Exponential OED

Exponential OED
State dimension:	1
Differential states:	2
Discrete control functions:	1

The Exponential OED problem was formulated as minimal design problem that to highlight one important difference between single and multiple shooting. We are interested in finding an optimal experimental design to determine the 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 $\dot{x}(t)=x(t)\cdot (x(t)+p),t\in [0,t_f],x(0)=x_0.$

We furthermore restrict the state to be in the interval $[0,200]$. We assume that we have one measurement at the end time point $t_f$. This allows to eliminate sampling function directly from the control problem and to use the objective function

$\text{trace}C(t_f)=\frac{1}{D(t_f)}=\frac{1}{H(0)+G(t_f{)}^2\cdot w_1}=\frac{1}{G(t_f{)}^2}$

Applying our transformation, we obtain the following experimental design control problem:

$\begin{array}{lll}\underset{q}{\min }& & 1/G(t_f{)}^2\\ \text{subject to}\\ \dot{x}(t)& =& x(t)\cdot (x(t)+p),\\ \dot{G}(t)& =& (p+2\cdot x(t))\cdot G(t)+x(t),\\ \dot{F}(t)& =& w(t)\cdot G(t{)}^2,\\ \dot{z}(t)& =& w(t),\\ x(0)& =& q,\\ G(0)& =& \epsilon ,\\ x(t)& \le & 200,\\ q& \in & {𝒟}\end{array}$

Parameters

These fixed values are used within the model:

$p=1;t_f=0.6;\epsilon =10^{-3}.$

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States and measurement control for different choices of $p$.

Miscellaneous and Further Reading

The Toy OED problem was introduced by Sebastian Sager in [Sager2013]Author: Sager, S.
Journal: SIAM Journal on Control and Optimization
Number: 4
Pages: 3181--3207
Title: Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin's Maximum Principle
Url: http://mathopt.de/PUBLICATIONS/Sager2013.pdf
Volume: 51
Year: 2013
, which contains further details.

References

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