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

We introduce a new topological method to naturally extend a partial function $h \colon X \rightharpoonup [-\infty, \infty]$ on a ``generalization'' of a metric space $X$ equipped with a dynamical system $f \colon X \rightharpoonup X$, to a function $h_f^{\varepsilon\text{-}\ell^p} \colon X \to [-\infty,\infty]$ with parameters $\varepsilon,p$, 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 $\mathop{\mathrm{dom}}h$, using this extended function, one can construct a sufficient condition for the existence of an optimal $\varepsilon$-perturbation path from any point into $\mathop{\mathrm{dom}}h$ that minimizes the value of $h$ under the condition $X = \mathop{\mathrm{dom}} f \sqcup \mathop{\mathrm{dom}}h = \bigsqcup_{n = 0}^\infty f^{-n}(\mathop{\mathrm{dom}}h)$. In addition, if the domain $X$ is finite, then the function $h_f^{\varepsilon\text{-}\ell^p} \colon X \to [-\infty,\infty]$ 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.