Galerkin approximationwith Legendre polynomials for a continuous-time nonlinear optimal control problem

We investigate the use of an approximation method for obtaining near-optimal solutions to a kind of nonlinear continuous-time (CT) system. The approach derived from the Galerkin approximation is used to solve the generalized Hamilton-Jacobi-Bellman (GHJB) equations. The Galerkin approximation with Legendre polynomials (GALP) for GHJB equations has not been applied to nonlinear CT systems. The proposed GALP method solves the GHJB equations in CT systems on some well-defined region of attraction. The integrals that need to be computed are much fewer due to the orthogonal properties of Legendre polynomials, which is a significant advantage of this approach. The stabilization and convergence properties with regard to the iterative variable have been proved. Numerical examples show that the update control laws converge to the optimal control for nonlinear CT systems.     

A matrix-free quasi-Newton method for solving large-scale nonlinear systems

One of the widely used methods for solving a nonlinear system of equations is the quasi-Newton method. The basic idea underlining this type of method is to approximate the solution of Newton’s equation by means of approximating the Jacobian matrix via quasi-Newton update. Application of quasi-Newton methods for large scale problems requires, in principle, vast computational resource to form and store an approximation to the Jacobian matrix of the underlying problem. Hence, this paper proposes an approximation for Newton-step based on the update of approximation requiring a computational effort similar to that of matrix-free settings. It is made possible by approximating the Jacobian into a diagonal matrix using the least-change secant updating strategy, commonly employed in the development of quasi-Newton methods. Under suitable assumptions, local convergence of the proposed method is proved for nonsingular systems. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with Newton’s, Chord Newton’s and Broyden’s methods.     

Combination of the sequential secant method and Broyden's method with projected updates

We introduce a new algorithm for solving nonlinear simultaneous equations, which is a combination of the sequential secant method with Broyden's Quasi-Newton method with projected updates as introduced by Gay and Schnabel. The new algorithm has the order of convergence of the sequential secant method and the choice of the first increments is justified by the minimum variation principles of Quasi-Newton methods. Two versions of the method are compared numerically with some well-known test problems.     

一种求解约束优化问题的混合算法

提出一种基于修改增广Lagrange函数和PSO的混合算法用于求解约束优化问题。将约束优化问题转化为界约束优化问题,混合算法由两层迭代结构组成,在内层迭代中,利用改进PSO算法求解界约束优化问题得到下一个迭代点。外层迭代主要修正Lagrange乘子和罚参数,检查收敛准则是否满足,重构下次迭代的界约束优化子问题,检查收敛准则是否满足。数值实验结果表明该混合算法的有效性。     

A BFGS trust-region method for nonlinear equations

In this paper, a new trust-region subproblem combining with the BFGS update is proposed for solving nonlinear equations, where the trust region radius is defined by a new way. The global convergence without the nondegeneracy assumption and the quadratic convergence are obtained under suitable conditions. Numerical results show that this method is more effective than the norm method.     

A genetic algorithm for optimization problems with fuzzy relation constraints using max-product composition

We consider nonlinear optimization problems constrained by a system of fuzzy relation equations. The solution set of the fuzzy relation equations being nonconvex, in general, conventional nonlinear programming methods are not practical. Here, we propose a genetic algorithm with max-product composition to obtain a near optimal solution for convex or nonconvex solution set. Test problems are constructed to evaluate the performance of the proposed algorithm showing alternative solutions obtained by our proposed model.     

粒子群算法在求解方程组中的应用

提出了采用粒子群算法求解线性方程组和非线性方程组的智能算法。采用粒子群算法求解方程组具有形式简单、收敛迅速和容易理解等特点,且能在一次计算中多次发现方程组的解,可以解决非线性方程组多解的求解问题,为线性方程组和非线性方程组的求解提供了一种新的方法。     

Stable symmetric secant methods with restart

Two secant type methods are proposed for solving systems of nonlinear equations with a symmetrical Jacobi matrix. Quasi-Newton shift formulas of rank 2 are used. Stability and superlinear convergence are proved.Translated from Kibernetika, No. 3, pp. 62–66, May–June, 1991.     

Sulla convergenza di un procedimento basato sul metodo delle secanti modificato

In this paper a method of E. Polak for solving systems of nonlinear equations is generalized and shown to have superlinear convergence under larger assumptions. The result is obtained giving a new criterion for the convergence of the modified secant method. The convergence does not depend upon the goodness of initial approximation of the solution, and is global for a wide class of functions. Numerical results are reported on some known test problems.      

Monotone iterates for solving coupled systems of nonlinear parabolic equations

This paper deals with numerical solutions of coupled nonlinear parabolic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear parabolic equations. This monotone convergence leads to existence-uniqueness theorems. An analysis of convergence rates of the monotone iterative method is given. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating is proposed. A convergence analysis of the monotone domain decomposition algorithm is presented. An application to a gas–liquid interaction model is given.     

