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24 Flow in a Channel
Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABORATORY
24.1 Problem Formulation
Find u(t) over t in [0; 1 ] to minimize
subject to:
After some transformation we get this problem:
Reference: [14]
24.2 Problem setup
toms t
p = tomPhase('p', t, 0, 1, 30);
setPhase(p);
tomStates x1 x2 x3 x4
x0 = icollocate({x1 == 3*t.^2 - 2*t.^3
x2 == 2*t - 6*t.^2
x3 == t - 12*t
x4 == -12});
% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0})
final({x1 == 1; x2 == 0})};
% Various constants and expressions
R = 10;
% ODEs and path constraints
ceq = collocate({dot(x1) == x2
dot(x2) == x3; dot(x3) == x4
dot(x4) == R*(x2.*x3-x1.*x4)});
% Objective
objective = 1; %(feasibility problem)
24.3 Solve the problem
options = struct;
options.name = 'Flow in a Channel Steering';
solution = ezsolve(objective, {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);
x4 = subs(collocate(x4),solution);
Problem type appears to be: lpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
Problem: --- 1: Flow in a Channel Steering f_k 1.000000000000000000
sum(|constr|) 0.000000000018584877
f(x_k) + sum(|constr|) 1.000000000018584900
f(x_0) 0.000000000000000000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 11 ConJacEv 11 Iter 9 MinorIter 91
CPU time: 0.062500 sec. Elapsed time: 0.078000 sec.
24.4 Plot result
figure(1)
plot(t,x2,'*-');
legend('x2');
title('Flow in a Channel state variables');
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