SP equation. Based on the semi-discrete SP equation, an integrable numerical scheme, i.e., a self-adaptive moving mesh scheme, is proposed and used for the numerical computation of the short pulse equation.

A piecewise variational iteration method for Riccati differential equations

Abstract: In this paper, we introduce a piecewise variational iteration method for Riccati differential equations, which is a modified variational iteration method (MVIM). 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. However, the solutions obtained using the MVIM give good approximations for a larger interval, rather than a local vicinity of the initial position. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method. Numerical results show that the method does not share the drawback of the conventional VIM and is a satisfactory method for Riccati differential equations.

Decentralized Coordinated Attitude Control for Satellite Formation Flying via the State-Dependent Riccati Equation Technique

Abstract: The goal of the present study is to develop a decentralized coordinated attitude control algorithm for satellite formation flying. To handle the non-linearity of the dynamic system, the problems of absolute and relative attitude dynamics are formulated for the state-dependent Riccati equation (SDRE) technique. The SDRE technique is for the first time utilized as a non-linear controller of the relative attitude control problem for satellite formation flying, and then the results are compared to those from linear control methods, mainly the PD and LQR controllers. The stability region for the SDRE-controlled system was obtained using a numerical method. This estimated stability region demonstrates that the SDRE controller developed in the present paper guarantees the globally asymptotic stability for both the absolute and relative attitude controls. Moreover, in order to complement a non-selective control strategy for relative attitude error in formation flying, a selective control strategy is suggested. This strategy guarantees not only a reduction in unnecessary calculation, but also the mission-failure safety of the attitude control algorithm for satellite formation. The attitude control algorithm of the formation flying was tested in the orbital-reference coordinate system for the sake of applying the control algorithms to Earth-observing missions. The simulation results illustrate that the attitude control algorithm based on the SDRE technique can robustly drive the attitude errors to converge to zero.

Stability of an Additive-Cubic-Quartic Functional Equation

Abstract: In this paper, we consider the additive-cubic-quartic functional equation and prove the generalized Hyers-Ulam stability of the additive-cubic-quartic functional equation in Banach spaces.

Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order

Abstract: The aim of this paper is to prove the stability in the sense of Hyers-Ulam of differential equation of second order . That is, if is an approximate solution of the equation , then there exists an exact solution of the equation near to .

Response to comments on "Computing the positive stabilizing solution to algebraic riccati equations with an indefinite quadratic term via a recursive method"

Abstract: An iterative algorithm to solve algebraic riccati equations with an indefinite quadratic term is proposed. The global convergence and local quadratic rate of convergence of the algorithm are guaranteed and a proof is given. Numerical examples are also provided to demonstrate the superior effectiveness of the proposed algorithm when compared with methods based on finding stable invariant subspaces of Hamiltonian matrices. A game theoretic interpretation of the algorithm is also provided.