一种求解非线性约束优化全局最优的新方法

本文提出了一种求解非线性约束优化的全局最优的新方法—它是基于利用非线性互补函数和不断增加新的约束来重复解库恩-塔克条件的非线性方程组的新方法。因为库恩-塔克条件是非线性约束优化的必要条件,得到的解未必是非线性约束优化的全局最优解,为此,本文首次给出了通过利用该优化问题的先验知识,不断地增加约束来限制全局最优解范围的方法,一些仿真例子表明提出的方法和理论有效的,并且可行的。     

Enclosing the solutions of nonlinear systems of equations

The quadratically convergent Newton-type methods and its variants are generally used for solving the nonlinear systems of equations. Most of these methods use the convexity conditions [7] of the involved bounded linear operators for their convergence. Alefeld and Herzberger [1] have proposed a quadratically convergent iterative method for enclosing the solutions of the special type of nonlinear system of equations arising from the discretization of nonlinear boundary value problems which do not require the convexity conditions but uses the subinverses of the bounded linear operators. In this paper, we have proposed a modification of this method which takes it further faster. The proposed method uses bom the superinverses and subinverses of bounded linear operators. At the expense of slightly more computations than used in [1], the rate of convergence of our method enhances from quadratic to cubic. Finally, the method is tested on a numerical example.     

Convergence of generalized-$\boldsymbol{\alpha}$ time integration for nonlinear systems with stiff potential forces

Generalized-\(\alpha\) time integration schemes, originally developed for application in structural dynamics, are increasingly popular throughout many branches of multibody system simulation. Their simple implementation and the opportunity to control the numerical dissipation make them highly appealing for use in broad fields of application.Initially introduced for the solution of linear ordinary differential equations, there have been several extensions to nonlinear structural dynamics and constrained multibody systems in various formulations.In the present paper, we consider the application to systems with very stiff potential forces (singular singularly perturbed systems) whose solution approaches in the limit case that of a constrained system (index-3 differential–algebraic equation). We give a convergence analysis comparing the highly oscillatory solutions of the stiff system to those of the associated constrained one and show that the classical second order convergence result holds for position coordinates as well as for appropriately projected errors on the velocity level.The theoretical results are verified by numerical experiments for a simple test example.     

Multilevel augmentation methods for nonlinear ill-posed problems

We propose multilevel augmentation methods for solving nonlinear ill-posed problems, involving monotone operators in the Hilbert space by using the Lavrentiev regularization method. This leads to a fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. The regularization parameter choice strategies considered by Pereverzev and Schock (2005) are introduced and the optimal convergence rates of the regularized solutions are obtained. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.     

Multigrid method for nonlinear poroelasticity equations

In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies.     

Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method

Solving systems of nonlinear equations is one of the most difficult numerical computation problems. The convergences of the classical solvers such as Newton-type methods are highly sensitive to the initial guess of the solution. However, it is very difficult to select good initial solutions for most systems of nonlinear equations. By including the global search capabilities of chaos optimization and the high local convergence rate of quasi-Newton method, a hybrid approach for solving systems of nonlinear equations is proposed. Three systems of nonlinear equations including the “Combustion of Propane” problem are used to test our proposed approach. The results show that the hybrid approach has a high success rate and a quick convergence rate. Besides, the hybrid approach guarantees the location of solution with physical meaning, whereas the quasi-Newton method alone cannot achieve this.     

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist. This work was partially supported by the NSF grants DMS-0410266 and DMS-0710831.     

Adaptive Impulsive Observers for a Class of Switched Nonlinear Systems with Unknown Parameter

This work investigates and solves the design of adaptive impulsive observers for a class of uncertain switched nonlinear systems with unknown parameter. Sufficient conditions are derived for designing such observers for each subsystem to reconstruct asymptotically and update system states in real time. The state observer is represented in terms of impulsive differential equations. The parameter estimation law is modelled by an impulse‐free, time‐varying differential equation associated with the impulse time sequence in order to determine when the observer estimated state is updated. The asymptotic convergence to zero of the observation errors is established by applying the method of multiple time‐varying Lyapunov functions. Sufficient conditions are derived that guarantee the convergence of parameter estimation. An example of switched Lorenz system along with numeric and simulation results is presented to demonstrate the effectiveness of the proposed method.     

A convergence analysis of the Adomian decomposition method for an abstract Cauchy problem of a system of first-order nonlinear differential equations

In this article, we study some fundamental results concerning the convergence of the Adomian decomposition method (ADM) for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Under certain conditions, we obtain upper estimates for the norm of solutions of this system. We also obtain results about the error estimates for the approximate solutions by the ADM and discuss their applications.     

Tracking in nonlinear differential-algebraic control systems with applications to constrained robot systems

We consider the design of a feedback control law for control systems described by a class of nonlinear differential-algebraic equations so that certain desired outputs track given reference inputs. The nonlinear differential-algebraic control system being considered is not in state variable form. Assumptions are introduced and a procedure is developed such that an equivalent state realization of the control system described by nonlinear differential-algebraic equations is expressed in a familiar normal form. A nonlinear feedback control law is then proposed which ensures, under appropriate assumptions, that the tracking error in the closed loop differential-algebraic system approaches zero exponentially. Applications to simultaneous contact force and position tracking in constrained robot systems with rigid joints, constrained robot systems with joint flexibility, and constrained robot systems with significant actuator dynamics are discussed.