NUMERICAL PROBLEMS INVOLVED IN FINDING OPTIMAL CONTROL STRATEGIES BY NONLINEAR PROGRAMMING TECHNIQUES (original) (raw)

Generalized B-spline functions method for solving optimal control problems

2014

In this paper we introduce a numerical approach that solves optimal control problems (OCPs) using collocation methods. This approach is based upon B-spline functions. The derivative matrices between any two families of B-spline functions are utilized to reduce the solution of OCPs to the solution of nonlinear optimization problems. Numerical experiments conrm our theoretical ndings.

Optimal control of dynamic systems: Application to spline approximations

Applied Mathematics and Computation, 1998

Generally, classical polynomial splines tend to exhibit unwanted undulations. In this work, we discuss a technique, based on control principles, for eliminating these undulations and increasing the smoothness properties of the spline interpolants. We give a generalization of the classical polynomial splines and show that this generalization is, in fact, a family of splines that covers the broad spectrum of polynomial, trigonometric and exponential splines. A particular element in this family is determined by the appropriate control data. It is shown that this technique is easy to implement. Several numerical and curve-fitting examples are given to illustrate the advantages of this technique over the classical approach. Finally, we discuss the convergence properties of the interpolant.

About an Algorithm of Function Approximation by the Linear Splines

2016

The actual application for the problem of best approximation of grid function by linear splines was formulated. A mathematical model and a method for its solution were developed. Complexity of the problem was that it was multi extremal and could not be solved analytically. The method was developed in order to solve the problem of dynamic programming scheme, which was extended by us. Given the application of the method to the problem of flow control in the pressure-regulating systems, the pipeline network for transport of substances (pipelines of water, oil, gas, and etc.) that minimizes the amount of substance reservoirs and reduces the discharge of substance from the system. The method and the algorithm developed here may be used in computational mathematics, optimal control and regulation system, and regressive analysis.

A numerical method for solving initial-value-problems with spline functions

BIT, 1979

A numerical method, using spline functions of degree five, for obtaining approximate solutions to initial value problems is presented. It is shown that the method is stable and the convergence is analysed. Some numerical experiments are included. Introduction. Recently we presented in [4] a method for construction of global approximations to the initial value problems in ordinary differential equations, using interpolate, piecewise polynomial functions of degree three, where we achieve convergence of order four. Now, we work on the same initial value problem with piecewise polynomial functions of degree five, belonging to C2[a, b] and with the collocation method. We prove the convergence and stability of the method achieving approximations of order six.

A survey on parametric spline function approximation

Applied Mathematics and Computation, 2005

This survey paper contains a large amount of material and indeed can serve as an introduction to some of the ideas and methods for the solution of ordinary and partial differential equations starting from SchoenbergÕs work [Quart. Appl. Math. 4 (1946) 345-369]. The parametric spline function which depends on a parameter x > 0, is reduces to the ordinary cubic or quintic spline for x = 0. A note on parametric spline function approximation, which is special case of this work has been published in [Comp. Math. Applics. 29 (1995) 67-73]. This article deals with the odd-order parametric spline relations.

Two-Stage Spline-Approximation with an Unknown Number of Elements in Applied Optimization Problem of a Special Kind

Mathematics and Statistics, 2021

Being a continuation of the paper published in Mathematics and Statistics, vol. 7, No. 5, 2019, this article describes the algorithm for the first stage of spline- approximation with an unknown number of elements of the spline and constraints on its parameters. Such problems arise in the computer-aided design of road routes and other linear structures. In this article we consider the problem of a discrete sequence approximation of points on a plane by a spline consisting of line segments conjugated by circular arcs. This problem occurs when designing the longitudinal profile of new and reconstructed railways and highways. At the first stage, using a special dynamic programming algorithm, the number of elements of the spline and the approximate values of its parameters that satisfy all the constraints are determined. At the second stage, this result is used as an initial approximation for optimizing the spline parameters using a special nonlinear programming algorithm. The dynamic programming algorithm is practically the same as in the mentioned article published earlier, with significant simplifications due to the absence of clothoids when connecting straight lines and curves. The need for the second stage is due to the fact that when designing new roads, it is impossible to implement dynamic programming due to the need to take into account the relationship of spline elements in fills and in cuts, if fills will be constructed from soils of cuts. The nonlinear programming algorithm is based on constructing a basis in zero spaces of matrices of active constraints and adjusting this basis when changing the set of active constraints in an iterative process. This allows finding the direction of descent and solving the problem of excluding constraints from the active set without solving systems of linear equations in general or by solving linear systems of low dimension. As an objective function, instead of the traditionally used sum of squares of the deviations of the approximated points from the spline, the article proposes other functions, taking into account the specifics of a specific project task.

Spline approximations for systems of ordinary differential equations

In first place, I am greatly indebted to my advisor, Emilio Defez, for his kind support and companionship, initiating my work in this area of mathematics and guiding the research for this thesis with him. Many thanks also go to my tutor, Antonio Hervás, for plenty of useful advise, and to Javier Ibáñez for his expertise on MATLAB in a fruitful collaboration with him.