Decoupling problems around a trajectory: from linearity to nonlinearity

Given a nonlinear control system for which an admissible statetrajectory is specified, we solve approximately the input outputdecoupling problem around this nominal trajectory. An approximatesolution for this problem is obtained by dealing with the linearizedsystem along this trajectory. An exact solution to the inputoutput decoupling problem for the linearization is shown tobe an approximate solution to the input output decoupling problemaround the nominal trajectory for the original nonlinear system.In a similar way, we provide an approximate solution to thedisturbance decoupling problem around a specified trajectoryof the nonlinear system. The nonlinear model of a two link robotmanipulator is used to illustrate the results on input outputdecoupling.     

AN EFFECTIVE CONTINUOUS ALGORITHM FOR APPROXIMATE SOLUTIONS OF LARGE SCALE MAX-CUT PROBLEMS

An effective continuous algorithm is proposed to find approximate solutions of NP-hardmax-cut problems.The algorithm relaxes the max-cut problem into a continuous nonlinearprogramming problem by replacing n discrete constraints in the original problem with onesingle continuous constraint.A feasible direction method is designed to solve the resultingnonlinear programming problem.The method employs only the gradient evaluations ofthe objective function,and no any matrix calculations and no line searches are required.This greatly reduces the calculation cost of the method,and is suitable for the solutionof large size max-cut problems.The convergence properties of the proposed method toKKT points of the nonlinear programming are analyzed.If the solution obtained by theproposed method is a global solution of the nonlinear programming problem,the solutionwill provide an upper bound on the max-cut value.Then an approximate solution to themax-cut problem is generated from the solution of the nonlinear programming and providesa lower bound on the max-cut value.Numerical experiments and comparisons on somemax-cut test problems(small and large size)show that the proposed algorithm is efficientto get the exact solutions for all small test problems and well satisfied solutions for mostof the large size test problems with less calculation costs.     

Method of weighted residuals as applied to nonlinear differential equations

A new method for solving a class of nonlinear boundary-value problems is presented. In this method, the nonlinear equation is linearized by guessing an initial solution and using it to evaluate the nonlinear terms. Next, a method of weighted residuals is applied to transform the linearized form of the boundary value problem to an initial value problem. The second (improved) solution is obtained by integrating the initial value problem by a fourth order Runge-Kutta scheme. The entire process is repeated until a desired convergence criterion is achieved.     

A FEASIBLE DIRECTION ALGORITHM WITHOUT LINE SEARCH FOR SOLVING MAX-BISECTION PROBLEMS

This paper concerns the solution of the NP-hard max-bisection problems. NCP func-tions are employed to convert max-bisection problems into continuous nonlinear program-ming problems. Solving the resulting continuous nonlinear programming problem generatesa solution that gives an upper bound on the optimal value of the max-bisection problem.From the solution, the greedy strategy is used to generate a satisfactory approximate so-lution of the max-bisection problem. A feasible direction method without line searches isproposed to solve the resulting continuous nonlinear programming, and the convergenceof the algorithm to KKT point of the resulting problem is proved. Numerical experimentsand comparisons on well-known test problems, and on randomly generated test problemsshow that the proposed method is robust, and very efficient.     

分数阶微分方程边值问题解的存在性

应用Gteen函数将分数阶微分方程边值问题可转化为等价的积分方程．近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性．讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Caratheodory条件,利用非紧性测度的性质和M6nch’s不动点定理证明解的存在性．     

Convergence and Error Bound for Perturbation of Linear Programs

In various penalty/smoothing approaches to solving a linear program, one regularizes the problem by adding to the linear cost function a separable nonlinear function multiplied by a small positive parameter. Popular choices of this nonlinear function include the quadratic function, the logarithm function, and the x ln(x)-entropy function. Furthermore, the solutions generated by such approaches may satisfy the linear constraints only inexactly and thus are optimal solutions of the regularized problem with a perturbed right-hand side. We give a general condition for such an optimal solution to converge to an optimal solution of the original problem as the perturbation parameter tends to zero. In the case where the nonlinear function is strictly convex, we further derive a local (error) bound on the distance from such an optimal solution to the limiting optimal solution of the original problem, expressed in terms of the perturbation parameter.     

