Showing papers on "Riccati equation published in 2011"

Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis

Abstract: -1. Introduction. -2. Additive Cauchy Equation (Behavior of additive functions, Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Stability on a restricted domain, Method of invariant means, Fixed point method, Composite functional congruences, Pexider equation, Remarks). -3. Generalized Additive Cauchy Equations (Functional equation f(ax+by)=af(x)+bf(y), Additive Cauchy equations of general form, Functional equation f(x+y)2=(f(x)+f(y))2). -4. Hossza"'s Functional Equation (Stability in the sense of Borelli, Hyers-Ulam stability, Generalized Hossza"'s equation is not stable on the unit interval, Hossza"'s functional equation of Pexider type). -5. Homogeneous Functional Equation(Homogeneous equation between Banach algebras, Superstability on a restricted domain, Homogeneous equation between vector spaces, Homogeneous equation of Pexide type). -6. Linear Functional Equations (A system for linear functions, Functional equation f(x+cy)=f(x)+cf(y), Stability for other equations).-7. Jensen's Functional Equation (Hyers-Ulam-Rassias stability, Stability on a restriced domain, Fixed point method, Lobacevskii s functional equation). -8. Quadratic Functional Equations (Hyers-Ulam-Rassias stability, Stability on a restricted domain, Fixedpoint method, Quadratic functional equation of other type, Quadratic functional equation of Pexider type). -9. Exponential Functional Equations (Superstability, Stability in the sense of Ger, Stability on a restricted domain, Exponential functional equation of other type). -10. Multiplicative Functional Equations (Superstability, delta-multiplicative functional, Theory of AMNM algebras, Functional equation f(xy)= f(x)y, Functional equation f(x+y)= f(x)f(y)f(1/x+1/y)). -11. Logarithmic Functional Equations (Functional equation f(xy)= yf(x), Superstability of equation f(xy)= yf(x), Functional equation of Heuvers). -12. Trigonometric Functional Equations (Cosine functional equation, Sine functional equation, Trigonometric equations with two unknowns, Butler-Rassias functional equation, Remarks). -13. Isometric Functional Equation (Hyers-Ulam stability, Stability on a restricted domain, Fixed point method, Wigner equation). -14. Miscellaneous (Associativity equation, Equation of multiplicative derivation, Gamma functional equation). -Bibliography. -Index.

Lyapunov Differential Equation Approach to Elliptical Orbital Rendezvous with Constrained Controls

Abstract: and energy. A parametric Lyapunov differential equation approach is proposed in this paper to solve this constrained regulation problem. After establishing the fact that the Tschauner-Hempel equations are both null controllable with controls of bounded magnitude and energy, this paper proves that the proposed linear periodic controllersemigloballystabilizesthesystem.Equivalently,forany fixedinitialconditions,themagnitudeandenergy of the control can be made as small as desired by tuning some free parameters in the feedback laws. In comparison with the existing quadratic-regulation-based approach, which requires solutions to nonlinear Riccati differential equations,thenewapproachrequiresonlythesolutionoflinearperiodicLyapunovdifferentialequations,whichare investigated inthepaperbyusingtheperiodicgenerator approach.Numericalsimulationsofthe nonlinearmodelof the spacecraft rendezvous instead of a linearized one show that both the magnitude and energy of the control can be reduced to an arbitrarily small level by reducing the values of some parameters in the controller and that the rendezvous mission can be accomplished satisfactorily.

Multivariable Newton-based extremum seeking

Abstract: We present a Newton-based extremum seeking algorithm for the multivariable case. The design extends the recent Newton-based extremum seeking algorithms for the scalar case and introduces a dynamic estimator of the Hessian matrix that removes the difficulty with the possible singularity of this matrix estimate. This estimator has the form of a differential Riccati equation. We prove local stability of the new algorithm for general nonlinear dynamic systems using averaging and singular perturbations. In comparison with the standard gradient-based multivariable extremum seeking, the proposed algorithm removes the dependence of the convergence rate on the unknown Hessian matrix and makes the convergence rate, of both the parameter estimates and of the estimates of the Hessian inverse, user-assignable. In particular, the new algorithm allows all the parameters to converge with the same speed, even with maps that have highly elongated level sets. In the parameter space, the new algorithms produces trajectories straight to the extremum, as opposed to non-direct “steepest descent” trajectories. Simulation results show the advantage of the proposed approach over gradient-based extremum seeking.

