$title Catalyst Mixing COPS 2.0 #14 (CATMIX,SEQ=242)

$onText
Determine the optimal mixing policy of two catalysts along the length of a
tubular plug flow reactor involving several reactions.

This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.

The number of discretization points can be specified using the command
line parameter --nh. COPS performance tests have been reported for nh
= 100, 200, 400, 800

Dolan, E D, and More, J J, Benchmarking Optimization
Software with COPS. Tech. rep., Mathematics and Computer
Science Division, 2000.

von Stryk, O, User's Guide for DIRCOL (Version 2.1):
A Direct Collocation Method for the Numerical Solution of
Optimal Control Problems. Tech. rep., Technische Universitt 
Mnchen, 1999.

Keywords: nonlinear programming, chemical engineering, mixing policy
$offText

$if     set n  $set nh %n%
$if not set nh $set nh 100

Set nh 'number of subintervals' / 0*%nh% /;

Alias (nh,i);

Scalar
   tf    'final time'               / 1 /
   x1_0  'initial condition for x1' / 1 /
   x2_0  'initial condition for x2' / 0 /
   alpha 'smoothing parameter'      / 0 /
   h;

h = tf/%nh%;

Variable
   u(nh)
   x1(nh)
   x2(nh)
   obj;

Positive Variable u;
u.up(nh) = 1;

Equation
   defobj 'objective function'
   ode1(nh)
   ode2(nh);

defobj..
   obj =e= -1 + x1['%nh%'] + x2['%nh%'] + alpha*h*sum{nh(i+1), sqr(u[i+1] - u[i])};

ode1(nh(i+1))..
   x1[i+1] =e= x1[i] + (h/2)*(u[i]*(10*x2[i]-x1[i]) + u[i+1]*(10*x2[i+1]-x1[i+1]));

ode2(nh(i+1))..
   x2[i+1] =e= x2[i] + (h/2)*(u[i]*(x1[i] - 10*x2[i])
            - (1 - u[i])*x2[i] + u[i+1]*(x1[i+1] - 10*x2[i+1])
            - (1 - u[i+1])*x2[i+1]);

x1.l[nh]   = 1;
x1.fx['0'] = x1_0;
x2.fx['0'] = x2_0;

Model catmix / all /;

$if set workSpace catmix.workSpace = %workSpace%

solve catmix minimizing obj using nlp;