Denbigh Reaction

Denbigh Reaction
State dimension:	1
Differential states:	3
Discrete control functions:	1

The Mountain Car problem s based on the system of chemical reactions initially considered by Denbigh (1958), which was also studied by Aris (1960) and more recently by Luus (1994):

$\begin{array}{rl}A+B& \to X\\ X& \to Q\\ X& \to Y\\ A+X& \to P\end{array}$

where X is an intermediate, Y is the desired product, and P and Q are waste products.

Its dynamics are given by a three-dimensional ODE model. The optimal integer control functions exhibits a bang bang structure.

Mathematical formulation

$\begin{array}{lll}\underset{u}{\min }& & t_f\\ \text{subject to}\\ \dot{x}(t)& =& v(t),\\ \dot{v}(t)& =& 0.001\cdot u(t)-0.0025\cdot \cos (3\cdot x(t)),\\ x(0)& =& -0.5,\\ v(0)& =& 0,\\ x(t_f)& =& 0.5,\\ v(t_f)& \ge & 0,\\ u(t)& \in & [-1,1]\mathrm{\forall }t\in [0,t_f]\end{array}$

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 and a detailed description can be found in [1].

References

[1] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/mountain_car/mountain_car.html
[2] Andrew William Moore. Efficient memory-based learning for robot control. Technical Report UCAM-CL-TR-209, University of Cambridge, Computer Laboratory, November 1990. URL: https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-209.pdf, doi:10.48456/tr-209.
[3] Alexey A Melnikov, Adi Makmal, and Hans J Briegel. Projective simulation applied to the grid-world and the mountain-car problem. arXiv preprint arXiv:1405.5459, 2014.