A numerical method for solving linear non-autonomous systems based on linear programming

doi:10.1016/j.amc.2005.11.046

Applied Mathematics and Computation

Volume 178, Issue 2, 15 July 2006, Pages 287-294

https://doi.org/10.1016/j.amc.2005.11.046 Get rights and content

Abstract

In this article we propose an approximate solution method for solving linear non-autonomous systems with an initial condition. In this method, the initial value problem is transformed into a linear programming problem to obtain an approximate solution. The existence and uniqueness criterion for solution is also proved in a linear programming manner. Finally we propose a method to find the fundamental matrix of these systems. Numerical examples are considered.

Introduction

We are going to find the approximate solution of the following initial value problem: $\dot{x} (t) = A (t) x (t) + g (t), t \in [a, b],$ $x (a) = x_{a},$ where x = (x₁, … , x_n) is a vector function with components in C¹([a, b]), and x_a = (x_1a, … , x_na) is given. Here, A(·) is an n × n matrix which its entries are continuous functions on [a, b], and g(·) = (g₁(·), … , g_n(·)) is an n × 1 vector function with continuous components on [a, b].

There are many numerical methods to solve this problem and even nonlinear cases. Some of this numerical methods are Euler, Runge–Kutta, and Taylor series method (see [7]).

By defining artificial controls, Alavi, et al. [1] have converted this type of problems to an optimal control problem and then solve it by the method of [9]. In this method the approximate solution is found by the solution of a linear programming (LP) problem.

In the present method, we convert the initial value problem to a calculus of variation problem. A straightforward discretization of the resulting problem leads to a finite dimensional linear programming problem. Solution of this LP problem determines the value of x(·) at some specified points.

Methods of this type were frequently and successfully applied to different problems, for example, the controllability of time-varying systems [5], and optimal guidance to a moving target [8].

LP formulation enables us to prove the existence theorem for initial value problem (1), (2) based on duality theorem. We also use sensitivity analysis to find the fundamental matrix of system (1). Error estimation and numerical examples are also given.

Section snippets

Metamorphosis

For our present purpose, it seems to be convenient if we could change the initial value problem (1), (2) to an optimization type problem. To this means, let define the function $F : R^{2 n} \to [0, + \infty)$ as follows: $F (x (t), \dot{x} (t)) = ‖ \dot{x} (t) - A (t) x (t) - g (t) ‖,$ where ∥·∥ is the L₁ norm defined as $‖ y ‖ = \sum_{i = 1}^{n} | y_{i} |, y = (y_{1}, \dots, y_{n}) \in R^{n} .$ Consider the following calculus of variations problem: $\min J (x (\cdot), \dot{x} (\cdot)) = \int_{a}^{b} F (x (t), \dot{x} (t)) d t$ $\begin{matrix} subject to : \\ x (\cdot) \in C^{1} ([a, b]), x (a) = x_{a} . \end{matrix}$ It can be shown that x(·) is the solution of (1), (2) if and only if x(·) is

Discretization

Let N be a positive integer, $Δ (N) = \frac{b - a}{N}$ , and consider the following subsets of integer numbers: $\begin{matrix} J = {1, 2, \dots, N + 1}, \\ J_{1} = {1, 2, 3, \dots, N}, \\ I = {1, 2, \dots, n} . \end{matrix}$ Let P(N) = {t₁, t₂, … , t_N+1} be a partition of [a, b], where $t_{j} = a + (j - 1) Δ (N) \forall j \in J .$ For large value of N we have ${\dot{x}}_{i} (t_{j}) \approx \frac{x_{i} (t_{j} + Δ (N)) - x_{i} (t_{j})}{Δ (N)} = \frac{x_{i} (t_{j + 1}) - x_{i} (t_{j})}{Δ (N)} .$ Let define x_ij = x_i(t_j) for all i ∈ I and j ∈ J. Thus we have ${\dot{x}}_{i} (t_{j}) \approx \frac{x_{ij + 1} - x_{ij}}{Δ (N)}, i \in I, j \in J_{1} .$ We set $f_{ij} = \frac{x_{ij + 1} - x_{ij}}{Δ (N)} - \sum_{k \in I} a_{ikj} x_{kj} - g_{ij} \forall i \in I, \forall j \in J_{1},$ where $a_{ikj} = A_{ik} (t_{j}), g_{ij} = g_{i} (t_{j}) \forall i, k \in I, \forall j \in J_{1} .$ Now we may discretize problem (3),

