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.

Miscellaneous and Further Reading

The Toy OED problem was introduced by Sebastian Sager in the paper [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

[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