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

$\begin{array}{llcl} \min_{x, w} & - 1 + x_{1} (t_{f}) + x_{2} (t_{f}) \\ s.t. & {\dot{x}}_{1} & = & w (t) (10 x_{2} (t) - x_{1} (t)), \\ {\dot{x}}_{2} & = & w (t) (x_{1} (t) - 10 x_{2} (t)) - (1 - w (t)) x_{2} (t), \\ x (t_{0}) & = & (1, 0)^{T}, \\ w (t) & \in & {0, 1} . \end{array}$

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
Results with relaxed controls and collocation from the COPS library
Optimal relaxed controls showing a bang-bang structure.

Source Code

Model descriptions are available in

AMPL/TACO code at Catalyst mixing problem (TACO)