PROPT Drug Displacement Problem

This page is part of the PROPT Manual. See PROPT Manual.

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

12.4.3 Example 3: The desired level of two drugs, warfarin and phenylbutazone, must be reached in a patients bloodstream in minimum time.

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

Problem Formulation

Find u over t in [0; t ] to minimize

$J=t_{F}$

subject to:

${\frac {dx_{1}}{dt}}=g_{1}*(g_{4}*(0.02-x_{1})+46.4*x_{1}*(u-2*x_{2}))$

${\frac {dx_{2}}{dt}}=g_{1}*(g_{3}*(u-2*x_{2})+46.4*(0.02-x_{1}))$

$g_{2}=1+0.2*(x_{1}+x_{2})$

$g_{3}=g_{2}^{2}+232+46.4*x_{2}$

$g_{4}=g_{2}^{2}+232+46.4*x_{1}$

$g_{1}={\frac {g_{2}^{2}}{g_{3}*g_{4}-2152.96*x_{1}*x_{2}}}$

$0<=u<=8$

x1 is the concentration of warfarin, and x2 of phenylbutazone. The initial and final condition are:

$x_{0}=[0.02\ 0]$

$x_{t_{f}}=[0.02\ 2.00]$

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

Problem setup

toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 50);
setPhase(p);

tomStates x1 x2
tomControls u

% Initial guess
x0 = {t_f == 300
    icollocate({
    x1 == 0.02; x2 == 2*t/t_f})
    collocate(u == 8-8*t/t_f)};

% Box constraints
cbox = { 1 <= t_f <= 500
    0 <= collocate(u) <= 8};

% Boundary constraints
cbnd = {initial({x1 == 0.02; x2 == 0})
    final({x1 == 0.02; x2 == 2})};

% General variables
g2 = 1+0.2*(x1+x2);
g3 = g2.^2+232+46.4*x2;
g4 = g2.^2+232+46.4*x1;
g1 = g2.^2./(g3.*g4-2152.96*x1.*x2);

% ODEs and path constraints
ceq = collocate({
    dot(x1) == g1.*(g4.*(0.02-x1)+46.4*x1.*(u-2*x2))
    dot(x2) == g1.*(g3.*(u-2*x2)+46.4*(0.02-x1))});

Solve the problem

options = struct;
options.name = 'Drug Displacement';
% Objective is first parameter
solution = ezsolve(t_f, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
u = subs(collocate(u),solution);

Problem type appears to be: lpcon
Time for symbolic processing: 0.41796 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Drug Displacement              f_k     221.333418113505790000
                                       sum(|constr|)      0.000000061270786413
                              f(x_k) + sum(|constr|)    221.333418174776570000
                                              f(x_0)    300.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   14 ConJacEv   14 Iter   10 MinorIter  256
CPU time: 0.046800 sec. Elapsed time: 0.048000 sec. 

Plot result

figure(1)
plot(t,u,'+-');
legend('u');
title('Drug Displacement control');