# Euler Buckling Problem

Problem 4: Miser3 manual

## Contents

## Problem description

Over t in [0; 1 ], minimize

subject to:

```
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
```

## Problem setup

toms t toms z1 p = tomPhase('p', t, 0, 1, 40); setPhase(p); % States tomStates x1 x2 x3 x4 % We don't need to introduce any control variables. % Initial guess x0 = {icollocate({ x1 == 0; x2 == 1 x3 == 0.5; x4 == t/40}) z1 == 10}; % Box constraints cbox = { icollocate({-10 <= x1 <= 10 -10 <= x2 <= 10; 0.5 <= x3 <= 10}) 0 <= z1 <= 500}; % Boundary constraints cbnd = {initial({x3 >= 0.5; x1 == 0 x2 == 1; x4 == 0}) final({x1 == 0; x4 == 1})}; % ODEs and path constraints ceq = {collocate({ dot(x1) == x2 dot(x2) == -z1*x1./x3.^2 x3 >= 0.5 % Path constr. % Integral constr. }) integrate(x3) == 1}; % Objective objective = z1;

## Solve the problem

options = struct; options.name = 'Euler Buckling'; solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); % Extract optimal states and controls from solution t = subs(collocate(t),solution); x1 = subs(collocate(x1),solution); x2 = subs(collocate(x2),solution); x3 = subs(collocate(x3),solution);

Problem type appears to be: lpcon ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Euler Buckling f_k 9.891564777222249900 sum(|constr|) 0.000000000136989172 f(x_k) + sum(|constr|) 9.891564777359239000 f(x_0) 10.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 40 ConJacEv 39 Iter 20 MinorIter 185 CPU time: 0.109375 sec. Elapsed time: 0.110000 sec.

## Plot result

figure(1) plot(t,x1,'*-',t,x2,'*-',t,x3,'*-'); legend('x1','x2','x3'); title('Euler Buckling state variables');

## Footnote

In the original [Miser3] problem formulation, it is requested to compute "u", equal to x3_t. u is not included in the optimization problem, thereby speeding up the solution process. x3_t can be obtained by simple numeric differentiation of x3.

Note, however, that because there was no constraint on u, and it was not included in the cost function, x3_t looks very strange.