Let $(w_i)_{i=1}^N$, $(c_i)_{i=1}^N$ and $(\sigma_i)_{i=1}^N$ be real-valued sequences. The function $f:\mathbb{R}\rightarrow\mathbb{R}$ given by

$$ f(s)=\frac{\sum_{i=1}^Nw_{i}e^{-\frac{1}{2\sigma_{i}^{2}}(s-c_{i})^{2}}}{\sum_{i=1}^Ne^{-\frac{1}{2\sigma_{i}^{2}}(s-c_{i})^{2}}} $$

is widely used in the context of Dynamic Movement Primitives (DMPs), a method of trajectory control/planning for robots.¹

Some questions regarding the stability of this method depends on the Lipchistz continuity of $f$. In particular, I was wondering the following question:

Is $f$ globally Lipschitz continuous for any choice of $N > 1$, $(w_i)_{i=1}^N$, $(c_i)_{i=1}^N$ and $(\sigma_i)_{i=1}^N$?

[1] Ijspeert, Auke Jan; Nakanishi, Jun; Hoffmann, Heiko; Pastor, Peter; Schaal, Stefan, Dynamical movement primitives: learning attractor models for motor behaviors, Neural Comput. 25, No. 2, 328-373 (2013). ZBL1269.92002.



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