Linear-Quadratic Optimal Actuator Location

Abstract: In control of vibrations, diffusion and many other problems governed by partial differential equations, there is freedom in the choice of actuator location. The actuator location should be chosen to optimize performance objectives. In this paper, we consider linear quadratic performance. Two types of cost are considered; the choice depends on whether the response to the worst initial condition is to be minimized; or whether the initial condition is regarded as random. In practice, approximations are used in controller design and thus in selection of the actuator locations. The optimal cost and location of the approximating sequence should converge to the exact optimal cost and location. In this work conditions for this convergence are given in the case of linear quadratic control. Examples are provided to illustrate that convergence may fail when these conditions are not satisfied.

Optimal Linear Estimators for Systems With Random Sensor Delays, Multiple Packet Dropouts and Uncertain Observations

Abstract: This paper is concerned with the optimal linear estimation problem for linear discrete-time stochastic systems with random sensor delays, packet dropouts and uncertain observations. We develop a unified model to describe the mixed uncertainties of random delays, packet dropouts and uncertain observations by three Bernoulli distributed random variables with known distributions. Based on the proposed model, the optimal linear estimators that only depend on probabilities are developed via an innovation analysis approach. Their solutions are given in terms of a Riccati equation and a Lyapunov equation. They can deal with the optimal linear filtering, prediction and smoothing for systems with random sensor delays, packet dropouts and uncertain observations in a unified framework. Simulation results show the effectiveness of the proposed optimal linear estimators.

An efficient approach for solving the Riccati equation with fractional orders

Abstract: The present study introduces a novel and simple analytical method for the solution of fractional order Riccati differential equation. In this approach, the solution considered as a Taylor series expansion converges rapidly to the nonlinear problem. New homotopy perturbation method (NHPM) depends only on two components of the homotopy series. The method is illustrated by applications and the results obtained are compared with those of the exact solution. Moreover, comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient.

Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics

Abstract: In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the K(m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.

On well-posedness of the Degasperis-Procesi equation

Abstract: It is shown in both the periodic and the non-periodic cases that the data-to-solution map for the Degasperis-Procesi (DP) equation is not a uniformly continuous map on bounded subsets of Sobolev spaces with exponent greater than 3/2. This shows that continuous dependence on initial data of solutions to the DP equation is sharp. The proof is based on well-posedness results and approximate solutions. It also exploits the fact that DP solutions conserve a quantity which is equivalent to the $L^2$ norm. Finally, it provides an outline of the local well-posedness proof including the key estimates for the size of the solution and for the solution's lifespan that are needed in the proof of the main result.

A modified block method for the direct solution of second order ordinary differential equations

Abstract: The direct solution of general second order ordinary differential equation is considered in this paper. The method is based on the collocation and interpolation of the power series approximate solution to generate a continuous linear multistep method. We modified the existing block method in order to accommodate the general nth order ordinary differential equation. The modified block is adopted to generate the independent solution at the grid points. This method was found to be efficient when tested on second order ordinary differential equation

Oscillation of second-order nonlinear delay dynamic equations on time scales

Abstract: By using the generalized Riccati transformation and the inequality technique, we establish one new oscillation criterion for the second-order nonlinear delay dynamic equations on time scales. Our results not only extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales.

On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity

Abstract: Considered in this paper is the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa-Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. In addition, the persistence properties of the strong solutions for the equation are obtained.