Error bounds for finite element solutions of mildly nonlinear elliptic boundary value problems

Summary The equivalence in a Hilbert space of variational and weak formulations of linear elliptic boundary value problems is well known. This same equivalence is proved here for mildly nonlinear problems where the right hand side of the differential equation involves the solution function. A finite element approximation to the solution of the weak problem ina finite dimensional subspace of the original Hilbert space is defined. An inequality bounding the error in this approximation over all functions of the space is derived, and in particular this holds for an interpolant to the weak solution. Thus this inequality, together with previously known, interpolation error bounds, produces a bound on the finite element solution to this nonlinear problem. An example of a mildly nonlinear Poisson problem is given.     

A new filled function method for an unconstrained nonlinear equation

In this paper, we transform an unconstrained system of nonlinear equations into a special optimization problem. A new filled function is constructed by employing the special properties of the transformed optimization problem. Theoretical and numerical properties of the proposed filled function are investigated and a solution of the algorithm is proposed. Under some conditions, we can find a solution or an approximate solution to the system of nonlinear equations in finite iterations. The implementation of the algorithm on six test problems is reported with satisfactory numerical results.     

一类非饱和流动的Fourier级数解(英)

本文研究带有抽取的非饱和流动中出现的一个非线性边值问题．利用互惠变换及HopfCole变换将问题化为一个移动边界问题，进而获得了Fourier级数解．     

Vector Eigenfunction Expansions for Plane Channel Flows

The full nonlinear initial-boundary value problem for the evolution of disturbances in plane Poiseuille flow is considered. The problem is formulated in vector form using the normal velocity and normal vorticity as components. The solution is presented as an expansion in linear eigenmodes. These modes consist of both Orr-Sommerfeld modes and modes of the normal vorticity (Squire) equation. The case of degenerating eigenmodes is also considered and it is shown that the Benney-Gustavsson normal velocity-normal vorticity resonance is a special case of a degeneracy between the vector eigenmodes. The solution to the nonlinear problem is presented as an expansion in the linear eigenmodes as well as in modes of the self-adjoint part of the linear equation. The full nonlinear solution is further reduced to small systems of coupled amplitude equations using the center manifold theorem.     

Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented.     

A Duality Approach to Nonlinear Diffusion Equations

We provide new existence results for a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity, under minimal growth and coercivity conditions. The results given in this paper prove that a generalized solution to the nonlinear equation is provided by a solution to an equivalent minimization problem for a convex functional involving the potential of the nonlinearity and its conjugate, in the case when the potential is time and space depending. If the potential is time depending only and it has a symmetry at infinity, the null minimizer in the minimization problem is found to coincide with a weak solution to the nonlinear equation.     

The Inverse Problem Method on the Half-Line and a Nested System of Riccati Equations

A mixed problem for the nonlinear Bogoyavlenskii system on the half-line is studied by the inverse problem method. The solution of the mixed problem is reduced to the solution of the inverse spectral problem of recovering a forth-order differential operator on the half-line from the Weyl matrix. We derive evolution equations for the elements of the Weyl matrix and give an algorithm for the solution of the mixed problem. Evolution equations of the elements of the Weyl matrix are nonlinear. It is shown that they can be reduced to a nested system of three successively solvable matrix Riccati equations.     

An approach for solving nonlinear programming problems

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.     

An exact penalty function method for nonlinear mixed discrete programming problems

In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. Then, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton methods. It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method.     

Numerical Solutions for Option Pricing Models Including Transaction Costs and Stochastic Volatility

The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation. The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. In this paper under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method and provide numerical solutions using the finite difference method.     

Initial trace for a doubly nonlinear parabolic equation

This paper studies the Cauchy problem for a doubly nonlinear parabolic equation. The main result shows that if there is a nonnegative solution of the Cauchy problem, then the initial trace of the solution is uniquely given as a nonnegative Borel measure satisfying an exponential growth condition. This extends the known result for the heat equation to the nonlinear case.     

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.     

微分变换法求解二维非线性Volterra积分微分方程

为了解决二维非线性Volterra积分微分方程的求解问题,本文给出微分变换法.利用该方法将方程中的微分部分和积分部分进行变换,这样简化了原方程,进而得到非线性代数方程组,从而将原问题转换为求解非线性代数方程组的解,使得计算更简便.文中最后数值算例说明了该方法的可行性和有效性.     

Lyapunov functions in justification theorems for asymptotics

A formal asymptotic solution is considered for a nonlinear system of ordinary differential equations in a neighborhood of a singular point. The problem of existence of an exact solution with such an asymptotics and the problem of stability of this solution are solved. The main tool in these studies is the Lyapunov function for a system linearized on a formal solution.