Tubular Reactor

Tubular Reactor
State dimension:	1
Differential states:	2
Discrete control functions:	1

The Tubular Reactor problem is a two-dimensional ODE model. It aims to maximize the value of the second differential state at the end of the time interval.

The optimal integer control functions exhibits a singular arc.

Mathematical formulation

$\begin{array}{lll}\underset{w}{\min }& & -x_2(1)\\ \text{subject to}\\ \dot{x_1}(t)& =& -(w(t)+\frac{1}{2}\cdot w(t{)}^2)\cdot x_1(t),\\ \dot{x_2}(t)& =& w(t)\cdot x_1(t),\\ x(0)& =& (1,0{)}^T,\\ w(t)& \in & [0,5]\end{array}$

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States and discretized control for a local optimum.

Miscellaneous and Further Reading

The Bryson-Denham problem is a variation of the double integrator problem [1]. This formulation detailed description can be found in [2].

References

[1] Arthur E Bryson and Yu-Chi Ho. Applied Optimal Control: Optimization, Estimation and Control. CRC Press, 1975.
[2] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/bryson_denham/bryson_denham.html