On variational perturbations of control problems: Minimum-time problem and minimum-effort problem

In order to obtain numerical solutions for an abstract optimal control problem, one approximates the abstract operations in a computationally feasible manner. After having found an approximate optimal solution, the question is whether a sequence of these approximate optimal solutions converges to an optimal solution of the original problem. In this work, we are concerned with this type of convergence on the time-optimal control problem for a class of linear systems with distributed parameters and on the minimum-effort problem.     

A new method for solving nonlinear second order partial differential equations

In this paper, a new method for finding the approximate solution of a second order nonlinear partial differential equation is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a distributed parameter control system, the theory of measure is used for obtaining the approximate solution of the original problem.     

A new approach for solving the Stokes problem

In this paper, a new approach for finding the approximate solution of the Stokes problem is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a distributed parameter control system, the theory of measure is used to approximate the velocity functions by piecewise linear functions. Then, the approximate values of pressure are obtained by a finite difference scheme.     

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.     

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method.     

On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups

In a Banach space, for the approximate solution of the Cauchy problem for the evolution equation with an operator generating an analytic semigroup, a purely implicit three-level semidiscrete scheme that can be reduced to two-level schemes is considered. Using these schemes, an approximate solution to the original problem is constructed. Explicit bounds on the approximate solution error are proved using properties of semigroups under minimal assumptions about the smoothness of the data of the problem. An intermediate step in this proof is the derivation of an explicit estimate for the semidiscrete Crank–Nicolson scheme. To demonstrate the generality of the perturbation algorithm as applied to difference schemes, a four-level scheme that is also reduced to two-level schemes is considered.     

An approximate solution for linear boundary-value problems with slowly varying coefficients

In this paper, an approximate closed-form solution for linear boundary-value problems with slowly varying coefficient matrices is obtained. The derivation of the approximate solution is based on the freezing technique, which is commonly used in analyzing the stability of slowly varying initial-value problems as well as solving them. The error between the approximate and the exact solutions is given, and an upper bound on the norm of the error is obtained. This upper bound is proportional to the rate of change of the coefficient matrix of the boundary-value problem. The proposed approximate solution is obtained for a two-point boundary-value problem and is compared to its solution obtained numerically. Good agreement is observed between the approximate and the numerical solutions, when the rate of change of the coefficient matrix is small.     

An inner approximation method incorporating a branch and bound procedure for optimization over the weakly efficient set

In this paper, we consider an optimization problem which aims to minimize a convex function over the weakly efficient set of a multiobjective programming problem. From a computational viewpoint, we may compromise our aim by getting an approximate solution of such a problem. To find an approximate solution, we propose an inner approximation method for such a problem. Furthermore, in order to enhance the efficiency of the solution method, we propose an inner approximation algorithm incorporating a branch and bound procedure.     

On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities

In this paper, we consider a vector optimization problem involving approximately star-shaped functions. We formulate approximate vector variational inequalities in terms of Fréchet subdifferentials and solve the vector optimization problem. Under the assumptions of approximately straight functions, we establish necessary and sufficient conditions for a solution of approximate vector variational inequality to be an approximate efficient solution of the vector optimization problem. We also consider the corresponding weak versions of the approximate vector variational inequalities and establish various results for approximate weak efficient solutions.     

INVERSE EIGENVALUE PROBLEM OF HERMITIAN GENERALIZED ANTI-HAMILTONIAN MATRICES

§1　IntroductionWe considerthe following inverse eigenvalue problem offinding an n-by-n matrix A∈S such thatAxi =λixi,i =1,2 ,...,m,where S is a given set of n-by-n matrices,x1 ,...,xm(m≤n) are given n-vectors andλ1 ,...,λmare given constants.Let X=(x1 ,...,xm) ,Λ=(λ1 ,λ2 ,...,λm) ,then the above inverse eigenvalue problemcan be written as followsProblem Given X∈Cn×m,Λ=(λ1 ,...,λm) ,find A∈S such thatAX =XΛ,where S is a given matrix set.We also discuss the so-called opti…     