LP formulation

Let define $R_{ij} = \max {f_{ij}, 0}, S_{ij} = - \min {f_{ij}, 0}$ for all i ∈ I and j ∈ J₁.

Then we have $| f_{ij} | = R_{ij} + S_{ij}, f_{ij} = R_{ij} - S_{ij} .$ Now we can linearize problem (5), (6), (7) by these new variables. Therefore we must solve the following LP. $\min \sum_{i \in I} \sum_{j \in J_{1}} (R_{ij} + S_{ij})$ $\begin{matrix} subject to : \\ R_{ij} - S_{ij} = f_{ij} \forall i \in I, \forall j \in J_{1}, \end{matrix}$ $x_{i 1} = x_{ia} \forall i \in I,$ $x_{ij} free \forall i \in I, \forall j \in J,$ $R_{j}, S_{j} ⩾ 0 \forall j \in J_{1} .$ To solve the above LP problem one may use a method for solving LP problems, for example revised simplex method ([2]) or interior point methods [12]. There are also many softwares for LP solving, a

Fundamental matrix

Let {e₁, e₂, … , e_n} be the standard basis for $R^{n}$ . If x_j(·) is the solution of (1) with e_j as initial value, then X(·) = [x₁(·)⋯x_n(·)] is called the fundamental matrix for this system.

The fundamental matrix has several interesting properties (see [10]). Except in special forms of (1) the fundamental matrix is very hard for calculation. By our method we can find this matrix approximately.

Let $x_{1}^{N} (.)$ is the linear approximation found by solving the corresponding LP with e₁ as initial value. If we

Examples

Now we study the performance of proposed method on two examples. To make a criterion for error estimation, let define a total error corresponding to N as: $TE (N) = \frac{1}{N} \sum_{j \in J} ‖ x (t_{j}) - x_{j}^{N} ‖_{\infty},$ where, x(t_j) denotes the exact solution at t_j, and $x_{j}^{N}$ is the approximation value for x(t_j) obtained by solving the corresponding LP problem for N. If $\ddot{x} (\cdot)$ be twice differentiable on [a, b], from a theorem on error estimation of Euler method, it could be shown that $TE (N) ⩽ (b - a) \frac{e^{(b - a) K} - 1}{2 KN} M_{2},$ where, $M_{2} = \max_{t \in [a, b]} ‖ \ddot{x} (t) ‖_{\infty}, K =$

Conclusions

A numerical method is derived to solve the initial value problem for a non-autonomous linear system. The approximate solution is found via solving a linear programming problem. The LP formulation has some aspects, for example in the proof of existence theorem, and in finding the fundamental matrix numerically.

The proposed method may be applied to solve boundary value problems as well.

In the case of nonlinear systems, the method can be applied on linearized form of the system [6].

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There are more references available in the full text version of this article.

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A numerical method for solving linear non-autonomous systems based on linear programming

Abstract

Introduction

Section snippets

Metamorphosis

Discretization

LP formulation

Fundamental matrix

Examples

Conclusions

Applied Mathematics and Computation

Solving nonlinear ordinary differential equations as a control problem by using measure theory

Scientia Iranica

Linear Programming and Network Flows

Numerical Analysis

Software survey: linear programming

OR/MS Today

A linear programming approach to the controllability of time varying systems

IUST-International Journal of Engineering Science