Toy OED

Toy OED
State dimension:	1
Differential states:	4
Discrete control functions:	1

The Toy OED problem looks for an optimal measurement strategy to determine a single parameter in a one-dimensional ODE model, where can directly measure the single state.

The optimal integer control functions shows bang bang behavior.

Mathematical formulation

For a single parameter $p$ the original initial value problem is given by $\dot{x}(t)=p\cdot x(t),t\in [0,t_f],x(0)=x_0.$

We assume both $x_0$ and $t_f$ to be fixed and are only interested in when to measure, with an upper bound $M$ on the measuring time. We can measure the state directly, i.e. $h(x(t))=x(t)$. Thus, the experimental design problem simplifies to:

$\begin{array}{lll}\underset{x,G,F,z,w}{\min }& & 1/F(t_f)\\ \text{subject to}\\ \dot{x}(t)& =& p\cdot x(t),\\ \dot{G}(t)& =& p\cdot G(t)+x(t),\\ \dot{F_{11}}(t)& =& w(t)\cdot G(t{)}^2,\\ \dot{z}(t)& =& w(t),\\ x(0)& =& x_0,\\ G(0)& =& F(0)=z(0)=0,\\ w(t)& \in & {𝒲},\\ 0& \le & M-z(t_f)\end{array}$

Reference Solutions

Here is one local solution to the above control problem.

Source Code

Model descriptions are available in

• AMPL at Lotka Experimental Design (AMPL)
• VPLAN code at Lotka Experimental Design (VPLAN)

Variants

There are several alternative formulations and variants of the above problem, in particular

• a prescribed time grid for the control function [Sager2006]Address: Heidelberg
  Author: S. Sager; H.G. Bock; M. Diehl; G. Reinelt; J.P. Schl\"oder
  Booktitle: Recent Advances in Optimization
  Editor: A. Seeger
  Note: ISBN 978-3-5402-8257-0
  Pages: 269--289
  Publisher: Springer
  Series: Lectures Notes in Economics and Mathematical Systems
  Title: Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem
  Volume: 563
  Year: 2009
  , see also Lotka Experimental Design (AMPL),
• no fishing, i.e., $u\equiv 0$,
• different fishing control functions for the two species,
• different parameters and start values.

Miscellaneous and Further Reading

The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper [Sager2006]Address: Heidelberg
Author: S. Sager; H.G. Bock; M. Diehl; G. Reinelt; J.P. Schl\"oder
Booktitle: Recent Advances in Optimization
Editor: A. Seeger
Note: ISBN 978-3-5402-8257-0
Pages: 269--289
Publisher: Springer
Series: Lectures Notes in Economics and Mathematical Systems
Title: Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem
Volume: 563
Year: 2009
and revisited in his PhD thesis [Sager2005]Address: Tönning, Lübeck, Marburg
Author: S. Sager
Editor: ISBN 3-89959-416-9
Publisher: Der andere Verlag
Title: Numerical methods for mixed--integer optimal control problems
Url: http://mathopt.de/PUBLICATIONS/Sager2005.pdf
Year: 2005
. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, [Sager2011d]Author: S. Sager
How published: University of Heidelberg
Month: August
Note: Habilitation
Title: On the Integration of Optimization Approaches for Mixed-Integer Nonlinear Optimal Control
Url: http://mathopt.de/PUBLICATIONS/Sager2011d.pdf
Year: 2011
.

References

[Sager2013]

Sager, S. (2013): Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin's Maximum Principle. SIAM Journal on Control and Optimization, 51, 3181--3207