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:

$\begin{array}{lll} \min_{x, G, F, z, w} & 1 / F (t_{f}) \\ subject to \\ \dot{x} (t) & = & p \cdot x (t), \\ \dot{G} (t) & = & p \cdot G (t) + x (t), \\ \dot{F_{11}} (t) & = & w (t) \cdot G (t)^{2}, \\ \dot{z} (t) & = & w (t), \\ x (0) & = & x_{0}, \\ G (0) & = & F (0) = z (0) = 0, \\ w (t) \in 𝒲, \\ 0 & \leq & M - z (t_{f}) \end{array}$

Reference Solutions

Here is one local solution to the above control problem.

Source Code

Model descriptions are available in

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