Catalyst mixing problem

Catalyst mixing problem
State dimension:	1
Differential states:	2
Continuous control functions:	1
Path constraints:	2
Interior point equalities:	2

The Catalyst mixing problem seeks an optimal policy for mixing two catalysts "along the length of a tubular plug ow reactor involving several reactions". (Cite and problem taken from the COPS library)

Mathematical formulation

The problem is given by

$\begin{array}{llcl} \displaystyle \min_{x, w} &-1 + x_1(t_f) + x_2(t_f) \\[1.5ex] \mbox{s.t.} & \dot{x}_1 & = & w(t) ( 10 x_2(t) - x_1(t)), \\ & \dot{x}_2 & = & w(t) ( x_1(t) - 10 x_2(t)) - (1 - w(t)) \, x_2(t) , \\ & x(t_0) &=& (1, 0)^T, \\ & w(t) &\in& \{0,1\}. \end{array}$

Parameters

In this model the parameters used are $t_0 = 0, \, \, t_f = 1$.

Reference Solution

If the problem is relaxed, i.e., we demand that w(t) be in the continuous interval [0, 1] instead of the binary choice \{0,1\}, the optimal solution can be determined by means of direct optimal control.

• Reference solution plots
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  Results with relaxed controls and collocation from the COPS library

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  Optimal relaxed controls showing a bang-bang structure.

Source Code

Model descriptions are available in

• AMPL/TACO code at Catalyst mixing problem (TACO)