On the optimal control of entry-exit phenomena in planar fast-slow dynamical systems
Abstract
We study a geometric singular perturbation theory (GSPT) approach to optimal control problems involving entry-exit phenomena in planar fast-slow dynamical systems. In contrast to previous work, we assume that the control acts only on the slow dynamics and we consider minimum-time optimal control problems (i.e., we aim at minimizing the time to reach a certain target). We apply this method to a simplified fast-slow toy model, characterized by a repetition of entry-exit processes, enabling a transparent analytical exploration of the control dynamics and allowing us to derive the key results explicitly. First, we analyze each non-autonomous entry-exit phenomenon separately, and we derive the optimal way to control them. Second, by artificially generating a stable limit cycle, we determine under which conditions it is possible to reach the target. Finally, under such conditions, we combine all entry-exit processes together and we demonstrate the existence of an optimal control. Our approach mainly exploits techniques of GSPT and dynamic programming. Indeed, we show that our problem can be formulated in a way resembling the Bellman equation, even though not in its standard setting. Numerical simulations, built on such Bellman-like formulation, illustrate our analytical results. In particular, the optimal control is derived with an algorithm that converges in a finite number of steps. Lastly, we extend this novel approach to more complex planar fast-slow dynamical systems.
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BibTeX
@misc{borsotti2026optimal,
title = {{On the optimal control of entry-exit phenomena in planar fast-slow dynamical systems}},
author = {Jacopo Borsotti and Christian Kuehn and Mattia Sensi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25747},
eprint = {2609.25747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25747},
}
RIS
TY - UNPB
TI - On the optimal control of entry-exit phenomena in planar fast-slow dynamical systems
AU - Jacopo Borsotti
AU - Christian Kuehn
AU - Mattia Sensi
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25747
UR - https://arxiv.org/abs/2609.25747
AB - We study a geometric singular perturbation theory (GSPT) approach to optimal control problems involving entry-exit phenomena in planar fast-slow dynamical systems. In contrast to previous work, we assume that the control acts only on the slow dynamics and we consider minimum-time optimal control problems (i.e., we aim at minimizing the time to reach a certain target). We apply this method to a simplified fast-slow toy model, characterized by a repetition of entry-exit processes, enabling a transparent analytical exploration of the control dynamics and allowing us to derive the key results explicitly. First, we analyze each non-autonomous entry-exit phenomenon separately, and we derive the optimal way to control them. Second, by artificially generating a stable limit cycle, we determine under which conditions it is possible to reach the target. Finally, under such conditions, we combine all entry-exit processes together and we demonstrate the existence of an optimal control. Our approach mainly exploits techniques of GSPT and dynamic programming. Indeed, we show that our problem can be formulated in a way resembling the Bellman equation, even though not in its standard setting. Numerical simulations, built on such Bellman-like formulation, illustrate our analytical results. In particular, the optimal control is derived with an algorithm that converges in a finite number of steps. Lastly, we extend this novel approach to more complex planar fast-slow dynamical systems.
ER -