H∞ control for stochastic systems with Poisson jumps

Abstract: This paper discusses the H ∞ control problem for a class of linear stochastic systems driven by both Brownian motion and Poisson jumps. The authors give the basic theory about stabilities for such systems, including internal stability and external stability, which enables to prove the bounded real lemma for the systems. By means of Riccati equations, infinite horizon linear stochastic state-feedback H ∞ control design is also extended to such systems.

Dispersion for the Schrödinger equation on networks

Abstract: In this paper, we consider the Schrodinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. We give an explicit description of the solution of the linear Schrodinger equation with constant coefficients. This allows us to prove dispersive estimates, which in turn are useful for solving the nonlinear Schrodinger equation. The proof extends also to the laminar case of positive step-function coefficients having a finite number of discontinuities.

Discrete Hamilton–Jacobi Theory

Abstract: We develop a discrete analogue of Hamilton–Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton–Jacobi equation is discrete only in time. We describe a discrete analogue of Jacobi's solution and also prove a discrete version of the geometric Hamilton–Jacobi theorem. The theory applied to discrete linear Hamiltonian systems yields the discrete Riccati equation as a special case of the discrete Hamilton–Jacobi equation. We also apply the theory to discrete optimal control problems, and recover some well-known results, such as the Bellman equation (discrete-time HJB equation) of dynamic programming and its relation to the costate variable in the Pontryagin maximum principle. This relationship between the discrete Hamilton–Jacobi equation and Bellman equation is exploited to derive a generalized form of the Bellman equation that has controls at internal stages.

On the numerical solution of large-scale sparse discrete-time Riccati equations

Abstract: We discuss the numerical solution of large-scale discrete-time algebraic Riccati equations (DAREs) as they arise, e.g., in fully discretized linear-quadratic optimal control problems for parabolic partial differential equations (PDEs). We employ variants of Newton's method that allow to compute an approximate low-rank factor of the solution of the DARE. The principal computation in the Newton iteration is the numerical solution of a Stein (aka discrete Lyapunov) equation in each step. For this purpose, we present a low-rank Smith method as well as a low-rank alternating-direction-implicit (ADI) iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.

A comparative study of numerical methods for solving quadratic Riccati differential equations

Abstract: In this paper, we use Legendre wavelet method for solving quadratic Riccati differential equations and perform a comparative study between the proposed method and other existing methods. Our results show that in comparison with other existing methods, the Legendre wavelet method provides a fast convergent series of easily computable components. The present study is illustrated by exploring two kinds of nonlinear Riccati differential equations that shows the pertinent features of the Legendre wavelet method.

Magnetic torque attitude control of a satellite using the state-dependent Riccati equation technique

Abstract: A non-linear attitude control method for a satellite with magnetic torque rods using the state-dependent Riccati equation (SDRE) technique has been developed. The magnetic torque caused by the interaction with the Earth's magnetic field and the magnetic moment of torque rods plays a role of the control torque. The detailed equations of motion for this system are presented using angular velocity and quaternions. The SDRE controller is developed for the non-linear systems which can be formed in pseudo-linear representations using the state-dependent coefficient (SDC) method without linearization procedure. The aim of this control system is to achieve a stable attitude within 5°, and minimize the control effort. The stability regions for the SDRE controlled satellite system are estimated through the investigation of the stability conditions developed for pseudo-linear systems and the application of Lyapunov's theorem. For comparisons, the Linear Quadratic Regulator (LQR) method using the solution of the algebraic Riccati equation (ARE) is also applied to this non-linear system. The performance of the SDRE non-linear control system demonstrates more robustness and stability than the LQR control system when subjected to a wide range of initial conditions.

Poincar\'e-Lelong equation via the Hodge Laplace heat equation

Abstract: In this paper, we develop a method of solving the Poincar\'e-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on $(1, 1)$-forms. The method is effective in proving an optimal result when $M$ has nonnegative bisectional curvature. It also provides an alternate proof of a recent gap theorem of the first author.