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Table 1 Numerical investiation of the space mapping procedure. For \(\sigma = 0.01\) no space mapping is needed, the optimal deterministic control is accepted. The number of space mapping steps increases with increasing stochastic strength. The \(L^{2}\)-error of the trajectory of the center of mass compared to the center of mass of the optimal deterministic solution increases as well for larger σ. The space mapping procedure is decreasing the error by a factor three for \(\sigma = 0.02\). As the stochastic starts to superimpose the crowd behaviour for \(\sigma \ge 0.03\), we see that the space mapping approach decreases the error only marginally. The second part of the table shows results obtained with 1000 Monte Carlo samples. The values change only slightly, which justifies to fix the number of MC samples to 100 for the following computations

From: Space mapping-based receding horizon control for stochastic interacting particle systems: dogs herding sheep

 

σ = 0.01

σ = 0.02

σ = 0.03

σ = 0.04

MC = 100

Space mapping iterations

0

1

3

–

\(L^{2}\) error with deterministic control

\(7.00\cdot e^{-3}\)

\(3.73\cdot e^{-2}\)

\(9.09\cdot e^{-2}\)

\(1.57\cdot e^{-1}\)

\(L^{2}\) error after space mapping

\(7.00\cdot e^{-3}\)

\(1.05\cdot e^{-2}\)

\(7.05 \cdot e^{-2}\)

–

MC = 1000

Space mapping iterations

0

1

3

–

\(L^{2}\) error with deterministic control

\(9.28\cdot e^{-3} \)

\(4.25\cdot e^{-2} \)

\(9.70\cdot e^{-2} \)

\(1.60\cdot e^{-1} \)

\(L^{2}\) error after space mapping

\(9.28\cdot e^{-3}\)

\(1.31\cdot e^{-2}\)

\(7,07\cdot e^{-2}\)

–