Multi-parameter persistence in dynamical systems for maximizing effects of control inputs

Abstract

We introduce a new topological method to naturally extend a partial function $h:X\rightharpoonup [-\infty ,\infty ]$ on a ``generalization'' of a metric space $X$ equipped with a dynamical system $f:X\rightharpoonup X$, to a function $h_f^{\epsilon \text{-}{\ell }^p}:X\to [-\infty ,\infty ]$ with parameters $\epsilon ,p$, which allows us to apply existing topological data analysis techniques to functions defined on the whole space. Moreover, given a function $h$ that evaluates the ``quality'' of points within $\mathrm{dom}h$, using this extended function, one can construct a sufficient condition for the existence of an optimal $\epsilon$-perturbation path from any point into $\mathrm{dom}h$ that minimizes the value of $h$ under the condition $X=\mathrm{dom}f\bigsqcup \mathrm{dom}h={⨆}_{n=0}^{\infty }{f}^{-n}(\mathrm{dom}h)$. In addition, if the domain $X$ is finite, then the function $h_f^{\epsilon \text{-}{\ell }^p}:X\to [-\infty ,\infty ]$ can be computed recursively. As an application, for a given partial evaluation function on a space equipped with a dynamical system, one can construct a three-parameter filtration associated with its extension, which naturally identifies minimal paths. This clarifies the relationship among three factors: the evaluation of the cost norm, the strength of control, and the resulting value.