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

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

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

This formulation detailed description can be found in [1].

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