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

$trace C (t_{f}) = \frac{1}{D (t_{f})} = \frac{1}{H (0) + G (t_{f})^{2} \cdot w_{1}} = \frac{1}{G (t_{f})^{2}}$

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

$\begin{array}{lll} \min_{q} & 1 / G (t_{f})^{2} \\ subject to \\ \dot{x} (t) & = & x (t) \cdot (x (t) + p), \\ \dot{G} (t) & = & (p + 2 \cdot x (t)) \cdot G (t) + x (t), \\ \dot{F} (t) & = & w (t) \cdot G (t)^{2}, \\ \dot{z} (t) & = & w (t), \\ x (0) & = & q, \\ G (0) & = & ε, \\ x (t) & \leq & 200, \\ q & \in & 𝒟 \end{array}$

with p = 1 and regularization start value ε = 10−3. As can be seen, the optimal solution will maximize the value of G(t f ) and hence also of x(t f ). For t f = 0.6 the optimal initial value is given by q∗ = 1.203 leading to state values of x(t f ) = 200 and G(t f ) = 6088 and an objective value of φ ∗ = 2.7 · 10−8. The main problem with direct single shooting is that a large part of the feasible domain Q of q will cause the integrator to run into a singularity before t f . Hence only initial guesses for the optimization variable q that are below a critical value of ≈ 1.23 will give rise to a successful optimization. For multiple shooting the situation is different, due to the decoupling of the integration

Parameters

These fixed values are used within the model:

$x_{0} = 1; t_{f} = 0.2; 𝒲 = [0, 1]; M = 0.2; p \in {- 0.5, - 2}$

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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