Bernstein polynomial basis for numerical solution of boundary value problems

The purpose of this paper is to propose a computational method for the approximate solution of linear and nonlinear two-point boundary value problems. In order to approximate the solution, the expansions in terms of the Bernstein polynomial basis have been used. The properties of the Bernstein polynomial basis and the corresponding operational matrices of integration and product are utilized to reduce the given boundary value problem to a system of algebraic equations for the unknown expansion coefficients of the solution. On this approach, the problem can be solved as a system of algebraic equations. By considering a special case of the problem, an error analysis is given for the approximate solution obtained by the present method. At last, five examples are examined in order to illustrate the efficiency of the proposed method.     

On the complexity of solving feasible systems of linear inequalities specified with approximate data

An algorithm that gives an approximate solution of a desired accuracy to a system of linear inequalities specified with approximate data is presented. It uses knowledge that the actual instance is feasible to reduce the data precision necessary to give an approximate solution to the actual instance. In some cases, this additional information allows problem instances that are ill-posed without the knowledge of feasibility to be solved.The algorithm is computationally efficient and requires not much more data accuracy than the minimal amount necessary to give an approximate solution of the desired accuracy. This work aids in the development of a computational complexity theory that uses approximate data and knowledge.     

On approximate solutions to some nonlinear diffusion problems

In the paper an approximate solution to a nonlinear diffusion problem in semiconductor production is considered. Some integral operator methods are used to estimate an error between this approximate solution and the exact one.This paper has been written when the author was a Visiting Professor in the Universidad de Extremadura.     

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.     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.     

HGAH矩阵的逆特征值问题

In this paper, the inverse eigenvalue problem of Hermitian generalized anti-Hamihonian matrices and relevant optimal approximate problem are considered. The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived. The solution of the relevant optimal approximate problem is given.     

<Emphasis Type="Italic">α</Emphasis>-Conservative approximation for probabilistically constrained convex programs

In this paper, we address an approximate solution of a probabilistically constrained convex program (PCCP), where a convex objective function is minimized over solutions satisfying, with a given probability, convex constraints that are parameterized by random variables. In order to approach to a solution, we set forth a conservative approximation problem by introducing a parameter α which indicates an approximate accuracy, and formulate it as a D.C. optimization problem.     

A Bundle Algorithm Applied to Bilevel Programming Problems with Non-Unique Lower Level Solutions

In the paper, the question is investigated if a bundle algorithm can be used to compute approximate solutions for bilevel programming problems where the lower level optimal solution is in general not uniquely determined. To give a positive answer to this question, an appropriate regularization approach is used in the lower level. In the general case, the resulting algorithm computes an approximate solution. If the problem proves to have strongly stable lower level solutions for all parameter values in a certain neighborhood of the stationary solutions of the bilevel problem, convergence to stationary solutions can be shown.     

Gradient-flow approach for computing a nonlinear-quadratic optimal-output feedback gain matrix

In this paper, an approach is proposed for solving a nonlinear-quadratic optimal regulator problem with linear static state feedback and infinite planning horizon. For such a problem, approximate problems are introduced and considered, which are obtained by combining a finite-horizon problem with an infinite-horizon linear problem in a certain way. A gradient-flow based algorithm is derived for these approximate problems. It is shown that an optimal solution to the original problem can be found as the limit of a sequence of solutions to the approximate problems. Several important properties are obtained. For illustration, two numerical examples are presented.This project was partially supported by a research grant from the Australian Research Council.     

Continuity of approximate solution mappings for parametric equilibrium problems

In this paper, we obtain sufficient conditions for Hausdorff continuity and Berge continuity of an approximate solution mapping for a parametric scalar equilibrium problem. By using a scalarization method, we also discuss the Berge lower semicontinuity and Berge continuity of a approximate solution mapping for a parametric vector equilibrium problem.